# Preference Distillation for Personalized Generative Recommendation

Jerome Ramos Bin Wu Aldo Lipani
University College London, London, UK
{jerome.ramos.20, bin.wu.23<EMAIL_ADDRESS>
###### Abstract

Recently, researchers have investigated the capabilities of Large Language
Models (LLMs) for generative recommender systems. Existing LLM-based
recommender models are trained by adding user and item IDs to a discrete
prompt template. However, the disconnect between IDs and natural language
makes it difficult for the LLM to learn the relationship between users. To
address this issue, we propose a PErsonAlized PrOmpt Distillation (PeaPOD)
approach, to distill user preferences as personalized soft prompts.
Considering the complexities of user preferences in the real world, we
maintain a shared set of learnable prompts that are dynamically weighted based
on the user’s interests to construct the user-personalized prompt in a
compositional manner. Experimental results on three real-world datasets
demonstrate the effectiveness of our PeaPOD model on sequential
recommendation, top-n recommendation, and explanation generation tasks.

Preference Distillation for Personalized Generative Recommendation

Jerome Ramos Bin Wu Aldo Lipani University College London, London, UK
{jerome.ramos.20, bin.wu.23<EMAIL_ADDRESS>

##  1 Introduction

Recommender systems Ge et al. (2022) have emerged as pivotal tools in guiding
users through vast amounts of items and tailoring personalized experiences
across various tasks, including sequential recommendation, top-n
recommendation and explanation generation. Recently, recommender systems
trained using large language models (LLMs), known as generative recommender
systems, have unlocked the ability to perform multiple tasks with a single
model. It has been shown to perform well in a zero- and few-shot setting
Sanner et al. (2023) and can be further fine-tuned on a specific domain to
improve performance Ramos et al. (2024).

P5 Geng et al. (2022) formulates recommendation as a text-to-text sequence
task and trains the model to perform generative recommendation based on
discrete prompt templates. User and item IDs serve as important identifiers
for discrete prompt templates, beyond the simple semantic meaning within their
word embeddings. Thus, the disconnect between IDs and natural language and the
inherent rigidity of discrete prompts makes it difficult for the LLM to learn
the relationship between users, limiting the effectiveness of this approach.
Prompt Distillation has been proposed to distill the knowledge in discrete
prompts into continuous prompt vectors which do not map to any specific words
Li et al. (2023). However, the existing work distills the knowledge into
global-shared prompts, which is in conflict with the fact that users’
personalized preferences greatly differ across the entire user population.
Given that each user has their own unique preferences (Wu et al., 2022), it is
imperative to design a method to effectively distill preferences for
personalized recommendation.

One intuitive way to introduce distilled preferences is to create unique
prompts per user based solely on their individual interaction history.
However, there are two key issues: (1) Distilling separated prompts for each
user is costly and difficult, where the number of prompts increases with the
number of the user and in most cases, each user’s interaction history is too
sparse to effectively personalize the recommender; and (2) This approach does
not allow the model to match users with similar interests, which has been
shown to increase performance for traditional recommender models in a warm-
start setting Koren et al. (2009); Mnih and Salakhutdinov (2007); Cheng et al.
(2016).

In this work, we propose PErsonAlized PrOmpt Distillation (PeaPOD), a more
flexible and effective framework to distill user preferences as personalized
soft prompts. Specifically, we construct a set of decomposed prompt
components, which maintains a many-to-many mapping between user preferences
and continuous prompts. This is due to the compositional nature of user
preferences, meaning that users might share similar interests but maintain
their distinction as well. By updating only a small set of prompt components,
we can efficiently share prompts among users rather than maintaining a unique
prompt per user.

Moreover, to construct the many-to-many mapping (Wu et al., 2023) between
users and the distilled prompt components, we propose a compositional strategy
to dynamically weight the prompts based on user preference. First, we use
matrix factorization Koren et al. (2009) to generate a user embedding, which
compresses a user’s interaction history into a high-dimensional space. The
user embeddings are used to identify user preferences, helping to create a
weighted summation of the decomposed prompt components, called the user-
personalized prompt. Importantly, the construction of the user-personalized
prompt are dynamic across users but only maintains a limited set of shared
prompt components. From experiments across three datasets, PeaPOD is able to
achieve state-of-the-art performance in three recommendation tasks. We provide
all data and code used at: https://github.com/jeromeramos70/peapod.

In summary, our key contributions are:

  1. 1.

We provide a novel architecture, namely PErsonAlized PrOmpt Distillation
(PeaPOD), to distill complex user preferences into a limited set of learnable
prompts.

  2. 2.

We propose a compositional strategy, dynamically weighting the decomposed
prompt components to construct the user-specific prompt based on user
preference.

  3. 3.

We conduct extensive experiments, showing the effectiveness and superiority of
our proposed PeaPOD model.

##  2 Related Works

###  2.1 LLMs for Recommendation

In recent years, researchers have explored the effectiveness of using LLMs for
recommendation. The two main methods of LLMs for recommender systems are (1)
using the LLM’s pre-trained knowledge to recommend items in a zero-shot or
few-shot setting and (2) fine-tuning the model for a given domain.

One advantage of LLMs is that they are pre-trained with a large amount of data
from various domains. Thus, Sanner et al. (2023) crowdsourced a dataset of
natural language summaries of user preferences and showed that prompting a
language model with these user summaries yielded competitive performance with
traditional baselines in a near cold-start setting. Ramos et al. (2024) showed
that a generative recommendation system trained solely on natural language
user profiles are also competitive with traditional recommender systems in a
warm-start setting, with the added benefit of transparency and scrutability.
In addition, the zero-shot prompting of GPT-4 has also beaten SOTA models in a
conversational recommender setting He et al. (2023). Although prompting LLMs
can lead to strong performance in recommender tasks, they are heavily reliant
on better content knowledge rather than collaborative knowledge He et al.
(2023); Wu et al. (2024). Thus, if the item metadata is not already learned in
the pre-trained knowledge base or if the target dataset users interact
differently than the users that the LLM was originally trained on, the
performance of the recommender system will suffer.

To mitigate these issues, the model can be further fine-tuned on target
datasets in order to train the model based on the given user population and
add or update its knowledge on the item dataset. P5 Geng et al. (2022) is the
first work that trains a T5-based Raffel et al. (2020) foundational
recommender model to unify various recommendation tasks in a shared framework.
VIP5 Geng et al. (2023) extended P5 to a multi-modal foundational recommender
system. POD Li et al. (2023) builds upon the P5 model and appends a task-
specific, continuous prompt to the input to improve performance. Although
these models pass in the user_id as inputs, they still do not leverage the
collaborative information contained from similar users. In particular, they do
not implement a user-specific component to identify and share knowledge
between users with similar preferences. Previous work in continual learning
for image classification has shown that an attention-based mechanism based on
the image input is able to learn similarities across similar images to achieve
state-of-the-art performance Smith et al. (2023). In this work, we focus on
adding an attention-based mechanism to our model in order to leverage the
shared collaborative knowledge between users to improve the recommendation
performance.

###  2.2 Structural Prompt

One issue with generative approaches to recommendation is how to effectively
encode user-item interactions to use with LLMs. Researchers have proposed
numerous methods to bridge the gap between item IDs and natural language. One
option is to add the user’s name and item title to the input prompt. However,
this data may be unavailable or non-unique, making it difficult to personalize
recommendations. Another option is to directly insert the user and item IDs to
the discrete prompt template. However, LLMs are trained on natural language
text rather than IDs, which limits the effectiveness of this approach.

To tackle these issues, many researchers have explored how to effectively
encode item IDs with LLMs. For example, Rajput et al. (2023) proposed using
RQ-VAE with item metadata to generate semantically meaningful tuples of tokens
to serve as the Semantic ID for each item. GPTRec uses a novel SVD
tokenization method to generate quantized item embeddings based on user-item
interactions Petrov and Macdonald (2023). Tan et al. (2024) proposed training
a textual ID generator alongside the LLM-based recommender to convert item ids
to meaningful text descriptions. Although these approaches have improved item
IDs encoding for generative recommendations, the effective encoding of user
information remains largely unexplored. In particular, existing works do not
tackle the problem of how to encode both a user’s personalized preferences and
the shared knowledge between similar users to an LLM. Thus, our work focuses
on how to develop user-personalized prompts to improve the performance of a
generative recommender system.

##  3 Preliminary

###  3.1 Recommendation Tasks

Following the existing works Geng et al. (2022), we focus mainly on three
important recommendation tasks: (1) Sequential Recommendation, (2) Top-n
Recommendation, and (3) Explanation Generation. Let $u\in U$ denote a user in
the user set and $i\in I$ denote an item in the item set across all three
tasks. We aim to train a unified model to complete the following tasks:

  * •

Sequential Recommendation: given a user $u$ and a list of historical
interactions in chronological order $i_{1},i_{2},...i_{n-1}$, the task is to
predict the next item $i_{n}$ that $u$ will interact with.

  * •

Top-n Recommendation: given a pool of randomly sampled items from $I$, the
model must select the top $N$ items that the user wants to interact with.

  * •

Explanation Generation: given a user $u$ and item $i$, the model must generate
an explanation $E_{u,i}$ that explains why $u$ likes/dislikes $i$.

Importantly, the scope of our study focuses only on user and item IDs for
recommendation and excludes additional information such as item metadata or
user profiles because this information may be sensitive or unavailable.

###  3.2 Prompt Distillation

In traditional recommender systems, user and item IDs are commonly used to
identify unique users and items. However, since these IDs are unique to the
dataset and are not natural language text, it is necessary to perform
additional fine-tuning to allow the LLM to effectively understand these IDs.
The most common methodology is to train the model with discrete prompt
templates. For example, in the case of explanation generation, the discrete
prompt could be “Explain why user_1234 enjoys item_5678”. When using IDs with
discrete prompt templates, recent research proposes adding an additional
embedding space, called whole-word embeddings, so that the model can recognize
which tokens belong to the appropriate ID Geng et al. (2022). In particular,
each ID consists of the same whole-word embedding and all other tokens are
tokenized as normal.

Although discrete prompts can lead to good performance, they are inflexible
and require manually crafted prompts. Thus, the performance of the model may
depend strongly on how well the instructions were written. On the other hand,
continuous prompts are dynamically adjusted and fine-tuned through techniques
like soft prompt tuning, which optimizes prompts by learning continuous
embeddings. This approach allows the model to learn regardless of the the
quality of the prompt template. Li et al. (2023) introduced Prompt
Distillation, which distills discrete prompts into continuous prompt vectors.
The authors used prompt distillation to add task-specific continuous prompts
per recommendation task globally shared across all users to improve the
model’s performance. Since this method adds a shared continuous prompt per
recommendation task, we will refer to these continuous prompts as task-
specific prompts. Although task-specific prompts were shown to improve
performance in our recommendation tasks, they cannot express personalization
because each user uses the exact same prompts. Thus, it is imperative to
construct a new method to distill each user’s preferences into a personalized
soft prompt.

##  4 User-Personalized Prompt

Figure 1: Method to create the user-personalized prompt. The user embedding
from matrix factorization is used to weight the decomposed prompt components.
The distillled user preferences are then compositionally combined into a
single soft prompt that can be passed to the model as input. Figure 2: Full
PeaPOD model architecture. PeaPOD prepends two continuous prompts to the
discrete prompt template: (1) a user-personalized prompt based on the user’s
preferences and (2) a global task-specific prompt shared across all users. We
add the additional whole-word embedding to distinguish tokens belonging to a
given ID and train an encoder-decoder model.

The goal of the user-personalized prompt is to distill user preferences into a
soft prompt that is able to represent the user’s personal interests while also
leveraging collaborative knowledge between similar users. We maintain a set of
decomposed prompt components, which represents a many-to-many mapping between
user preferences and continuous prompt vectors. This allows us to efficiently
maintain a limited group of prompts and share knowledge across similar users.
The decomposed prompt vectors are then dynamically weighted and combined based
on the user’s interaction history to form a single continuous prompt. Using
this methodology, user with similar preferences should have similar weight
distributions. For example, if a decomposed prompt component contained
information about action movies, then all users with an affinity to actions
movies will contribute significant weight to this prompt component.

###  4.1 Query Function

Since we want to weight the decomposed prompt components based on each user’s
preferences, we introduce a query function $q(x)$. The purpose of $q(x)$ is to
obtain an embedding that effectively captures a user’s personalized interests.
In our case, we choose to use a user embedding generated from matrix
factorization Koren et al. (2009) as the query function. In particular, given
a user-item feedback matrix $A\in\mathbb{R}^{|U|\times|I|}$, the model learns
a user embedding $U\in\mathbb{R}^{|U|\times D}$ and item embedding
$U\in\mathbb{R}^{|I|\times D}$, where $D$ is the dimension size and $|U|$ and
$|I|$ are the number of users and items, respectively. By using a user
embedding as the query function, we can effectively encode user preferences
into the shared set of prompt components.

###  4.2 Prompt-Component Weighting

We use an attention-based architecture Vaswani et al. (2017) to attend the
prompt components based on the user’s personalized preferences. Let
$A\in\mathbb{R}^{M\times D}$ be a learnable attention vector, where $M$ is the
number of decomposed prompt components. We calculate the element-wise
multiplication between q(x) and $A$ to obtain the attended query, denoted as
$\lambda$. The attended query allows the model to focus on the most important
aspects of the user’s interaction history. To obtain the attended query we
calculate:

$\lambda=q(x)\odot A$ (1)

where $\odot$ is the element-wise product. We then calculate the cosine
similarity, denoted as $\gamma$, between $\lambda$ and
$K\in\mathbb{R}^{M\times D}$, where $k_{M}\in K$ is a learnable key that has a
corresponding attention vector $A_{M}$ and prompt component $p_{M}$. The
cosine similarity between the attended query $\lambda$ and $K$ is used to
determine the magnitude that each prompt component should receive, which we
define as:

$\alpha=\gamma(\lambda,K)$ (2)

where $\alpha\in\mathbb{R}^{M\times D}$ is a weighting vector. Since $\alpha$
is updated during training, the model is able to update its representation of
the user embedding rather than fixing it to a single representation. $\alpha$
can then be used to weight the prompt components based on attended query
function. Thus, we can determine how much each preference should contribute to
the user’s overall user-personalized prompt.

###  4.3 Prompt Composition

Finally, we introduce the user-personalized prompt $P$, which is calculated as
a weighted summation over the set of decomposed prompt components:

$P=\sum_{M}\alpha_{M}p_{M}$ (3)

where $p\in\mathbb{R}^{L\times D\times M}$ is a decomposed prompt component
and $L$ is the length of each prompt component. Each decomposed prompt
component is a soft prompt that represents information that can be
collaboratively shared across similar users. We base this idea on the
assumption that users may share similar interests and this shared
collaborative knowledge can be used to improve personalization. Our many-to-
many mapping between user preferences and soft prompts are dynamically
weighted using $\alpha$ based on each user’s interactions, distilling user
preferences to a user-personalized prompt in a compositional manner. Notably,
our model can capture the complexities of a personalized user profile while
maintaining a fixed number of prompts. Since the user-personalized prompt
focuses on the most important parts of the user embedding, our method is
similar high dimension clustering, meaning that users with similar preferences
will have similar user-personalized prompts. We show how to obtain the user-
personalized prompt in Figure 1.

###  4.4 Orthogonal Initialization

To reduce interference in knowledge between prompt components, we can
initialize $P$, $K$, and $A$ orthogonally using QR Decomposition. Namely, let
$Z\in\mathbb{R}^{R\times C}$ be a real matrix with $R$ rows and $C$ columns.
$Z$ can be broken into an orthogonal matrix $Q$ and upper matrix $R$. The
columns of $Q$ will thus be a set of $C$ orthogonal unit vectors which we can
use as the initial vectors. We can repeat this initialization process for
matrices $P$, $K$, and $A$.

###  4.5 Training and Generative Recommendation

Using the task-alternated training strategy proposed by Li et al. (2023), we
alternate the task trained for each batch. This allows us to save time during
training because the samples in the batch have the same data format and will
therefore be a similar length to one another. Importantly, task-alternated
training did not show any degradation in generative recommendation performance
versus traditional random batching strategies Li et al. (2023). We combine
user-personalized prompts, task-specific prompts, and discrete prompts as
input during training. We also add whole-word embeddings to help the model
identify which tokens belong to a particular ID. We show the whole mode
architecture in Figure 2.

As all the generative recommendation tasks are posed as a sequence-to-sequence
task, we use the Negative Log-Likelihood (NLL) loss to optimize the model
parameters $\Theta$.

$\mathcal{L}_{\Theta}=\frac{1}{|\mathcal{D}|}\sum_{(X,Y)\in\mathcal{D}}\frac{1}{|Y|}\sum_{t=1}^{|Y|}-\log
p(y_{t}|Y_{<t},X)$ (4)

where $\mathcal{D}$ is the training set that consists of input output pairs
(X,Y) and $\log p(y_{t}|Y_{<t})$ is the probability of generating token
$y_{t}$, given input sequence $X$ and the tokens previously generated.

In addtion, the model must output a sequence of text that forms an item ID (or
a natural language text in the case of explanation generation). For this task,
we choose beam search, which is commonly used for its effectiveness in
sequence-to-sequence generation. Suppose we set the number of beams to $b$. At
each time step, are $b$ candidate sequences. In the next step, any word in the
vocabulary $V$ are appended to the end of the candidate sequences, resulting
in $b\times V$ combinations. We can then select $b$ sequences that have the
maximum log-likelihood. The LLM can continue this process until the candidate
sequences reach a predefined maximum length. For sequential recommendation and
top-$N$ recommendation, the $b$ candidate sequences form the recommendation
list. For explanation, we select the sequence with the largest log-likelihood
from the candidates.

##  5 Experimental Setup

###  5.1 Datasets

For experimentation, we use the Sports, Beauty, and Toys & Games Dataset from
Amazon 111https://jmcauley.ucsd.edu/data/amazon/. We reuse the preprocessing
steps and the 8:1:1 train/validation/test splits created by Geng et al.
(2022). Each sample consists of a user ID, an item ID, a text review, a
rating, and a time stamp. We provide the dataset statistics in Table 1. For
explanation generation, the Sentires toolkit Zhang et al. (2014) was used to
extract ground-truth explanations for a given (user, item) pair. For
sequential recommendation, the last item in a user’s history is part of the
test set, the penultimate item is part of the validation set, and all other
items are included in the train set. To prevent data leakage, the same
training split used for sequential recommendation is used for top-n
recommendation. For reproducibility, we provide all code used in the
supplementary materials section of our submission and will release the code
publicly upon publication.

Dataset | Sports | Beauty | Toys
---|---|---|---
#Users | 35,598 | 22,363 | 19,412
#Items | 18,357 | 12,101 | 11,924
#Reviews | 296,337 | 198,502 | 167,597
#Sparsity (%) | 0.0453 | 0.0734 | 0.0724
Table 1: Statistics of the datasets.

###  5.2 Baselines

For sequential recommendation, we compare our model with the following
baselines: Caser Tang and Wang (2018), HGN Ma et al. (2019), GRU4Rec Jannach
and Ludewig (2017), BERT4Rec Sun et al. (2019), FDSA Zhang et al. (2019),
SASRec Kang and McAuley (2018), P5 Geng et al. (2022), VIP5 Geng et al.
(2023), and POD Li et al. (2023).

For top-n recommendation, we compare our method with: MF Koren et al. (2009),
MLP Cheng et al. (2016), P5 Geng et al. (2022), VIP5 Geng et al. (2023), and
POD Li et al. (2023).

Finally, for explanation generation, we use the following baselines: Att2Seq
Dong et al. (2017):, NRT Li et al. (2020), PETER Li et al. (2021), and POD Li
et al. (2023). For a fair comparison, we exclude P5 and VIP5 because we are
focused exclusively on models that only use user and item IDs rather than
additional user or item metadata.

Table 2: Performance comparison on sequential recommendation.

Methods | Sports | Beauty | Toys
---|---|---|---
HR@5 | NDCG@5 | HR@10 | NDCG@10 | HR@5 | NDCG@5 | HR@10 | NDCG@10 | HR@5 | NDCG@5 | HR@10 | NDCG@10
Caser | 0.0116 | 0.0072 | 0.0194 | 0.0097 | 0.0205 | 0.0131 | 0.0347 | 0.0176 | 0.0166 | 0.0107 | 0.0270 | 0.0141
HGN | 0.0189 | 0.0120 | 0.0313 | 0.0159 | 0.0325 | 0.0206 | 0.0512 | 0.0266 | 0.0321 | 0.0221 | 0.0497 | 0.0277
GRU4Rec | 0.0129 | 0.0086 | 0.0204 | 0.0110 | 0.0137 | 0.0099 | 0.0283 | 0.0137 | 0.0097 | 0.0059 | 0.0176 | 0.0084
BERT4Rec | 0.0115 | 0.0075 | 0.0191 | 0.0099 | 0.0203 | 0.0124 | 0.0347 | 0.0170 | 0.0116 | 0.0071 | 0.0203 | 0.0099
FDSA | 0.0182 | 0.0122 | 0.0288 | 0.0156 | 0.0267 | 0.0163 | 0.0407 | 0.0208 | 0.0228 | 0.0140 | 0.0381 | 0.0189
SASRec | 0.0233 | 0.0154 | 0.0350 | 0.0192 | 0.0387 | 0.0249 | 0.0605 | 0.0318 | 0.0463 | 0.0306 | 0.0675 | 0.0374
S3-Rec | 0.0251 | 0.0161 | 0.0364 | 0.0204 | 0.0387 | 0.0247 | 0.0607 | 0.0327 | 0.0443 | 0.0294 | 0.0640 | 0.0376
P5 | 0.0272 | 0.0169 | 0.0361 | 0.0198 | 0.0530 | 0.0370 | 0.0659 | 0.0421 | 0.0460 | 0.0567 | 0.0709 | 0.0587
VIP5 | 0.0412 | 0.0345 | 0.0475 | 0.0365 | 0.0556 | 0.0427 | 0.0677 | 0.0467 | 0.0662 | 0.0577 | 0.0749 | 0.0604
POD | 0.0496 | 0.0396 | 0.0576 | 0.0419 | 0.0537 | 0.0395 | 0.0688 | 0.0443 | 0.0691 | 0.0599 | 0.0742 | 0.0610
PeaPOD | 0.0505 | 0.0400 | 0.0611 | 0.0432 | 0.0588 | 0.0445 | 0.0738 | 0.0493 | 0.0692 | 0.0583 | 0.0787 | 0.0609

Table 3: Performance comparison on top-n recommendation.

Methods | Sports | Beauty | Toys
---|---|---|---
HR@1 | HR@5 | NDCG@5 | HR@10 | NDCG@10 | HR@1 | HR@5 | NDCG@5 | HR@10 | NDCG@10 | HR@1 | HR@5 | NDCG@5 | HR@10 | NDCG@10
MF | 0.0314 | 0.1404 | 0.0848 | 0.2563 | 0.1220 | 0.0311 | 0.1426 | 0.0857 | 0.2573 | 0.1224 | 0.0233 | 0.1066 | 0.0641 | 0.2003 | 0.0940
MLP | 0.0351 | 0.1520 | 0.0927 | 0.2671 | 0.1296 | 0.0317 | 0.1392 | 0.0848 | 0.2542 | 0.1215 | 0.0252 | 0.1142 | 0.0688 | 0.2077 | 0.0988
P5 | 0.0567 | 0.1514 | 0.1049 | 0.2196 | 0.1269 | 0.0571 | 0.1566 | 0.1078 | 0.2317 | 0.1318 | 0.0451 | 0.1322 | 0.0889 | 0.2023 | 0.1114
VIP5 | 0.0699 | 0.1882 | 0.1304 | 0.2717 | 0.1572 | 0.0615 | 0.1655 | 0.1147 | 0.2407 | 0.1388 | 0.0433 | 0.1301 | 0.0875 | 0.2037 | 0.1110
POD | 0.0895 | 0.2086 | 0.1506 | 0.2873 | 0.1756 | 0.0829 | 0.1926 | 0.1391 | 0.2670 | 0.1629 | 0.0567 | 0.1433 | 0.1009 | 0.2082 | 0.1215
PeaPOD | 0.1212 | 0.2741 | 0.2007 | 0.3616 | 0.2286 | 0.1097 | 0.2441 | 0.1780 | 0.3260 | 0.2039 | 0.0728 | 0.1618 | 0.1190 | 0.2315 | 0.1411

Table 4: Performance comparison on explanation generation.

Methods | Sports | Beauty | Toys
---|---|---|---
BLEU-4 | ROUGE-1 | ROUGE-2 | ROUGE-L | BLEU-4 | ROUGE-1 | ROUGE-2 | ROUGE-L | BLEU-4 | ROUGE-1 | ROUGE-2 | ROUGE-L
Att2Seq | 0.5305 | 12.2800 | 1.2107 | 9.1312 | 0.7889 | 12.6590 | 1.6820 | 9.7481 | 1.6238 | 13.2245 | 2.9942 | 10.7398
NRT | 0.4793 | 11.0723 | 1.1304 | 7.6674 | 0.8295 | 12.7815 | 1.8543 | 9.9477 | 1.9084 | 13.5231 | 3.6708 | 11.1867
PETER | 0.7112 | 12.8944 | 1.3283 | 9.8635 | 1.1541 | 14.8497 | 2.1413 | 11.4143 | 1.9861 | 14.2716 | 3.6718 | 11.7010
POD | 1.0013 | 14.0168 | 2.0436 | 11.1236 | 1.0630 | 15.2517 | 1.5737 | 11.3283 | 2.3053 | 12.2889 | 3.8512 | 10.3923
PeaPOD | 0.9148 | 11.2685 | 1.7870 | 8.8078 | 1.0050 | 13.4532 | 1.5863 | 10.3238 | 2.4295 | 13.5537 | 3.9567 | 11.3863

###  5.3 Evaluation Metrics

For sequential and top-n recommendation, we evaluate models using two commonly
used metrics, normalized Discounted Cumulative Gain (nDCG) and Hit Ratio (HR).
nDCG is a metric used to evaluate the quality of ranked lists by considering
the position of relevant items, while HR measures the fraction of relevant
items that appear in the top-n positions of a ranked list. In particular, we
measure HR@{5,10} and nDCG@{5,10} for sequential recommendation and
HR@{1,5,10} and nDCG@{5,10} for top-n recommendation. For explanation
generation, we evaluate models using BLEU Papineni et al. (2002) and ROUGE Lin
and Hovy (2003). BLEU evaluates machine translation quality using n-gram
precision against reference translations, while ROUGE assesses text
summarization and translation by comparing n-gram overlaps with reference
texts. We report the F1 scores for BLEU-4, ROUGE-1, ROUGE-2, and ROUGE-L. For
all metrics, a greater score indicates better performance. In all tables, the
best performing model for each metric is in bold and the second best model is
underlined.

###  5.4 Implementation Details

To ensure a fair comparison, we compare PeaPOD with POD Li et al. (2023), P5
Geng et al. (2022), and VIP5 Geng et al. (2023) using T5-small (60.5 M
parameters) Raffel et al. (2020) as the pre-trained LLM. On average, each
experiment took 6 hours to run. Since the main focus of our study is not on
formulating discrete prompts, we reuse the same discrete prompts as Geng et
al. (2022). We adapt task-alternated training, as proposed by Li et al.
(2023), to speed up training time by alternating each task per batch. Notably,
both VIP5 and P5 both mention that fine-tuning with the T5-small pre-trained
checkpoint often outperforms fine-tuning with T5-base.

For all models, we train the model on an Nvidia RTX 3090 for 30 epochs, a
batch size of 64, the AdamW optimizer Kingma and Ba (2015), and a learning
rate of 5e-3. We save the best performing model on the validation set. The
length of the task-specific prompts and prompt components are both set to 3.
We set the number of prompt components to 20. For our hyperparameter search,
we tested prompt lengths of {3, 5, 8, 10} and the number of prompts of {10,
20, 30, 100}.To obtain the user embedding, we train a probabalistic matrix
factorization model Mnih and Salakhutdinov (2007), a variant of matrix
factorization, for 100 iterations with an embedding size of 512, learning rate
of 1e-3, and a lambda regularization of 1e-3. During inference time, the
number of beams is set to 20 and the inference batch size is set to 32 for all
tasks.

##  6 Results

###  6.1 Sequential Recommendation

In Table 2, we compare our models with our selected baseline models. Our
results show that PeaPOD outperforms all baselines in the Sports and Beauty
datasets and is either the first or second best model for all metrics in the
Toys dataset. Notably, the only difference between our method and POD, which
is the second best performing model, is the additional personalized-user
prompt that is prepended to the input. Interestingly, even though the user
embedding passed to the attention mechanism is trained on a top-n recommender
model, the additional information still provides a performance boost to
sequential recommendation. This performance gain may be attributed to the way
the data was split. In particular, the sequential and top-n recommendation
contains the same user-item interactions, meaning that information about the
users can be shared to improve performance in both tasks.

###  6.2 Top-n Recommendation

In Table 3, we report the performance of the top-n recommendation task. We
note that PeaPOD performs the best in all metrics for all datasets, with a
significant improvement in performance. We note that matrix factorization,
which is used to generate the user embedding, is trained for top-n
recommendation. Thus, the boost in performance may be more significant when
compared to the other two tasks because of the user embedding selected as the
query function. Nevertheless, we also find that PeaPOD outperforms matrix
factorization itself, meaning that combining traditional collaborative
filtering methods with LLMs leads to increased performance.

Figure 3: Comparison between PeaPOD and an ablated version (PeaPOD-X) trained
without the task-specific prompts on our three tasks.

###  6.3 Explanation Generation

In Table 4, we report the performance of all models on the explanation
generation task. Interestingly PeaPOD performs particularly well on both
Sports and Toys datasets, but performs significantly worse in Beauty when
compared to the top baseline models. Nevertheless, PeaPOD ranks first or
second best on six out of the twelve evaluations, meaning that it is still
competitive in explanation generation.

Upon manual inspection across all models, we notice that there are a
significant amount of duplicates across explanations, meaning that the same
explanations are generated regardless of the (user, item) pair. Thus, we posit
that limiting the explanation generation to exclusively user and item IDs
hurts performance. Similar to how performance for sequential and top-n
recommendation was improved by providing collaborative user embeddings, the
performance of item explanations can be improved by providing additional item
metadata to the model. Previous research Geng et al. (2023); Tan et al.
(2024); Rajput et al. (2023) has already explored how discrete item IDs can be
transformed to provide more informative tokens to the model. However, we leave
the additional experiments of combining enhanced item IDs with PeaPOD to
future work.

###  6.4 Training without Task-Specific Prompts

To understand the effectiveness of user-personalized prompts versus globally
shared prompts, we fine-tune the model with the same architecture and
hyperparameters, but omit the task-specific prompts. We refer to the ablated
model as ‘PeaPOD-X’.

As shown in Figure 3, we find that in all three datasets, the performance of
the sequential recommendation marginally decreases, but the top-n
recommendation performance improves. For explanation generation, the original
model performs better for the Beauty dataset, but worse for Sports and Toys.
We attribute the boost in top-n recommendation performance to the user
embedding used for the query function. In particular, matrix factorization is
a top-n recommendation model, meaning that the model is most likely to excel
in this task. However, the lack of a task-specific prompt for sequential
recommendation hinders the performance on this task. Nevertheless, we observe
for the sequential recommendation task, the ablated model is still better than
solely using the discrete prompt and has similar performance to POD. Thus, we
show the effectiveness of a user-personalized prompt that utilizes preferences
shared between similar users to improve recommendation performance. We report
all metrics for this experiment in Appendix B.

##  7 Conclusion

In this paper, we propose PeaPOD, a novel recommender model that dynamically
weights a set of decomposed prompt components to distill user preferences in a
compositional manner. We maintain a many-to-many mapping between user
preferences and continuous prompts to enable knowledge sharing among users
with similar preferences. We utilize an attention-based mechanism to weight
prompt components based on the user’s interaction history, thus creating a
user-personalized prompt that distills both the user’s and similar users’
preferences into a learnable soft prompt. In our experiments, we show that our
user-personalized prompt is effective in increasing the performance for three
common recommendation tasks. Our model shows the effectiveness of preference
distillation to improve personalized generative recommendation.

##  8 Limitations and Ethical Concerns

One limitation to our approach is that we limit our study to exclusively use
user and item IDs. Thus, we do not incorporate any additional metadata such as
user profiles or item descriptions. We find that this may hinder performance
on explanation generation, which relies on having a strong understanding of
both the user and item to provide a personalized response. We predict that
providing additional metadata to the model can improve performance,
particularly if it is given in natural language text as the LLM is already
pre-trained on such data. We plan to extend PeaPOD to utilize this information
in future work.

In addition, our model takes a two-step approach to generate user-personalized
prompts. Namely, the user embedding is generated using matrix factorization,
which is then passed to the model as the query function. However, the models
are not connected end-to-end, which means that the user embedding cannot be
continuously updated during training. For future work, we plan to combine
these two models into a single end-to-end framework.

Another concern is the issue of fairness in generative recommender systems.
For example, prior research has shown that LLM-based recommenders skew towards
popular items, which causes a disparity in recommendations and can lead to a
bias-amplification loop Yoon et al. (2024). In addition, past research has
investigated the issue of fairness in the LLM itself Salutari et al. (2023).
Thus, careful consideration must be made when deploying a generative
recommender model to mitigate unfair recommendations.

## References

  * Cheng et al. (2016) Heng-Tze Cheng, Levent Koc, Jeremiah Harmsen, Tal Shaked, Tushar Chandra, Hrishi Aradhye, Glen Anderson, Greg Corrado, Wei Chai, Mustafa Ispir, Rohan Anil, Zakaria Haque, Lichan Hong, Vihan Jain, Xiaobing Liu, and Hemal Shah. 2016.  Wide & Deep Learning for Recommender Systems.  In _Proceedings of the 1st Workshop on Deep Learning for Recommender Systems_ , pages 7–10, Boston MA USA. ACM.
  * Dong et al. (2017) Li Dong, Shaohan Huang, Furu Wei, Mirella Lapata, Ming Zhou, and Ke Xu. 2017.  Learning to Generate Product Reviews from Attributes.  In _Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 1, Long Papers_ , pages 623–632, Valencia, Spain. Association for Computational Linguistics.
  * Ge et al. (2022) Yingqiang Ge, Shuchang Liu, Zuohui Fu, Juntao Tan, Zelong Li, Shuyuan Xu, Yunqi Li, Yikun Xian, and Yongfeng Zhang. 2022.  A Survey on Trustworthy Recommender Systems.  _Preprint_ , arxiv:2207.12515.
  * Geng et al. (2022) Shijie Geng, Shuchang Liu, Zuohui Fu, Yingqiang Ge, and Yongfeng Zhang. 2022.  Recommendation as Language Processing (RLP): A Unified Pretrain, Personalized Prompt & Predict Paradigm (P5).  In _Proceedings of the 16th ACM Conference on Recommender Systems_ , pages 299–315, Seattle WA USA. ACM.
  * Geng et al. (2023) Shijie Geng, Juntao Tan, Shuchang Liu, Zuohui Fu, and Yongfeng Zhang. 2023.  VIP5: Towards Multimodal Foundation Models for Recommendation.  In _Findings of the Association for Computational Linguistics: EMNLP 2023_ , pages 9606–9620, Singapore. Association for Computational Linguistics.
  * He et al. (2023) Zhankui He, Zhouhang Xie, Rahul Jha, Harald Steck, Dawen Liang, Yesu Feng, Bodhisattwa Prasad Majumder, Nathan Kallus, and Julian Mcauley. 2023.  Large Language Models as Zero-Shot Conversational Recommenders.  In _Proceedings of the 32nd ACM International Conference on Information and Knowledge Management_ , pages 720–730, Birmingham United Kingdom. ACM.
  * Jannach and Ludewig (2017) Dietmar Jannach and Malte Ludewig. 2017.  When Recurrent Neural Networks meet the Neighborhood for Session-Based Recommendation.  In _Proceedings of the Eleventh ACM Conference on Recommender Systems_ , pages 306–310, Como Italy. ACM.
  * Kang and McAuley (2018) Wang-Cheng Kang and Julian McAuley. 2018.  Self-attentive sequential recommendation.  In _2018 IEEE international conference on data mining (ICDM)_ , pages 197–206. IEEE.
  * Kingma and Ba (2015) Diederik P. Kingma and Jimmy Ba. 2015.  Adam: A method for stochastic optimization.  In _ICLR (Poster)_.
  * Koren et al. (2009) Yehuda Koren, Robert Bell, and Chris Volinsky. 2009.  Matrix Factorization Techniques for Recommender Systems.  _Computer_ , 42(8):30–37.
  * Li et al. (2020) Lei Li, Yongfeng Zhang, and Li Chen. 2020.  Generate Neural Template Explanations for Recommendation.  In _Proceedings of the 29th ACM International Conference on Information & Knowledge Management_, pages 755–764, Virtual Event Ireland. ACM.
  * Li et al. (2021) Lei Li, Yongfeng Zhang, and Li Chen. 2021.  Personalized Transformer for Explainable Recommendation.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 4947–4957, Online. Association for Computational Linguistics.
  * Li et al. (2023) Lei Li, Yongfeng Zhang, and Li Chen. 2023.  Prompt Distillation for Efficient LLM-based Recommendation.  In _Proceedings of the 32nd ACM International Conference on Information and Knowledge Management_ , pages 1348–1357, Birmingham United Kingdom. ACM.
  * Lin and Hovy (2003) Chin-Yew Lin and Eduard Hovy. 2003.  Automatic evaluation of summaries using N-gram co-occurrence statistics.  In _Proceedings of the 2003 Conference of the North American Chapter of the Association for Computational Linguistics on Human Language Technology - NAACL ’03_ , volume 1, pages 71–78, Edmonton, Canada. Association for Computational Linguistics.
  * Ma et al. (2019) Chen Ma, Peng Kang, and Xue Liu. 2019.  Hierarchical Gating Networks for Sequential Recommendation.  In _Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, pages 825–833, Anchorage AK USA. ACM.
  * Mnih and Salakhutdinov (2007) Andriy Mnih and Russ R Salakhutdinov. 2007.  Probabilistic Matrix Factorization.  In _Advances in Neural Information Processing Systems_ , volume 20. Curran Associates, Inc.
  * Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002.  BLEU: A method for automatic evaluation of machine translation.  In _Proceedings of the 40th Annual Meeting on Association for Computational Linguistics_ , ACL ’02, pages 311–318, USA. Association for Computational Linguistics.
  * Petrov and Macdonald (2023) Aleksandr V. Petrov and Craig Macdonald. 2023.  Generative Sequential Recommendation with GPTRec.  _Preprint_ , arxiv:2306.11114.
  * Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2020.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _The Journal of Machine Learning Research_ , 21(1):140:5485–140:5551.
  * Rajput et al. (2023) Shashank Rajput, Nikhil Mehta, Anima Singh, Raghunandan Hulikal Keshavan, Trung Vu, Lukasz Heldt, Lichan Hong, Yi Tay, Vinh Tran, Jonah Samost, Maciej Kula, Ed Chi, and Maheswaran Sathiamoorthy. 2023.  Recommender Systems with Generative Retrieval.  _Advances in Neural Information Processing Systems_ , 36:10299–10315.
  * Ramos et al. (2024) Jerome Ramos, Hossen A. Rahmani, Xi Wang, Xiao Fu, and Aldo Lipani. 2024.  Natural Language User Profiles for Transparent and Scrutable Recommendations.  _Preprint_ , arxiv:2402.05810.
  * Salutari et al. (2023) Flavia Salutari, Jerome Ramos, Hossein A Rahmani, Leonardo Linguaglossa, and Aldo Lipani. 2023.  Quantifying the bias of transformer-based language models for african american english in masked language modeling.  In _Pacific-Asia Conference on Knowledge Discovery and Data Mining_ , pages 532–543. Springer Nature Switzerland Cham.
  * Sanner et al. (2023) Scott Sanner, Krisztian Balog, Filip Radlinski, Ben Wedin, and Lucas Dixon. 2023.  Large Language Models are Competitive Near Cold-start Recommenders for Language- and Item-based Preferences.  _Preprint_ , arxiv:2307.14225.
  * Smith et al. (2023) James Seale Smith, Leonid Karlinsky, Vyshnavi Gutta, Paola Cascante-Bonilla, Donghyun Kim, Assaf Arbelle, Rameswar Panda, Rogerio Feris, and Zsolt Kira. 2023.  CODA-Prompt: COntinual Decomposed Attention-Based Prompting for Rehearsal-Free Continual Learning.  In _2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , pages 11909–11919, Vancouver, BC, Canada. IEEE.
  * Sun et al. (2019) Fei Sun, Jun Liu, Jian Wu, Changhua Pei, Xiao Lin, Wenwu Ou, and Peng Jiang. 2019.  BERT4Rec: Sequential Recommendation with Bidirectional Encoder Representations from Transformer.  In _Proceedings of the 28th ACM International Conference on Information and Knowledge Management_ , pages 1441–1450, Beijing China. ACM.
  * Tan et al. (2024) Juntao Tan, Shuyuan Xu, Wenyue Hua, Yingqiang Ge, Zelong Li, and Yongfeng Zhang. 2024.  IDGenRec: LLM-RecSys Alignment with Textual ID Learning.  In _SIGIR_. arXiv.
  * Tang and Wang (2018) Jiaxi Tang and Ke Wang. 2018.  Personalized Top-N Sequential Recommendation via Convolutional Sequence Embedding.  In _Proceedings of the Eleventh ACM International Conference on Web Search and Data Mining_ , pages 565–573, Marina Del Rey CA USA. ACM.
  * Vaswani et al. (2017) Ashish Vaswani, Noam M. Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  In _Neural Information Processing Systems_.
  * Wu et al. (2023) Bin Wu, Jinyuan Fang, Xiangxiang Zeng, Shangsong Liang, and Qiang Zhang. 2023.  Adaptive compositional continual meta-learning.  In _International Conference on Machine Learning_ , pages 37358–37378. PMLR.
  * Wu et al. (2022) Bin Wu, Zaiqiao Meng, Qiang Zhang, and Shangsong Liang. 2022.  Meta-learning helps personalized product search.  In _Proceedings of the ACM Web Conference 2022_ , pages 2277–2287.
  * Wu et al. (2024) Bin Wu, Zhengyan Shi, Hossein A Rahmani, Varsha Ramineni, and Emine Yilmaz. 2024.  Understanding the role of user profile in the personalization of large language models.  _arXiv preprint arXiv:2406.17803_.
  * Yoon et al. (2024) Se-eun Yoon, Zhankui He, Jessica Maria Echterhoff, and Julian McAuley. 2024.  Evaluating Large Language Models as Generative User Simulators for Conversational Recommendation.  _Preprint_ , arxiv:2403.09738.
  * Zhang et al. (2019) Tingting Zhang, Pengpeng Zhao, Yanchi Liu, Victor S. Sheng, Jiajie Xu, Deqing Wang, Guanfeng Liu, and Xiaofang Zhou. 2019.  Feature-level Deeper Self-Attention Network for Sequential Recommendation.  In _Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence_ , pages 4320–4326, Macao, China. International Joint Conferences on Artificial Intelligence Organization.
  * Zhang et al. (2014) Yongfeng Zhang, Guokun Lai, Min Zhang, Yi Zhang, Yiqun Liu, and Shaoping Ma. 2014.  Explicit factor models for explainable recommendation based on phrase-level sentiment analysis.  In _Proceedings of the 37th International ACM SIGIR Conference on Research & Development in Information Retrieval_, pages 83–92, Gold Coast Queensland Australia. ACM.
  * Zhou et al. (2020) Kun Zhou, Hui Wang, Wayne Xin Zhao, Yutao Zhu, Sirui Wang, Fuzheng Zhang, Zhongyuan Wang, and Ji-Rong Wen. 2020.  S3-Rec: Self-Supervised Learning for Sequential Recommendation with Mutual Information Maximization.  In _Proceedings of the 29th ACM International Conference on Information & Knowledge Management_, pages 1893–1902, Virtual Event Ireland. ACM.

##  Appendix A Baselines

For each of our three tasks, we compare our model with the following
baselines.

###  A.1 Sequential Recommendation

Sequential recommendation predicts a user’s next preferred items based on
their previous interactions, which are given in historical order.

  * •

Caser: ConvolutionAl Sequence Embedding Recommendation Tang and Wang (2018)
uses a convolutional neural network to learn the sequential pattern of the
user interaction history.

  * •

HGN Ma et al. (2019): Hierarchical Gating Network uses two gating modules to
model a user’s short and long term interests.

  * •

GRU4Rec Jannach and Ludewig (2017): Gated Recurrent Unit (GRU) for
Recommendation uses a GRU to process a user’s recommender session.

  * •

BERT4Rec Sun et al. (2019): Fine-tunes a bidirectional encoder representations
from transformers model for sequential recommendation.

  * •

FDSA Zhang et al. (2019): Feature-level Deeper Self-Attention network.
Integrates item features as feature sequences, which are then combined with
item sequences to generate recommendations.

  * •

SASRec Kang and McAuley (2018): Self-Attentive based Sequential Recommendation
model. Uses self-attention mechanisms for short-term semantics and Recurrent
Neural Networks for long-term semantics.

  * •

$\mathbf{S^{3}}$-Rec Zhou et al. (2020): Self-Supervised learning for
Sequential Recommendation. Uses four self-supervised learning objectives to
capture the correlations between items in sequential data.

  * •

P5 Geng et al. (2022): Pretrain, Personalized Prompt, and Predict Paradigm.
Trains a model for multi-task generative recommendation using discrete prompt
templates (e.g. “Which item of the following to recommend for user_{}?
item_list:{}”

  * •

VIP5 Geng et al. (2023): An extension of P5 that trains on multimodal data.

  * •

POD Li et al. (2023): An extension of P5 that uses a task-specific, distilled
prompt as part of the input.

###  A.2 Top-n Recommendation

Top-n recommendation predicts a list of the top-n items based on the user’s
interaction history.

  * •

MF Koren et al. (2009): Matrix Factorization is a collaborative filtering
technique that decomposes a user-item interaction matrix into latent user and
item embeddings.

  * •

MLP Cheng et al. (2016): Multi-Layer Perceptron is a neural network trained to
learn captures non-linear interactions between users and items.

  * •

P5 Geng et al. (2022): See definition in Section A.1.

  * •

VIP5 Geng et al. (2023): See definition in Section A.1.

  * •

POD Li et al. (2023): See definition in Section A.1.

###  A.3 Explanation Generation

Given a user ID and item ID, the model must return a natural language
explanation describing why the user might like/dislike the target item.
Notably, we exclude P5 and VIP5 because we are exclusively focused on models
that exclusively use only user and item IDs rather than item titles or other
item metadata.

  * •

Att2Seq Dong et al. (2017): an attention based model that learns to generate
product reviews from item attributes.

  * •

NRT Li et al. (2020): Neural Ratings and Tips generation simultaneously
returns a rating and a tip based on a neural network.

  * •

PETER Li et al. (2021): a transformer based model that performs rating
prediction and explanation generation.

##  Appendix B Training without Task-Specific Prompts

We report the original model including task-specific prompts (denoted as
PeaPOD) versus training the model without them (denoted as PeaPOD-X). All
inputs and hyperparameters are otherwise the same.

Table 5: Performance comparison on sequential recommendation.

Methods | Sports | Beauty | Toys
---|---|---|---
HR@5 | NDCG@5 | HR@10 | NDCG@10 | HR@5 | NDCG@5 | HR@10 | NDCG@10 | HR@5 | NDCG@5 | HR@10 | NDCG@10
PeaPOD | 0.0505 | 0.0400 | 0.0611 | 0.0432 | 0.0588 | 0.0445 | 0.0738 | 0.0493 | 0.0692 | 0.0583 | 0.0787 | 0.0609
PeaPOD-X | 0.0480 | 0.0381 | 0.0581 | 0.0412 | 0.0565 | 0.0402 | 0.0732 | 0.0456 | 0.0705 | 0.0557 | 0.0835 | 0.0594

Table 6: Performance comparison on top-n recommendation.

Methods | Sports | Beauty | Toys
---|---|---|---
HR@1 | HR@5 | NDCG@5 | HR@10 | NDCG@10 | HR@1 | HR@5 | NDCG@5 | HR@10 | NDCG@10 | HR@1 | HR@5 | NDCG@5 | HR@10 | NDCG@10
PeaPOD | 0.1212 | 0.2741 | 0.2007 | 0.3616 | 0.2286 | 0.1097 | 0.2441 | 0.1780 | 0.3260 | 0.2039 | 0.0728 | 0.1618 | 0.1190 | 0.2315 | 0.1411
PeaPOD-X | 0.1232 | 0.2744 | 0.2017 | 0.3662 | 0.2309 | 0.1130 | 0.2480 | 0.1820 | 0.3304 | 0.2079 | 0.0790 | 0.1795 | 0.1308 | 0.2513 | 0.1536

Table 7: Performance comparison on explanation generation.

Methods | Sports | Beauty | Toys
---|---|---|---
BLEU-4 | ROUGE-1 | ROUGE-2 | ROUGE-L | BLEU-4 | ROUGE-1 | ROUGE-2 | ROUGE-L | BLEU-4 | ROUGE-1 | ROUGE-2 | ROUGE-L
PeaPOD | 0.9148 | 11.2685 | 1.7870 | 8.8078 | 1.0050 | 13.4532 | 1.5863 | 10.3238 | 2.4295 | 13.5537 | 3.9567 | 11.3863
PeaPOD-X | 0.9561 | 12.1112 | 1.7613 | 9.4255 | 0.9598 | 11.3673 | 1.7317 | 8.8236 | 2.5306 | 15.3696 | 3.8183 | 12.6750


# Galois representations and composite moduli

Subham Bhakta  Mathematisches Institut, Georg-August-Universität Göttingen,
Germany<EMAIL_ADDRESS>
###### Abstract.

It is known that for any elliptic curve $E/\mathbb{Q}$ and any integer $m$ co-
prime to $30,$ the induced Galois representation
$\rho_{E,m}:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\text{GL}_{2}(\mathbb{Z}/m\mathbb{Z})$
is surjective if and only if $\rho_{E,\ell}$ is surjective for any prime
$\ell|m.$ In this article, we shall discuss some generalizations,
applications, and variants of this phenomenon.

###### Key words and phrases:

Linear recurrence sequence, exponential sums, modular forms

###### 2010 Mathematics Subject Classification:

Primary 11F30, 11L07; Secondary 11P05, 11B37, 11F80

###### Contents

  1. 1 Introduction
  2. 2 Background on finite groups
  3. 3 Proof of the main statements
  4. 4 Further remarks

##  1\. Introduction

Let $E/\mathbb{Q}$ be an elliptic curve. Serre introduced the following
representation,

$\rho_{E,n}:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\text{Aut}_{\mathbb{C}}(E[n])\cong\text{GL}_{2}\big{(}\mathbb{Z}/n\mathbb{Z}\big{)},$

where $E[n]$ is the set of $n$-torsion points in $E(\mathbb{C}).$ Serre’s open
image theorem says that, if $E$ is without complex multiplication, then there
exists a constant $c_{E}>0$ such that for any prime $\ell>c_{E},$ the
associated representation $\rho_{E,\ell}$ is surjective. It is conjectured
that $c_{E}$ is uniformly bounded. For the known bounds on $c_{E},$ the reader
may refer to [2] and [12]. When $E$ has complex multiplication, the
surjectivity is not true for large primes; see page 12 in [1]. In general,
whether the elliptic curve $E$ is without complex multiplication or not, Serre
showed that for any $m\in\mathbb{N}$ with $(m,30)=1$, $\rho_{E,m}$ is
surjective if and only if $\rho_{E,\ell}$ is surjective for any prime
$\ell\mid m$. For reference, the reader may look at [2] and [10]. We shall
call this phenomenon local-global property of Galois representations.

We first consider a similar problem over arbitrary number fields. Let $K$ be a
number field, and $E$ be an elliptic curve defined over $K.$ We now have an
associated Galois representation defined by

$\rho_{E,n}:\text{Gal}(\overline{K}/K)\longrightarrow\text{Aut}_{\mathbb{C}}(E[n])\cong\text{GL}_{2}(\mathbb{Z}/n\mathbb{Z}).$

The absolute Galois group $\text{Gal}(\overline{K}/K)$ is contained in
$\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),$ but since $E$ is not
necessarily defined over $\mathbb{Q},$ one can not possibly extend
$\rho_{E,n}$ to $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}).$ Following
Cojocaru’s arguments, and the ramification theory for cyclotomic fields, it is
not hard to show that a similar result holds for any $m$ with $(m,30d_{K})=1,$
or $(\phi(m),D_{k})=1,$ where $d_{K}$ and $D_{k}$ respectively are the
discriminant and degree of $K$ over $\mathbb{Q}.$ We shall have a detailed
discussion on this in Theorem 1.

Let us call a natural number $m,\textit{bad}$ if there exists a finite
extension $K_{m}$ of $K,$ and an elliptic curves $E$ over $K_{m}$ such that
$\text{im}(\rho_{E,m})\neq\text{GL}_{2}(\mathbb{Z}/m\mathbb{Z})$ but
$\text{im}(\rho_{E,\ell})=\text{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ for any
prime $\ell\mid m.$ Moreover, we call such a number field as native to $m$ and
such an elliptic curve as exceptional elliptic curve for the pair $(m,K_{m}).$
This is how we measure the failure of local-global property for Galois
representations.

For any arbitrary number $K,$ we arrange the elliptic curves over $K$ with
respect to the usual height $h(E)=||(a,b)||,$ where we consider the usual norm
in $\mathbb{R}\otimes\mathcal{O}_{K}^{2}.$ Here $E$ is in the Weierstrass form
given by $E_{(a,b)}:y^{2}=x^{3}+ax+b$ with $a,b\in\mathcal{O}_{K},$ the ring
of integers of $K.$ Denote $S_{K}(x)=\\{(a,b)\in\mathcal{O}^{2}_{K}\mid
h(E_{(a,b)})\leq x\\}.$ It can be shown that $|S_{K}(x)|=c_{K}x^{2D_{K}},$ for
some constant $c_{K}>0.$

Let us now state the main result of this article.

######  Theorem 1.

Let $K$ be a number field with discriminant $d_{K}$ and degree $D_{k}$ over
$\mathbb{Q}.$ Consider $E/K$ to be an elliptic curve and $m$ be any natural
number co-prime to $30.$

  1. (a)

The induced Galois representation $\rho_{E,m}$ is surjective if and only if
$\rho_{E,\ell}$ is surjective for any prime $\ell|m,$ provided that $K$
contains no proper abelian extension of $\mathbb{Q},$ or if $m$ is co-prime to
the discriminant $D_{K}.$

  2. (b)

Any integer $m$ co-prime to $30,$ that is not square-free, is bad. Moreover,
for any number field, $K_{m}$ native to $m,$ almost all the elliptic curves
over $K_{m}$ are exceptional.

Following the same argument, one can show that for any arbitrary number field
$K,\text{im}(\rho_{E,m})$ contains $\text{SL}_{2}(\mathbb{Z}/m\mathbb{Z})$, if
and only if $\text{im}(\rho_{E,\ell})$ contains
$\mathrm{SL}_{2}(\mathbb{Z}/m\mathbb{Z})$ for any prime $\ell|m,$ assuming
that $(m,30)=1.$ In particular, it follows from Proposition 5.2 of [11] that

######  Corollary 2.

Let $K$ be any arbitrary number field and $m$ be any given integer co-prime to
$30.$ Then for almost all the elliptic curves $E$ over
$K,~{}\mathrm{im}(\rho_{E,m})$ contains
$\mathrm{SL}_{2}(\mathbb{Z}/m\mathbb{Z}).$

Following the notations introduced by Grant in [4], we denote
$\mathcal{E}_{m}(X)$ to be the number of elliptic curves (up-to isomorphism)
over $\mathbb{Q}$ of height at most $X$, such that the corresponding
representation $\rho_{E,m}$ is not surjective. Let $m$ be a natural number
with $(m,210)=1,$ then it follows from Cojocaru’s and Grant’s result that,

$\mathcal{E}_{m}(X)=\sum_{\ell|m}\mathcal{E}_{\ell}(X)\leq\omega(m)X^{1+\varepsilon},$

where $\omega(m)$ denotes the number of distinct prime factors of $m.$

Zywina later considered a related problem over arbitrary number fields in
[11]. The interesting part comes when we want to make Zywina’s bounds
effective. For instance, if we look at Proposition 5.7 in [11], it is clear
that the absolute constant is given by the number of conjugacy classes of
$\text{GL}_{2}(\mathbb{Z}/m\mathbb{Z})$ with determinant $1$. Therefore, the
absolute constant is about $\frac{m^{3}}{\phi(m)}.$ By Theorem 1, the constant
is of order $\sum_{\ell\mid m}\frac{\ell^{3}}{\phi(\ell)},$ for any
$m\in\mathbb{N}$ with $(m,30d_{K})=1.$ Of course, this constant is better when
$m$ is not square-free, and the prime factors are with sufficiently large
valuations.

Serre introduced a representation associated with the pair of elliptic curves
(analogously for the arbitrary tuple as well) as,

$\rho_{E_{1}\times E_{2},n}(\sigma)=\left({\rho_{E_{1},n}(\sigma)\atop
0}{0\atop\rho_{E_{2},n}(\sigma)}\right)~{}\mathrm{for}~{}\mathrm{all}~{}\sigma\in\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}).$

In this case, Serre showed an analog of his open image theorem. To be more
precise, Serre showed that $\text{im}(\rho_{E_{1}\times
E_{2},\ell})=\Delta(\ell)$ for all but finitely many primes $\ell$, where the
diagonal sub-group $\Delta(\ell)$ is given by,

$\Delta(\ell)=\left\\{\left({g_{1}\atop 0}{0\atop
g_{2}}\right)\mid\text{det}(g_{1})=\text{det}(g_{2}),\big{(}g_{1},g_{2}\big{)}\in\text{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})\times\text{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})\right\\}.$

Jones in [6] considered this topic and proved an asymptotic estimate analogous
to Grant’s main result in [4]. Grant’s work was based on counting rational
points on certain modular curves. Jones’s approach was by studying the
distribution of Frobenius symbols using the multi-dimensional version of
Gallagher’s large sieve.

######  Theorem 3.

Let $K$ be a number field with discriminant $d_{K}$ and degree $D_{k}$ over
$\mathbb{Q}.$ Consider $(E_{1},E_{2})$ be any pair of elliptic curves over
$K$, and $m$ be a natural number, and $m$ be any natural number co-prime to
$30.$ The induced Galois representation $\rho_{E_{1}\times E_{2},m}$ has image
$\Delta(m)$ if and only if, $\rho_{E,\ell}$ has image $\Delta(\ell)$ for any
prime $\ell|m,$ provided that $K$ contains no proper abelian extension of
$\mathbb{Q},$ or if $m$ is co-prime to the discriminant $D_{K}.$

In Section 4, we shall discuss a modular analog of the local-global
phenomenon. We shall see that Theorem 1 corresponds to newforms, and Theorem 3
correspond to any arbitrary cusp form.

For polarized abelian varieties, it is believed that (see [5]) an analog of
Serre’s open image theorem is true. If we trust this, we should have an analog
of Theorem 1. Indeed, we have the following

######  Theorem 4.

. Let $K$ be any arbitrary number field, and $A/K$ be a polarized abelian
variety of dim $g$. For any integer, $m\in\mathbb{N},$ the induced Galois
representation $\rho_{A,m}$ is surjective if and only if $\rho_{A,\ell}$ is
surjective for any prime $\ell|m$, provided that one of the conditions $(a)$
or $(b)$ in Theorem 1 holds.

Another paper by Jones (see [7]) focused on the required axioms, under which
$\mathrm{H}=\mathrm{G}$ can be obtained from certain local conditions, where
$\mathrm{H}\subseteq\mathrm{G}\subseteq\mathrm{GL_{n}}(\widehat{\mathbb{Z}})$
be algebraic groups. Our article primarily considers the vertical version of
such a result. In [2] Cojocaru first raised and answered such a question with
$\mathrm{G}=\text{GL}_{2}(\mathbb{Z}/n\mathbb{Z})$ and
$\mathrm{H}=\text{im}(\rho_{E})$ for an elliptic curve $E$ (better to say
without complex multiplication) over $\mathbb{Q}.$ We first generalize the
same over number fields. Jones applied his main result to the cases of the
pair of elliptic curves and polarized abelian varieties. Taking inspiration
from the same, we discuss a further generalization in section 4 for a general
class of algebraic groups. More precisely, our last section is about a
discussion around the following question. First of all, let us denote
$\pi_{m,\ell^{r}}$ to be the natural projection from $\mathbb{Z}/m\mathbb{Z}$
to $\mathbb{Z}/\ell^{r}\mathbb{Z},$ when $\ell^{r}\mid m.$ In the similar
scenario, we denote $\pi_{m/\ell^{r}}$ to be the natural projection from
$\mathbb{Z}/m\mathbb{Z}$ to $\mathbb{Z}/\frac{m}{\ell^{r}}\mathbb{Z}.$ For any
integer $m$ and prime $\ell$ we also denote
$\mathrm{pr}_{\ell}=\pi_{\ell^{r},\ell}\circ\text{ker}\circ\pi_{m/\ell^{r}},$
where $\ell^{r}||m.$

######  Question 1.

Let $A$ be an alegebraic group, and $m$ be an integer. Take $G$ be a subgroup
of $A(\mathbb{Z}/m\mathbb{Z}).$ Then is it true that

$\mathrm{G}=A(\mathbb{Z}/m\mathbb{Z})~{}\mathrm{if~{}and~{}only~{}if}~{}\mathrm{pr}_{\ell}(G)=A(\mathbb{Z}/\ell\mathbb{Z})~{}\mathrm{for~{}any~{}prime}~{}\ell\mid
m?$

We already have some examples of $A$ and $G$ for which we have a positive
answer to the question for any integer $m$ co-prime to $30.$ For example, it
follows from Theorem 4 that we can take $A=\mathrm{GSp}_{2g}$ and $G$ to be an
image of the Galois representation associated with some polarized abelian
variety,

$\rho_{A,m}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\mathrm{GSp}_{2g}(\mathbb{Z}/m\mathbb{Z}).$

In Section 4 we shall discuss this problem for a larger family.

###  1.1. Overview of the article

The article is organized as follows. In Section 2 we discuss the main results
from finite groups. Nothing original is guaranteed in this section. Rather, it
gives the reader a quick introduction to the subject and provides a notational
setup for the article. In Section 3, we first prove one of our main results
Theorem 1. One of the crucial results needed to prove this result is Lemma 7.

Moreover, we also see how to extend these ideas to study more general cases
involving pairs of elliptic curves and abelian varieties. We record these
results in Theorem 3 and Theorem 4. In this process, Lemma 16 plays the
important role, which generalizes Lemma 7. Finally, in Section 4 we quickly
discuss the possible generalizations of the results from previous sections,
including the modular analogs.

###  1.2. Notations

We denote $\mathbb{R}$ to be the set of all real numbers, $\mathbb{Q}$ to be
the of all rational numbers, $\mathbb{N}$ to be the of all natural numbers and
$\mathbb{Z}$ to be the of all integers.

Let $f,g$ be two functions supported on a subset of $\mathbb{R}.$ We write
$f\ll g$ for $|f|\leq c|g|$ where $c$ is a constant irrespective of the
domains of $f$ and $g$, often $f=O(g)$ is written to denote the same. In this
article, such a constant is absolute, unless otherwise specified. Moreover, we
write $f=o(g)$ to say that $\lim_{x\to\infty}\frac{|f(x)|}{|g(x)|}=0.$

For any integer $m,$ we denote $\phi(m)$ to be the Euler’s totient function,
and $\omega(n)$ be the number of distinct prime factors of $m.$

##  2\. Background on finite groups

This section discusses the main group theoretic tools required to prove the
main results of this article. Although there may be many resources available
on this, it serves the reader a quick introduction to the subject.

To begin with, let $\mathrm{G}$ be a finite group. Denote
$\text{Occ}(\mathrm{G})$ to be the isomorphism classes of non-abelian simple
groups coming as the quotient of composite factors of $\mathrm{G}$. First, we
have the following immediate facts.

######  Lemma 5.

For any finite group $\mathrm{G}$,

  1. (1)

$\mathrm{Occ}(\mathrm{G})=\phi$ if and only if $\mathrm{G}$ is solvable.

  2. (2)

If $\mathrm{H}$ is normal in $G$, then
$\mathrm{Occ}(\mathrm{G})=\mathrm{Occ}(\mathrm{H})\bigcup\mathrm{Occ}(\mathrm{G}/\mathrm{H})$

As a consequence, we have

######  Lemma 6.

For any integer $m,$

$\mathrm{Occ}(\mathrm{GL}_{2}(\mathbb{Z}/m\mathbb{Z}))=\bigcup_{\begin{subarray}{c}\ell,\mathrm{prime}\\\
\ell\mid
m\end{subarray}}\mathrm{Occ}(\mathrm{PSL}_{2}(\mathbb{Z}/\ell\mathbb{Z})).$

###  2.1. A criteria for containing the simple linear groups of degree $2$

Let us start by recalling the crucial results from [2]. The following lemma is
originally due to Serre ([10]).

######  Lemma 7.

Let $\ell$ be a prime and $\mathrm{H}$ be any subgroup subgroup of
$\mathrm{SL}_{2}(\mathbb{Z}/\ell^{r}\mathbb{Z}),$ then we have the following.

$\mathrm{pr}_{\ell}(\mathrm{H})=\mathrm{SL}_{2}(\mathbb{Z}/\ell\mathbb{Z})\implies\mathrm{H}=\mathrm{SL}_{2}(\mathbb{Z}/\ell^{r}\mathbb{Z}),$

where $\text{pr}_{\ell}$ denotes the canonical projection of
$\mathrm{SL}_{2}(\mathbb{Z}/\ell^{r}\mathbb{Z})\to\mathrm{SL}_{2}(\mathbb{Z}/\ell\mathbb{Z}).$

Later, we shall prove this generalization in Section 3. Now the following
result helps in computing the Occ of the finite groups.

######  Corollary 8.

Let $\ell>5$ be any prime, and $\mathrm{G}$ be any arbitrary subgroup of
$\mathrm{GL}_{2}(\mathbb{Z}/\ell^{r}\mathbb{Z})$. Then the following holds.

$\mathrm{PSL}_{2}(\mathbb{Z}/\ell\mathbb{Z})\in\mathrm{Occ}(\mathrm{G})~{}\text{if
and only
if}~{}\mathrm{SL}_{2}(\mathbb{Z}/\ell^{r}\mathbb{Z})\subseteq\mathrm{G}.$

Combining all these results, we have the following.

######  Corollary 9.

Let $m$ be any integer co-prime to $30,$ and $G$ be any subgroup of
$\mathrm{GL}_{2}(\mathbb{Z}/m\mathbb{Z}).$ Then the following holds.

$\mathrm{SL}_{2}(\mathbb{Z}/m\mathbb{Z})\subseteq\mathrm{G}~{}\mathrm{if~{}and~{}only~{}if}~{}\mathrm{PSL}_{2}(\mathbb{Z}/\ell\mathbb{Z})\in\mathrm{Occ}(\mathrm{G}),$

for every prime $\ell|m.$

###  2.2. Generalizations to the Symplectic groups

Let us start by recalling the standard definition of Symplectic groups.
Consider the groups,

$\mathrm{GSp}_{2g}(\mathbb{Z}/m\mathbb{Z})=\Big{\\{}\mathrm{g}\in\mathrm{GL}_{2g}(\mathbb{Z}/m\mathbb{Z})\mid\mathrm{g}\begin{pmatrix}0&I\\\
-I&0\\\ \end{pmatrix}\mathrm{g}^{t}=\begin{pmatrix}0&\lambda\\\ -\lambda&0\\\
\end{pmatrix},\lambda\in(\mathbb{Z}/m\mathbb{Z})^{*}\Big{\\}}$

and

$\mathrm{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z})=\Big{\\{}\mathrm{g}\in\mathrm{GL}_{2g}(\mathbb{Z}/m\mathbb{Z})\mid\mathrm{g}\begin{pmatrix}0&I\\\
-I&0\\\ \end{pmatrix}\mathrm{g}^{t}=\begin{pmatrix}0&I\\\ -I&0\\\
\end{pmatrix}\Big{\\}}.$

We then have the following fact when we want to compute Occ.

######  Lemma 10.

For any integer $m$, we have the following.

$\mathrm{Occ}\Big{(}\mathrm{GSp}_{2g}(\mathbb{Z}/m\mathbb{Z})\Big{)}=\bigcup_{\begin{subarray}{c}\ell~{}\mathrm{prime}\\\
\ell\mid
m\end{subarray}}\mathrm{Occ}\Big{(}\mathrm{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})\Big{)}.$

###### Proof.

Since
$\text{GSp}_{2g}(\mathbb{Z}/m\mathbb{Z})/\text{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z})$
is abelian,

$\text{Occ}\Big{(}\mathrm{GSp}_{2g}(\mathbb{Z}/m\mathbb{Z})\Big{)}=\text{Occ}\Big{(}\mathrm{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z})\Big{)}=\bigcup_{\ell^{r}||m}\text{Occ}\Big{(}\mathrm{Sp}_{2g}(\mathbb{Z}/\ell^{r}\mathbb{Z})\Big{)}.$

The last equality above follows from $(2)$ of Lemma 5. On the other hand,

$\mathrm{Occ}\Big{(}\mathrm{Sp}_{2g}(\mathbb{Z}/\ell^{r}\mathbb{Z})\Big{)}=\mathrm{Occ}\Big{(}\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})\Big{)}=\mathrm{Occ}\Big{(}\mathrm{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})\Big{)}$

since kernel of the projection map
$\mathrm{Sp}_{2g}(\mathbb{Z}/\ell^{r}\mathbb{Z})\rightarrow\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}),$
is a $\ell$-group, hence solvable, and Occ of the solvable part is empty. ∎

As an immediate consequence

######  Corollary 11.

For any prime $\ell>5$ not dividing $m,$

$\mathrm{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})\not\in\mathrm{Occ}\Big{(}\mathrm{GSp}_{2g}(\mathbb{Z}/m\mathbb{Z})\Big{)}.$

###### Proof.

It follows immediately from Lemma 10 and Lemma 6.1 of [7]. Basically, Sylow’s
theorem shows that
$\mathrm{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})\not\in\mathrm{Occ}\Big{(}\text{PSp}_{2g}(\mathbb{Z}/\ell^{\prime}\mathbb{Z})\Big{)},$
for any primes $\ell\neq\ell^{\prime}.$ ∎

Throughout the article, let us denote $\mathrm{comm}(G)$ to be the commutator
of $G$ for any finite group $G.$ We then have,

######  Lemma 12.

For any $m\in\mathbb{N}$ and $g>1,$

$\mathrm{comm}(\mathrm{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z}))=\mathrm{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z}).$

###### Proof.

Let us first note that
$\mathrm{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z})=\prod_{\ell^{r}||m}\mathrm{Sp}_{2g}(\mathbb{Z}/\ell^{r}\mathbb{Z}).$
It is enough to prove the statement when $m$ is the power of a prime. After
using Lemma 16, which we shall prove in the next section, it is enough to show
that
$\mathrm{comm}(\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}))=\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}).$
Since $\mathrm{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$ is simple and non-
abelian, for any prime $\ell$ and $g>1$, we have that
$\mathrm{comm}(\mathrm{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}))=\mathrm{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}).$
In particular, we have

$\mathrm{comm}(\pm\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}))=\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}).$

We now need to show if $\mathrm{H}$ is a subgroup of
$\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$ such that $\pm\mathrm{H}$ is the
full group, then $\mathrm{H}$ itself is the full group
$\text{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$. One can see that the symplectic
form $J=\begin{pmatrix}&I\\\ -I&\\\ \end{pmatrix}$ is in
$\text{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$. In particular, at least one of
$J$ and $-J$ is in $\mathrm{H},$ because $\pm
H=\text{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}).$ In either cases, we have
$J^{2}=-I\in H.$ This concludes the proof, since $\pm
H=H=\text{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}).$ ∎

######  Lemma 13.

Let $A\in\mathrm{M}_{n}(\mathbb{Z}/\ell\mathbb{Z})$ be an arbitrary matrix
with $\mathrm{tr}(A)=0$. Then $A$ is the sum of four square-zero matrices.

###### Proof.

See Theorem 1.1 in [3]. ∎

##  3\. Proof of the main statements

In this section, we shall prove the main results of this section. To begin
with, let $E$ be an elliptic curve over any arbitrary number field $K$.

###  3.1. Proof of Theorem 1

###  3.2. Proof of part (a)

Let $m$ be an integer co-prime to $30.$ If $\rho_{E,m}$ is surjective, then
$\rho_{E,\ell}=\text{pr}_{m,\ell}\circ\rho_{E,m}$ is surjective for any
$\ell\mid m,$ where

$\mathrm{pr}_{m,\ell}:\mathrm{GL}_{2}(\mathbb{Z}/m\mathbb{Z})\longrightarrow\mathrm{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z}),$

is the natural projection.

For the converse, it follows from the given hypothesis that
$\text{im}(\rho_{E,\ell})=\text{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ is a
quotient of $\text{im}(\rho_{E,m})$ for any prime $\ell|m$. In particular,

$\text{PSL}_{2}(\mathbb{Z}/\ell\mathbb{Z})\in\text{Occ}(\mathrm{G})$

for any prime $\ell|m$, where $\mathrm{G}=\text{im}(\rho_{E,m}).$ It follows
from Corollary 9 that, $\text{SL}_{2}(\mathbb{Z}/m\mathbb{Z})$ is contained in
$\mathrm{G}$. We then have that
$\text{SL}_{2}(\mathbb{Z}/m\mathbb{Z})=\mathrm{G}^{\prime},$ and in particular
$[G:G^{\prime}]\mid\phi(m).$ On the other hand, the Weil pairing gives
$[K(\zeta_{m}):K]\mid[G:G^{\prime}].$ It is now enough to ensure that
$[K(\zeta_{m}):K]=\phi(m)$. Note that,
$[K(\zeta_{m}):K]=[\mathbb{Q}(\zeta_{m}):K\cap\mathbb{Q}(\zeta_{m})],$ and
hence it is enough to ensure that $K\cap\mathbb{Q}(\zeta_{m})=\mathbb{Q}.$ We
shall see the imposed conditions on $m$ or $K$ gives us that privilege.111This
is not true in general. For instance, one may consider
$K=\mathbb{Q}(\sqrt{-15})$ and then we have $[K(\zeta_{15}):K]=4\neq\phi(15)$.

First of all since $K\cap\mathbb{Q}(\zeta_{m})$ is an abelian extension of
$\mathbb{Q}$ contained in $K,$ the first imposed condition on $K$ forces the
intersection to be trivial. On the other hand, since
$K\cap\mathbb{Q}(\zeta_{m})$ is an extension of $\mathbb{Q}$ contained in both
$K$ and $\mathbb{Q}(\zeta_{m}),$ it is evident that the condition
$(\phi(m),D_{K})=1$ implies $K\cap\mathbb{Q}(\zeta_{m})=\mathbb{Q}.$ Moreover,
the assumption $(m,d_{K})=1$ immediately implies that the discriminant of
$K\cap\mathbb{Q}(\zeta_{m})$ is only $1.$ In particular, in all the cases we
have $K\cap\mathbb{Q}(\zeta_{m})=\mathbb{Q},$ as desired.

###  3.3. Proof of part (b)

Take any integer $m$ that is not square-free. Write
$m=\prod_{i=1}^{d}\ell_{i}^{e_{i}}$ and without loss of generality let us
assume that $e_{1}>1.$ Consider $F$ to be an extension of $\mathbb{Q}$
contained in $\mathbb{Q}(\zeta_{\ell_{1}^{e_{1}}})$ of degree $\ell_{1}.$ We
can do that, because $e_{1}>1$ by the assumption. It is evident that,
$F\cap\mathbb{Q}(\zeta_{\ell_{1}})=\mathbb{Q}.$ Now for any $i>1,$ we have
$\mathbb{Q}(\zeta_{\ell_{i}})\cap\mathbb{Q}(\zeta_{\ell_{1}^{e_{1}}})=\mathbb{Q},$
and in particular, we have
$F\cap\mathbb{Q}(\zeta_{\ell_{1}})=\mathbb{Q},~{}\forall 1\leq i\leq d.$
Hence, for any $1\leq i\leq d$ we have

$[F(\zeta_{\ell_{i}}):F]=[\mathbb{Q}(\zeta_{\ell_{i}}):F\cap\mathbb{Q}(\zeta_{\ell_{i}})]=\ell_{i}-1,$

where the last implication is true because
$F\cap\mathbb{Q}(\zeta_{\ell_{i}})=\mathbb{Q}.$ Let us now denote $K_{m}=F,$
and show that the pair $(m,K_{m})$ satisfies all the necessary conditions for
$m$ to be a potentially bad number. First, we need to show that there exists
at least one elliptic curve $E$ over $K_{m},$ for which

$\text{im}(\rho_{E,m})\neq\text{GL}_{2}(\mathbb{Z}/m\mathbb{Z})~{}\mathrm{but}~{}\text{im}(\rho_{E,\ell_{i}})=\text{GL}_{2}(\mathbb{Z}/\ell_{i}\mathbb{Z}),~{}\forall~{}1\leq
i\leq d.$

We know from [11] that, there exists at least one elliptic curve $E/K_{m}$ for
which
$\text{im}(\rho_{E,\ell_{i}})\supset\text{SL}_{2}(\mathbb{Z}/\ell_{i}\mathbb{Z}),~{}\forall~{}1\leq
i\leq d.$ In fact, this holds for almost all elliptic curves over $K_{m}$.
From the construction, we know that
$[K_{m}(\zeta_{\ell_{i}}):K_{m}]=\ell_{i}-1,~{}\forall 1\leq i\leq d.$ Now it
follows from the argument of part (a) that, i.e., due to the Weil pairing
that,
$\text{im}(\rho_{E,\ell_{i}})=\text{GL}_{2}(\mathbb{Z}/\ell_{i}\mathbb{Z}),~{}\forall~{}1\leq
i\leq d.$

On the other hand, for any elliptic curve $E/K_{m},$ if the image of
$\rho_{E,m}$ is $\text{GL}_{2}(\mathbb{Z}/m\mathbb{Z}),$ then we must have
that $|K_{m}(\zeta_{m}):K_{m}|=\phi(m).$ This is because, it follows from
Weil-pairing that $\zeta_{m}\in K_{m}(E[m])$ and
$\sigma(\zeta_{m})=\zeta_{m}^{\det(\rho_{E,m}(\sigma))},$ where $\zeta_{m}$ is
the primitive $m^{\text{th}}$ root of unity. In particular, the fixed field of
$\text{SL}_{2}(\mathbb{Z}/m\mathbb{Z})$ correspond to $K(\zeta_{m}).$ This
shows that $[K_{m}(\zeta_{m}):K_{m}]=\phi(m).$ Instead, we have

$[K_{m}(\zeta_{m}):K_{m}]=[\mathbb{Q}(\zeta_{m}):K_{m}\cap\mathbb{Q}(\zeta_{m})]\leq[\mathbb{Q}(\zeta_{m}):K_{m}]<\phi(m),$

since $K_{m}$ is a non-trivial extension of $\mathbb{Q}$ contained in
$\mathbb{Q}(\zeta_{m}),$ a contradiction. ∎

######  Remark 1.

In part $(b)$ of Theorem 1, we assume that $m$ is square-free. Note that this
assumption is necessary. Otherwise, we immediately have the result because

$\text{GL}_{2}(\mathbb{Z}/\mathbb{Z})=\prod_{\begin{subarray}{c}\ell,~{}\mathrm{prime}\\\
\ell\mid m\end{subarray}}\text{GL}_{2}(\mathbb{Z}/\mathbb{Z}).$

Let us now discuss some interesting consequences of Theorem 1. If one wants to
make Proposition 5.7 in [11] effective, one can see the effective constant is
given by $\frac{m^{3}}{\phi(m)}.$ First let us recall the definition of the
set $B_{K,m}(x)$ from [11].

######  Corollary 14.

For any $m\in\mathbb{N}$ with $(m,30d_{K})=1,$ the effective constant is given
by

$\sum_{\ell\mid m}\frac{\ell^{3}}{\phi(\ell)}.$

###### Proof.

It follows from the proof of Proposition 5.7 in [11], that
$|B_{K,\ell}(x)|\leq\frac{\ell^{3}}{\phi(\ell)}\frac{\log
x}{x^{{\frac{[K:\mathbb{Q}]}{2}}}}.$ It is now enough to show that

$B_{K,m}(x)\subseteq\bigcup_{\ell\mid m}B_{K,\ell}(X).$

It follows from Theorem 1 that $\bigcap_{\ell\mid m}B_{K,\ell}(x)^{c}\supseteq
B_{K,m}(x)^{c},$ and this completes the proof. ∎

Theorem 1 motivates us to study many generalizations. The immediate ones that
come to our mind are Theorem 3 and 4, which we will discuss now.

###  3.4. Proof of Theorem 3

###  3.5. Proof of part (a)

One direction is obvious. For the other direction, by Theorem 1 we have that

$\text{im}(\rho_{E_{1},m})=\text{im}(\rho_{E_{2},m})=\text{GL}_{2}(\mathbb{Z}/m\mathbb{Z}).$

Let $G$ be $\text{im}(\rho_{E_{1}\times E_{2},m})\subseteq\Delta(m)$. It
follows from the given condition that $\Delta(\ell)$ is a quotient of $G$, for
any prime $\ell\mid m.$ Denote $S\Delta(\ell)$ to be the set of elements in
$\Delta(\ell)$ whose each block has determinant $1,$ and $G^{(\ell)}$ be
$\ker\circ\pi_{m/\ell^{r}}(G)\subseteq\Delta(\ell^{r})$, where $r$ is the
maximum power of $\ell$ dividing $m$, and $\pi_{m/\ell^{r}}$ be the natural
projection $\Delta(m)\to\Delta(m/\ell^{r}).$

Moreover, we consider
$G^{\prime}=\text{pr}_{\ell}(G^{(\ell)})\subseteq\Delta(\ell)$, and set

$G^{\prime}_{1}=\Big{\\{}g\in\text{GL}_{2}(\mathbb{F}_{\ell}):\begin{pmatrix}I&\\\
&g\\\ \end{pmatrix}\in
G^{\prime}\Big{\\}},~{}G^{\prime}_{2}=\Big{\\{}g\in\text{GL}_{2}(\mathbb{F}_{\ell}):\begin{pmatrix}g&\\\
&I\\\ \end{pmatrix}\in G^{\prime}\Big{\\}}.$

For the given condition, Since $\Delta(\ell)=G^{\prime},$ for every prime
$\ell\mid m.$ In particular, $\text{comm}(G^{\prime})=\text{S}\Delta(\ell).$
It now follows from Lemma 16, which we shall prove in the next section, that
$\text{S}\Delta(\ell^{r})\subseteq G^{(\ell)}.$

According to Lemma 3.3 in [6], $G\neq\Delta(\mathbb{Z}/m\mathbb{Z})$ implies
there exist a set $C_{1}\times
C_{2}\subseteq\text{GL}_{2}(\mathbb{Z}/m\mathbb{Z})\times\text{GL}_{2}(\mathbb{Z}/m\mathbb{Z})$,
closed under conjugation such that $\text{det}(C_{1})=\text{det}(C_{2})=1$
with $G\cap(C_{1}\times C_{2})=\phi$. This implies that, there exists at least
one element $(c_{1},c_{2})\not\in G$ for which $\det(c_{1})=\det(c_{2})=1.$ In
particular, there exists a prime $\ell\mid m$ such that the component of
$(c_{1},c_{2})$ associated to $\ell$ is not in $G^{(\ell)}$, which implies
$S\Delta(\ell^{r})\not\subseteq G^{(\ell)}.$ This contradicts the conclusion
of the previous paragraph. Let $E$ be an elliptic curve over an arbitrary
number field $K$, and consider the following invariant

$A(E)=30\prod_{\ell\in M_{E}}\ell,$

where $M_{E}$ is the set of primes $\ell\geq 7$ such that $\rho_{E,\ell}$ is
not surjective. Now for a pair of elliptic curves $E_{1}\times E_{2}$ over
$K$, let us consider

$A(E_{1}\times E_{2})=30\prod_{\ell\in M_{E_{1}\times E_{2}}}\ell,$

and $M_{E_{1}\times E_{2}}$ is the set of primes $\ell$ for which
$\text{im}(\rho_{E_{1}\times E_{2},\ell})\neq\Delta(\ell).$ It is clear that

$\text{lcm}(A(E_{1}),A(E_{2}))\mid A(E_{1}\times E_{2}).$

If they are not equal, then there exists a prime $\ell$ such that
$\text{im}(\rho_{E_{i},\ell})=\text{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ for
$i\in\\{1,2\\}$, and $\text{im}(\rho_{E_{1}\times
E_{2},\ell})\neq\Delta(\ell).$ Now by Lemma 5.1 of [8], $\rho_{E_{1},\ell}$
and $\rho_{E_{2},\ell}$ are conjugate up-to a quadratic character of
$\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. In fact, it follows from
Proposition 1 in [8] and Theorem 3 the following.

######  Corollary 15.

Let $K$ be any number field satisfying one the conditions in part (a) of
Theorem 1, and $E_{1},~{}E_{2}/K$ be two elliptic curves without complex
multiplication, which are not isogenous over $\overline{K}$. then we have the
following equality

$\mathrm{im}(\rho_{E_{1}\times
E_{2},m})=\mathrm{GL}_{2}(\mathbb{Z}/m\mathbb{Z}),$

for any integer $m$ co-prime to $A(E_{1}\times
E_{2})\ll\max\\{h_{1},h_{2}\\}^{O(1)},$ where $h_{1}$ and $h_{2}$ respectively
be the heights of $E_{1}$ and $E_{2}.$

###### Proof.

It follows from Theorem 3 that $\mathrm{im}(\rho_{E_{1}\times
E_{2},m})=\mathrm{GL}_{2}(\mathbb{Z}/m\mathbb{Z})$, for any integer $m$ co-
prime to $A(E_{1}\times E_{2}).$ For an upper bound on $A(E_{1}\times E_{2}),$
the reader may look at Proposition 1 in [8]. ∎

######  Remark 2.

Jones in [6] showed that almost all pairs of elliptic curves over
$\overline{\mathbb{Q}}$ are pairwise non-isogenous.

###  3.6. Proof of Theorem 4

We need the following generalization of Lemma 7.

######  Lemma 16.

Let $p$ be a prime, $r$ be any integr, and $\mathrm{G}$ be an algebraic
subgroup of $\mathrm{SL}_{n}(\mathbb{F}_{p^{r}}).$ Suppose that, $\mathrm{H}$
is a subgroup of $\mathrm{G}$ with
$\mathrm{pr}_{p}(\mathrm{H})=\mathrm{pr}_{p}(\mathrm{G}),$ then we have
$\mathrm{H}=\mathrm{G}.$ More generally, if
$\mathrm{H}\subseteq\mathrm{G}(\mathbb{Q}_{p}),$ with
$\mathrm{pr}_{p}(\mathrm{H})=\mathrm{pr}_{p}(\mathrm{G})$, then
$\mathrm{H}=\mathrm{G}(\mathbb{Q}_{p}).$

###### Proof.

Let us start with proving the first part because the second part is just a
rewording of the former. We proceed by induction, to show that for any $2\leq
k\leq r,$

$\text{pr}_{p^{k}}(\mathrm{H})=\text{pr}_{p^{k}}(\mathrm{G}).$

For each such $k$, consider the exact sequence

$1\longrightarrow\Big{\\{}A\in\text{G}(\mathbb{F}_{p^{k}}):A\text{ mod
}p^{k-1}\equiv
I\Big{\\}}\longrightarrow\text{G}(\mathbb{F}_{p^{k}})\longrightarrow\text{G}(\mathbb{F}_{p^{k-1}})\longrightarrow
1.$

Proceeding by induction, it is enough to show that

$\mathrm{pr}_{p^{k}}(\mathrm{H})\supseteq\Big{\\{}A\in\text{G}(\mathbb{F}_{p^{k}}):A\text{
mod }p^{k-1}\equiv I\Big{\\}}.$

Take an element $A=I+p^{k-1}U\in G(\mathbb{F}_{p^{k}})$ with
$U\in\text{M}_{n}(\mathbb{F}_{p})$, and we want to show
$A\in\mathrm{pr}_{p^{k}}(H).$ Note that $\text{det}(A)=1$ in
$\mathbb{F}_{p^{k}},$ implies $\text{tr}(U)=0$ in $\mathbb{F}_{p}$. By lemma
13, we can write $U$ as a sum of four square-zero matrices. It follows from
induction that there exists $h\in\mathrm{pr}_{p^{k}}(\mathrm{H})$ such that

$h=I+p^{k-2}U+p^{k-1}V$

for some $V\in M_{n}(\mathbb{F}_{p}),$ provided that
$I+p^{k-2}U\in\mathrm{G}(\mathbb{F}_{p^{k-1}})$. Since $\mathrm{G}$ is an
algebraic group, it is given by zeroes of some polynomials, say by
$f_{1},f_{2},\cdots,f_{t}$. The identity matrix $I_{n\times n}$ is an element
of $G$, in particular, we have

(1) $f_{i}\big{(}I_{n\times n}\big{)}=0,\hskip 2.84544pt\text{for all}\hskip
4.26773pt1\leq i\leq t.$

Since $I+p^{r-1}U\in\mathrm{G}(\mathbb{F}_{p^{r}})$, we have

$f_{s}\Big{(}\big{(}\delta_{ij}+p^{k-1}U_{ij}\big{)}_{1\leq i,j\leq
n^{2}}\Big{)}=0\text{ mod }p^{k},\text{ for all }1\leq s\leq t.$

Now any such $f_{s}$ is sum of the monomials of form $\gamma(\prod
g_{i,i})^{\alpha_{i}}(\prod_{i\neq j}g_{i,j}^{\beta_{i,j}}).$ Putting
$g_{i,j}=\delta_{i,j}+p^{k-1}U_{i,j}$ in each such monomial, and considering
modulo $p^{k},$ we get the following sum

(2) $\prod(1+\alpha_{i}U_{i,i}p^{k-1})(\prod_{i\neq
j}(p^{k-1}U_{i,j})^{\beta_{i,j}}.$

If all of the $\beta_{i,j}$’s are zero, then the above summation is congruent
to

$\prod(1+\alpha_{i}U_{i,i}p^{k-1})\equiv
1+p^{k-1}\sum\alpha_{i}U_{i,i}\pmod{p^{k}}$

provided that $k\geq 2.$ Now if some $\beta_{i,j}$ is more than $1$, the
equation at (2) is already congruent to $0$ modulo $p^{k},$ provided $k\geq
2.$ Same goes if two of the $\beta_{i,i}$’s are exactly $1$. Now, if only one
of the $\beta_{i,j}$ is $1$ and others are zero, the equation at (2) is
congruent to $p^{k-1}U_{i,j}$ modulo $p^{k}$ provided $k\geq 2.$ Combining all
these observations, one can see the equation at $(\ref{eqn:cond})$ is
congruent to

$f_{s}\big{(}I_{n\times n}\big{)}+p^{k-1}(\text{some term only
involving}~{}\alpha_{i},U_{i,j}^{\prime}s),\forall 1\leq s\leq t$

modulo $p^{k}$. A similar argument also shows

$f_{s}\Big{(}(\delta_{ij}+p^{k-2}U_{ij})_{1\leq i,j\leq n^{2}}\Big{)}=0\text{
mod }p^{k-1},~{}\forall~{}1\leq i\leq k,$

provided that $k\geq 3.$ So $I+p^{k-2}U\in\text{G}(\mathbb{F}_{p^{k-1}}),$ as
desired. By induction, we now have such a desired
$h\in\mathrm{pr}_{p^{k}}(H)$. And hence,

$A=h^{p}\in\mathrm{pr}_{p^{k}}(H),\forall~{}2\leq k\leq r.$

∎

######  Remark 3.

The lemma above is not necessarily true for algebraic groups not containing
$\text{SL}_{n}$. For a justification, two examples are addressed in the next
section. Roughly speaking, this is the case because $p\mid\alpha-1$ does not
imply $p^{r}\mid\alpha-1,$ while on the other hand, $p^{r}\not|~{}\alpha-1$
implies $p\not|~{}\alpha-1.$

We are now finished with all the preparations. Let us now prove the main
result about Abelian varieties.

###  3.7. Proof of the Theorem 4

###  3.8. Proof of part (a)

Again, one direction is immediate. For the other direction, let $\mathrm{G}$
be the image of $\rho_{A,m}$. It follows from the given condition that,
$\text{GSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$ is a quotient of $\mathrm{G}$ for
any prime $\ell\mid m$. In particular,
$\text{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})\in\text{Occ}(\mathrm{G}),\text{
for any prime }\ell|m,$ since $\text{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$ is
simple and non-abelian for any $g>1$. From lemma 10 and corollary 11, we have
that

$\mathrm{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})\in\text{Occ}(\text{pr}_{\ell}(G^{(\ell)})),~{}\mathrm{where}~{}\mathrm{G}^{(\ell)}=\text{ker}_{m/\ell^{r}}\mathrm{G}.$

For the case $g=1$ Cojocaru argued from this point by showing
$\text{pr}_{\ell}(\mathrm{G}^{(\ell)})$ contains
$\text{SL}_{2}(\mathbb{Z}/\ell\mathbb{Z}).$ Her argument was based on the fact
that a subgroup of $\text{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ not containing
$\text{SL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ is solvable. However, the same
argument does not work for the general cases. For example
$\text{Sp}_{2g-2}(\mathbb{Z}/\ell\mathbb{Z})$ is not a solvable subgroup of
$\text{GSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}).$ But on the other hand,
$\frac{\text{pr}_{\ell}(G^{(\ell)})}{\text{pr}_{\ell}(G^{(\ell)})\cap\text{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})}\subseteq\mathbb{F}_{\ell}^{*}$
is abelian. In particular, we have
$\text{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})\in\text{
Occ}(\text{pr}_{\ell}(G^{(\ell)})\cap\text{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}))$.
Comparing cardinality of them, we have
$\text{pr}_{\ell}(G^{(\ell)})\cap\text{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})=\text{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$,
and hence
$\text{pr}_{\ell}(G^{(\ell)})\supseteq\text{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}).$
It now follows from Lemma 12 that,
$\text{comm}(\text{pr}_{\ell}(G^{(\ell)}))=\text{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z}).$

On the other hand, $\text{Sp}_{2g}\subset\text{SL}_{2g}$ is an algebraic
group, and hence it follows from the previous lemma that

$\text{Sp}_{2g}(\mathbb{Z}/\ell^{r}\mathbb{Z})=\text{comm}(G^{(\ell)}).$

This shows that, $\text{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z})\subseteq
G\subseteq\text{GSp}_{2g}(\mathbb{Z}/m\mathbb{Z}).$ It again follows from
Lemma 12 that $\text{comm}(G)=\text{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z}).$

It is now enough to show that $\phi(m)\mid[G:G^{\prime}].$ Since $A$ is
equipped with a polarization, due to Weil pairing the primitive
$m^{\text{th}}$ root of unity $\zeta_{m}$ is in $K(A[m]),$ and hence,
$|\text{Gal}(K(A[m]):K))|$ divides $[G:G^{\prime}].$ Now the argument is now
essentially same as part (a) of Theorem 1, we only need to ensure that
$K(\zeta_{m})$ is an extension of $K$ of degree $\phi(m).$

###  3.9. Proof of part (b)

Let $m$ be a number co-prime to $30$ which is not squarefree. We can then
consider $S$ to be the set of prime factors of $m.$ By the assumption, there
exists s simply polarized abelian variety over $K$ of dimension $g$ for which
$\text{im}(\rho_{A,\ell})\supseteq\text{Sp}_{2g}(\mathbb{Z}/\ell),\forall\ell\in
S.$ We can now consider the same $K_{m}$ as in the part (a) of Theorem 1 to
ensure that $m$ is indeed a potential bad number in this settings. ∎

##  4\. Further remarks

In this section, we shall discuss the analogous situations for modular forms
and other algebraic groups. More precisely, we shall discuss the local-global
property for the relevant Galois representations.

###  4.1. The modular analog

Let $f(z)$ be any newform of weight $k$ and level $N$. It is known due to
Deligne-Serre correspondence that, for any integer $m$ we have an associated
Galois representation

$\rho_{f,m}:\text{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\longrightarrow\mathrm{GL}_{2}\left(\mathbb{Z}/m\mathbb{Z}\right),$

such that
$a(p)\pmod{m}\equiv\text{tr}\left(\rho_{f,m}(\text{Frob}_{p})\right)$ for any
prime $p\nmid Nm.$ When $f$ is without CM, it follows from [9] that, there
exists an integer $M_{f}$ such that for any integer $m$ co-prime to $M_{f},$
the image of this representation is given by

$\Delta_{k,m}=\left\\{A\in\text{GL}_{2}\left(\mathbb{Z}/m\mathbb{Z}\right)\mid\det(A)\in((\mathbb{Z}/m\mathbb{Z})^{*})^{k-1}\right\\}.$

Following the proof of Theorem 1, one could see that for any integer $m$ co-
prime to $30,$ to guarantee $\text{im}(\rho_{f,m})$ contains
$\text{SL}_{2}(\mathbb{Z}/m\mathbb{Z})$ if and only if,
$\text{im}(\rho_{f,\ell})$ contains $\text{SL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$
for any prime $\ell\mid m.$ In particular, one could perhaps get a smaller
$M_{f}$, as long as we want the image to contain only
$\text{SL}_{2}(\mathbb{Z}/m\mathbb{Z}).$ In this direction, we ask the
following.

######  Question 2.

Is it true that

$\mathrm{im}(\rho_{f,m})=\Delta_{k,m}~{}\mathrm{if~{}and~{}only~{}if},~{}\mathrm{im}(\rho_{f,\ell})=\Delta_{k,\ell},$

for any prime $\ell\mid m$?

It is not hard to notice that the answer to this question is yes, provided
that $\zeta_{m}^{(k-1,\phi(m))}$ is in the field corresponding to
$\text{ker}(\rho_{f,m}).$

Moreover, any cuspform $f(z)$ can be uniquely written as
$c_{1}f_{1}+c_{2}f_{2}+\cdots c_{r}f_{r},$ where
$c_{1},c_{2},\cdots,c_{r}\in\overline{\mathbb{Q}}.$ One can attach a Galois
representation
$\rho_{f,m}:\mathrm{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\to\mathrm{GL}_{2r}\left(\mathbb{Z}/m\mathbb{Z}\right)$
defined by the map

(3) $\sigma\mapsto\begin{pmatrix}\rho_{f_{1,\ell}}(\sigma)&&\\\
&\rho_{f_{2,\ell}}(\sigma)&&\\\ &&\ddots&\\\ &&&\rho_{f_{r,\ell}}(\sigma)\\\
\end{pmatrix}.$

In this case, the image is contained in
$\Delta_{k_{1},k_{2},\cdots,k_{r}}(m),$ where
$\Delta_{k_{1},k_{2},\cdots,k_{r}}(m)$ denotes the set of all block matrices
of size $2\times 2$ in $\mathrm{GL}_{2}\left(\mathbb{Z}/m\mathbb{Z}\right)$ in
which determinant of each block is a $k_{i}-1^{th}$ power of some element in
the multiplicative group $(\mathbb{\mathbb{Z}}/{m\mathbb{Z}})^{*}.$ Arguing
similarly as in the proof of Theorem 3 one can see that,
$\text{im}(\rho_{f,m})$ contains $\text{SL}_{2}(\mathbb{Z}/m\mathbb{Z})^{r}$
if and only if, $\text{im}(\rho_{f,\ell})$ contains
$\text{SL}_{2}(\mathbb{Z}/\ell\mathbb{Z})^{r}$ for any prime $\ell\mid m.$ One
can show that when $f_{1},f_{2},\cdots,f_{r}$ are not related by pairwise
Dirichlet characters, $\text{im}(\rho_{f,m})$ contains
$\text{SL}_{2}(\mathbb{Z}/m\mathbb{Z})^{r}$ for all but finitely many primes
$\ell.$ Horeover, we again as this stronger question.

######  Question 3.

Is it true that

$\mathrm{im}(\rho_{f,m})=\Delta_{k_{1},k_{2},\cdots,k_{r},m}~{}\mathrm{if~{}and~{}only~{}if},~{}\mathrm{im}(\rho_{f,\ell})=\Delta_{k_{1},k_{2},\cdots,k_{r},\ell},$

for any prime $\ell\mid m$?

###  4.2. On the local-global phenomenon

Motivating from the previous section let us discuss the most general situation
in this direction. Let $A$ be an algebraic subgroup of $\mathrm{GL}_{n}$ and
$\mathrm{G}$ be an arbitrary subgroup of $A(\mathbb{Z}/m\mathbb{Z})$ such that
$\mathrm{pr}_{\ell}(G)=A(\mathbb{Z}/\ell\mathbb{Z})~{}\mathrm{for~{}any~{}prime}~{}\ell$
dividing $m.$ Then what can be said about the relation between $G$ and $A$? To
be more precise, let us now recall Question 1 that we raised in the
introduction. In the previous section we already had an affirmative answer
when $A=\mathrm{GSp}_{2g}$ and $G$ is image of the mod-$\ell$ Galois
representation associated to some polarized abelian variety, and $m$ is an
integer satisfying certain coprimality conditions. However this is not true in
general. Here are a few examples to justify that.

###  4.3. Examples

#### 4.3.1.

Let $A=\mathrm{GL}_{2}$ and $\mathrm{G}$ be a subgroup of
$\mathrm{GL}_{2}(\mathbb{Z}/\ell^{2}\mathbb{Z})$ defined by

$\mathrm{G}=\Big{\\{}\begin{pmatrix}a&b\\\ c&d\\\ \end{pmatrix}\mid
a,b,c,d\in\mathbb{Z}/\ell\mathbb{Z},ad-bc\neq 0\Big{\\}}.$

It is clear that $\mathrm{G}$ is a subgroup of
$\mathrm{GL}_{2}(\mathbb{Z}/\ell^{2}\mathbb{Z})$ with

$\text{pr}_{\ell}(G)=\mathrm{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z}),\hskip
5.69046pt~{}\text{while}~{}\mathrm{G}\neq\mathrm{GL}_{2}(\mathbb{Z}/\ell^{2}\mathbb{Z}).$

In the similar vein, we can also consider $A=\mathrm{GL}_{n}$ and $G$ be a
subgroup of $A(\mathbb{Z}/\ell^{2}\mathbb{Z})$ given by a copy of
$\mathrm{GL}_{n}(\mathbb{Z}/\ell\mathbb{Z})$. For a more non trivial example,
one may consider $A$ to be the set of Borel elements in $\mathrm{GL}_{2}$ and
$G$ to be the set of all Borel elements over $\mathbb{Z}/\ell\mathbb{Z}.$

###  4.4. Positive answer in general terms

We first give an affirmative answer to Question 1 under the following
conditions.

  1. (1)

$\mathrm{comm}(A(\mathbb{Z}/\ell\mathbb{Z}))\subseteq\mathrm{pr}_{\ell}(G).$

  2. (2)

$|\mathrm{comm}(G)|=\frac{|A(\mathbb{Z}/m\mathbb{Z})|}{|\mathrm{comm}(A(\mathbb{Z}/m\mathbb{Z}))|}.$

Since $G$ is a subgroup of $A(\mathbb{Z}/m\mathbb{Z}),$ it follows immediately
from $(1)$ that

$\text{pr}_{\ell}(\mathrm{comm}(G^{(\ell)}))=\mathrm{comm}(A(\mathbb{Z}/\ell\mathbb{Z})),$

where $G^{(\ell)}=\text{ker}\circ\pi_{m/\ell^{r}}(G).$ It now follows from
Lemma 16 that
$\mathrm{comm}(G^{(\ell)})=\mathrm{comm}(A(\mathbb{Z}/\ell^{r}\mathbb{Z}),$
because $\mathrm{comm}(A(\mathbb{Z}/m\mathbb{Z}))$ is contained in
$\text{SL}_{n}(\mathbb{Z}/m\mathbb{Z}).$ In particular, we have

$\mathrm{A(\mathbb{Z}/m\mathbb{Z})}\subseteq\mathrm{G}\subseteq
A(\mathbb{Z}/m\mathbb{Z}).$

Condition $(1)$ and $(2)$ together now shows
$\mathrm{G}=A(\mathbb{Z}/m\mathbb{Z}).$

Note that we are putting an extra local condition at $(2)$, and that is not
exactly our question was originally about. From the arguments presented in the
previous section, we see that $(2)$ is actually true for the cases related to
abelian varieties. In general we do not necessarily have that luxury because,
once we have $A(\mathbb{Z}/\ell\mathbb{Z})$ is a quotient of $\mathrm{G},$ we
perhaps will not get a simple non-abelian quotient of
$A(\mathbb{Z}/\ell\mathbb{Z})$. So to avoid $(2),$ we might try to put the
condition that

(4) $\mathrm{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z})\subseteq
A(\mathbb{Z}/m\mathbb{Z}),$

for some $g\leq n/2,$ say. This condition implies that
$\text{PSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})\in\text{Occ}\Big{(}G^{(\ell)}\Big{)}.$
Now arguing similarly as the proof of Theorem 1.3, we have

$\text{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z})\subseteq\mathrm{G}\subseteq
A(\mathbb{Z}/m\mathbb{Z}).$

In particular,

$|\mathrm{G}/\mathrm{comm}(\mathrm{G})|\leq|\text{Sp}_{2g}(\mathbb{Z}/m\mathbb{Z})/|\text{Sp}_{2k}(\mathbb{Z}/m\mathbb{Z})|=O(m^{6(g-k)}\phi(m)).$

Summarizing the discussion, we see that we can remove $(2)$ at the cost of two
more conditions. One of them is $(\ref{eqn:cond1}),$ and the other one is if
we assume that $|\mathrm{G}/\mathrm{comm}(\mathrm{G})|$ is sufficiently larger
than $m^{6(g-k)}\phi(m).$

## Acknowledgement

I first thank the Mathematics Institute of Georg-August-Universität Göttingen
for providing a beautiful environment for study and research. I am also
indebted to my supervisor, Harald Helfgott, and Jitendra Bajpai, for many
helpful discussions. I always found them kind and helpful in all my needs.
This work is financially supported by ERC Consolidator grant 648329 (GRANT).

## References

  * [1] Francesco Campagna, _Cyclic reduction of elliptic curves_ , ALGANT Master Thesis (2018).
  * [2] Alina Carmen Cojocaru, _On the surjectivity of the Galois representations associated to non-CM elliptic curves_ , Canad. Math. Bull. 48 (2005), no. 1, 16–31, With an appendix by Ernst Kani. MR 2118760
  * [3] Clément de Seguins Pazzis, _A note on sums of three square-zero matrices_ , Linear Multilinear Algebra 65 (2017), no. 4, 787–805. MR 3606833
  * [4] David Grant, _A formula for the number of elliptic curves with exceptional primes_ , Compositio Math. 122 (2000), no. 2, 151–164. MR 1775416
  * [5] Chris Hall, _An open-image theorem for a general class of abelian varieties_ , Bull. Lond. Math. Soc. 43 (2011), no. 4, 703–711, With an appendix by Emmanuel Kowalski. MR 2820155
  * [6] Nathan Jones, _Pairs of elliptic curves with maximal Galois representations_ , J. Number Theory 133 (2013), no. 10, 3381–3393. MR 3071819
  * [7] by same author, _${\rm GL}_{2}$ -representations with maximal image_, Math. Res. Lett. 22 (2015), no. 3, 803–839. MR 3350106
  * [8] D. W. Masser and G. Wüstholz, _Galois properties of division fields of elliptic curves_ , Bull. London Math. Soc. 25 (1993), no. 3, 247–254. MR 1209248
  * [9] Kenneth A. Ribet, _On $l$-adic representations attached to modular forms. II_, Glasgow Math. J. 27 (1985), 185–194. MR 819838
  * [10] Jean-Pierre Serre, _Abelian $l$-adic representations and elliptic curves_, W. A. Benjamin, Inc., New York-Amsterdam, 1968, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
  * [11] David Zywina, _Elliptic curves with maximal Galois action on their torsion points_ , Bull. Lond. Math. Soc. 42 (2010), no. 5, 811–826. MR 2721742
  * [12] by same author, _On the surjectivity of mod $\ell$ representations associated to elliptic curves_, arXiv (2015).