# On the torsion locus of the Ceresa normal function

Matt Kerr  Department of Mathematics, Washington University in St. Louis, 1
Brookings Drive, St. Louis, MO 63130, USA<EMAIL_ADDRESS>and  Salim Tayou
Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, MA
02138, USA<EMAIL_ADDRESS>
###### Abstract.

We prove that the positive-dimensional part of the torsion locus of the Ceresa
normal function in $\mathcal{M}_{g}$ is not Zariski dense when $g\geq 3$.
Moreover, it has only finitely many components with generic Mumford-Tate group
equal to $\mathrm{GSp}_{2g}$; these components are defined over
$\overline{\mathbb{Q}}$, and their union is closed under the action of
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. More generally, we study the
distribution of the torsion locus of arbitrary admissible normal functions.

##  1\. Introduction

Let $g\geq 1$ and let $C$ be a smooth complex projective curve of genus $g$. A
point $x\in C(\mathbb{C})$ defines the Abel-Jacobi embedding $C\hookrightarrow
J(C)$, where $J(C)$ is the Jacobian of $C$. Let $\iota\colon J(C)\rightarrow
J(C)$ be the involution $y\mapsto-y$. Then the cycle
$Z_{C}:=[C]-[\iota(C)]\in\mathrm{CH}^{g-1}(J(C))$ is a homologically trivial
cycle on $J(C)$, and has a well defined image in $\mathrm{Griff}_{1}(J(C))$,
the group of homologically trivial cycles modulo the group of algebraically
trivial cycles (of dimension one). A famous theorem of Ceresa [Cer83] states
that for $g\geq 3$, a very general smooth projective curve $C$ of genus $g$
has non-torsion Ceresa cycle in $\mathrm{Griff}_{1}(J(C))$.

One can also consider the Abel-Jacobi class of $Z_{C}$ in the intermediate
Jacobian associated to the primitive cohomology
$V_{C}:=H^{2g-3}_{\text{pr}}(J(C),\mathbb{Z}(g-1))$. Doing this over all of
${\mathcal{M}}_{g}$ produces an admissible normal function
$\nu\in\mathrm{ANF}_{{\mathcal{M}}_{g}}({\mathcal{V}})$, where ${\mathcal{V}}$
is the VHS over ${\mathcal{M}}_{g}$ with fiber $V_{C}$ over
$[C]\in{\mathcal{M}}_{g}$. Informally, this is the graph of
$[C]\mapsto\mathrm{AJ}_{J(C)}(Z_{C})$. Ceresa’s proof shows that this normal
function is nontorsion.

In this note, we prove the following theorem (see §4.1).

######  Theorem 1.1.

Let $g\geq 3$ and let $H_{g}$ denote the hyperelliptic locus in
$\mathcal{M}_{g}$. There exists a finite set of subvarieties $T_{i}$ of
dimension $\leq 2g-2$, and a finite set of preimages of Shimura subvarieties
$X_{j}$, such that if $U$ is the complement of
$(\bigcup_{i}T_{i})\cup(\bigcup_{j}X_{j})\cup H_{g}$ in $\mathcal{M}_{g}$,
then the locus in $U(\mathbb{C})$ where $\nu$ is torsion is a countable set of
points. Moreover, any positive dimensional component of the torsion locus of
$\nu$ with Mumford-Tate group $\mathrm{GSp}_{2g}$ is defined over
$\overline{\mathbb{Q}}$.

######  Remark 1.2.

The locus where $Z_{C}$ itself is torsion is contained in the torsion locus of
$\nu$. In particular, by the standard spreading-out argument, the torsion
locus of $Z_{C}$ in $U$ is a countable subset of $U(\overline{\mathbb{Q}})$.
We note that a similar result has been recently obtained by S-W. Zhang.

Theorem 1.1 is a particular instance of the following more general theorem
which describes the distribution of the torsion locus of normal functions, see
Theorem 3.2 for the most general version and §§4.2-4.3 for applications to
other normal functions.

######  Theorem 1.3.

Let $\mathcal{M}$ be a smooth complex quasi-projective variety, and
$\mathcal{V}$ be a non-isotrivial $\mathbb{Q}$-simple $\mathbb{Z}$-VHS of
negative weight on $\mathcal{M}^{\text{an}}$ with simple adjoint Mumford–Tate
group $G$. Let $\phi\colon\mathcal{M}^{\text{an}}\to X=\Gamma\backslash
D=\Gamma\backslash G_{\mathbb{R}}/K$ denote the induced period map. We assume
that $D$ is a Hermitian symmetric domain and $X$ a connected Shimura variety,
that $\phi$ is quasi-finite onto its image (which is necessarily Hodge-generic
in $X$), and that either ${\mathcal{V}}\neq{\mathcal{F}}^{-1}{\mathcal{V}}$ or
$\dim({\mathcal{M}})\leq\tfrac{1}{2}\,\mathrm{rk}({\mathcal{V}})$.

Let $\nu\in\mathrm{ANF}_{\mathcal{M}}(\mathcal{V})$ be a nontorsion admissible
normal function, and write $\mathrm{TL}_{\text{pos}}(\nu)$ for the union of
the (countably many) positive-dimensional components of its torsion locus.

Then
$\mathcal{T}:=\overline{\mathrm{TL}_{\text{pos}}(\nu)}^{\text{Zar}}\subset\mathcal{M}$
is a proper algebraic subvariety. Moreover, any component of $\mathcal{T}$ is
either contained in the $\phi$-preimage of a proper locally symmetric
subvariety of $X$, or is a component of $\mathrm{TL}_{\text{pos}}(\nu)$.

######  Remark 1.4.

If ${\mathcal{V}}$ and $\nu$ arise from a family of varieties and a family of
cycles thereon, with both (as well as ${\mathcal{M}}$) defined over a number
field, then these latter components of $\mathrm{TL}_{pos}(\nu)$ (not contained
in the Hodge locus of ${\mathcal{V}}$) are defined over
$\overline{\mathbb{Q}}$; see Theorem 4.8.

We also prove a sufficient condition for the analytic density of the torsion
locus, see Theorem 3.9 for the most general version of the following theorem.

######  Theorem 1.5.

Assume that ${\mathcal{V}}$ is a $\mathbb{Q}$-simple $\mathbb{Z}$-VHS of
weight $-1$ and level111Recall that the _level_ of a Hodge structure $V$ (or
variation thereof) is the difference of the largest and smallest $p$ for which
$\mathrm{Gr}_{F}^{p}V\neq\\{0\\}$. $1$ over $\mathcal{M}$, such that its
period map is quasi-finite and its generic adjoint Mumford-Tate group is
simple. We assume furthermore that ${\mathcal{V}}$ has rank $2r\leq
2\dim(\mathcal{M})$. Let $\nu\in\mathrm{ANF}_{\mathcal{M}}(\mathcal{V})$ be a
nontorsion admissible normal function. Then $\mathrm{TL}(\nu)$ is analytically
dense in $\mathcal{M}$ and equidistributed with respect to the Betti form
$\omega_{Betti}$, i.e., for every $U\subset\mathcal{M}$ relatively compact
subset with zero measure boundary and for every differential form $\alpha$, we
have:

$\int_{\\{x\in
U|\,n\cdot\nu(x)=0\\}}\alpha\underset{n\rightarrow\infty}{\sim}\frac{n^{2r}}{\mathrm{vol}({\mathcal{J}}(V_{\mathbb{R}}))}\cdot\int_{U}\alpha\wedge\omega_{Betti}^{r}~{}.$

###  1.1. Ideas of the proofs

The proofs of the previous theorems rely on arithmetic and functional
transcendence results in Hodge theory, namely the Manin–Mumford conjecture,
Ax–Schanuel Theorem for variations of mixed Hodge structures recently
established by Gao–Klingler [GK24], and recent results on the study of the
Hodge locus for variations of mixed Hodge structures proved by Baldi–Urbanik
[BU24]. The equidistribution part in Theorem 1.5 relies on the computation of
the generic rank of normal functions with big monodromy and general
equidistribution results established in [TT23].

###  1.2. Acknowledgement

The present work started during the conference “The Ceresa Cycle in Arithmetic
and Geometry” at ICERM. We thank the organizers for their efforts and ICERM
for its hospitality, and Philip Griffiths and Jef Laga for valuable
discussions. M.K. was supported by NSF grant DMS-2101482, and S.T. was
supported by NSF grant DMS-2302388.

##  2\. Preliminaries on atypical Hodge loci and normal functions

###  2.1. Weak Hodge loci of variations of mixed Hodge structure

Let $S$ be a smooth quasi-projective algebraic variety and let
${\mathcal{E}}=\\{{\mathbb{E}}_{\mathbb{Z}},W_{\bullet},{\mathcal{F}}^{\bullet},\nabla\\}$
be an admissible graded-polarizable variation of mixed Hodge structure over
$S$.222Here ${\mathbb{E}}_{\mathbb{Z}}$ is a torsion-free $\mathbb{Z}$-local
system, the weight filtration $W_{\bullet}$ is defined on
${\mathbb{E}}:={\mathbb{E}}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Q}$ and
the Hodge filtration ${\mathcal{F}}^{\bullet}$ is defined on
${\mathcal{E}}:={\mathbb{E}}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathcal{O}_{S}$.
We make no notational distinction between the holomorphic vector bundle and
its sheaf of sections (both are denoted by ${\mathcal{E}}$). Let
$({\mathcal{G}},{\mathcal{D}}_{{\mathcal{G}}})$ be the generic mixed Hodge
datum, where ${\mathcal{G}}$ is the generic mixed Mumford-Tate group and
${\mathcal{D}}_{{\mathcal{G}}}$ is the mixed Mumford-Tate period domain. Let
$({\mathcal{H}},{\mathcal{D}}_{{\mathcal{H}}})$ be the monodromy datum of the
variation: here ${\mathcal{H}}$ is the _geometric monodromy group_ of
${\mathcal{E}}$, i.e., the neutral component of the $\mathbb{Q}$-Zariski
closure of the monodromy group ${\mathit{\Gamma}}$ associated to
${\mathbb{E}}_{\mathbb{Z}}$. Then ${\mathcal{G}}$ and ${\mathcal{H}}$ are
algebraic groups over $\mathbb{Q}$ and by a well-known theorem of Deligne and
André [And92], ${\mathcal{H}}$ is a normal subgroup of ${\mathcal{G}}^{der}$.
Associated to this data, we have a holomorphic period map

$\Phi:S\rightarrow{\mathit{\Gamma}}\backslash{\mathcal{D}}_{{\mathcal{G}}}~{}.$

Let
$(\overline{{\mathcal{G}}},\overline{{\mathcal{D}}}_{\overline{{\mathcal{G}}}})$
denote the quotient mixed Hodge datum where
$\overline{{\mathcal{G}}}:={\mathcal{G}}/{\mathcal{H}}$ and
$\overline{{\mathcal{D}}}_{{\mathcal{G}}/{\mathcal{H}}}$ is the mixed period
domain associated to $\overline{{\mathcal{G}}}$. Then the monodromy period
domain ${\mathcal{D}}_{{\mathcal{H}}}$ is isomorphic to a fiber of the
projection
${\mathcal{D}}_{{\mathcal{G}}}\rightarrow\overline{{\mathcal{D}}}_{\overline{{\mathcal{G}}}}$
and the period map $\Phi$ factors, up to a finite étale cover, as
$\Phi=\iota\circ\Phi_{{\mathcal{H}}}$ where:

$S\xrightarrow{\Phi_{{\mathcal{H}}}}{\mathit{\Gamma}}_{{\mathcal{H}}}\backslash{\mathcal{D}}_{{\mathcal{H}}}\xhookrightarrow{\iota}{\mathit{\Gamma}}\backslash{\mathcal{D}}_{{\mathcal{G}}}\xrightarrow{\pi}\overline{{\mathit{\Gamma}}}\backslash\overline{{\mathcal{D}}}_{\overline{{\mathcal{G}}}}~{}.$

We refer to $\Phi_{{\mathcal{H}}}$ as the _monodromy period map_.

Let $Y\subseteq S$ be a closed irreducible sub-variety of $S$. Then we have
similarly a generic Hodge datum
$({\mathcal{G}}_{Y},{\mathcal{D}}_{{\mathcal{G}}_{Y}})$, monodromy datum
$({\mathcal{H}}_{Y},{\mathcal{D}}^{0}_{Y})$, and quotient mixed datum
$(\overline{{\mathcal{G}}_{Y}},\overline{{\mathcal{D}}}_{\overline{{\mathcal{G}}_{Y}}})$.
The restriction $\Phi|_{Y}$ of the period map $\Phi$ to $Y$ factors similarly,
up to finite étale cover, as the composition of the first two arrows in

$Y\rightarrow{\mathit{\Gamma}}_{{\mathcal{H}}_{Y}}\backslash{\mathcal{D}}^{0}_{Y}\hookrightarrow{\mathit{\Gamma}}_{Y}\backslash{\mathcal{D}}_{{\mathcal{G}}_{Y}}\xrightarrow{\pi_{Y}}\overline{{\mathit{\Gamma}}}_{Y}\backslash\overline{{\mathcal{D}}}_{\overline{{\mathcal{G}}_{Y}}}~{}.$

######  Definition 2.1 ([KO21]).

Let $Y\subseteq S$ be a closed irreducible subvariety. We say that:

  1. (1)

$Y$ has _positive period dimension_ if $\Phi(Y)$ is not zero dimensional.

  2. (2)

$Y$ is _special_ if $Y$ is maximal among closed subvarieties with Mumford-Tate
group ${\mathcal{G}}_{Y}$. Equivalently, $Y$ is a component of
$\Phi^{-1}({\mathit{\Gamma}}_{Y}\backslash{\mathcal{D}}_{{\mathcal{G}}_{Y}})$.

  3. (3)

$Y$ is _weakly special_ if $Y$ is maximal among closed subvarieties with
monodromy group equal to ${\mathcal{H}}_{Y}$. Equivalently, under the period
map $\Phi$, $Y$ is a component of $\Phi^{-1}(\pi_{Y}^{-1}(\\{x\\}))$ where
$x\in\overline{{\mathit{\Gamma}}}_{Y}\backslash\overline{{\mathcal{D}}}_{\overline{{\mathcal{G}}_{Y}}}$
is a Hodge-generic point.

######  Definition 2.2.

Let $Y$ be a weakly special subvariety. We say that $Y$ is monodromically
atypical if

$\mathrm{codim}_{\Phi(S)}\left(\Phi(Y)\right)<\mathrm{codim}_{{\mathcal{D}}_{{\mathcal{H}}}}{\mathcal{D}}^{0}_{Y}.$

Our main tool in this paper is the following result recently established by
Baldi and Urbanik [BU24, Theorem 7.1]:

######  Theorem 2.3.

There exists a finite set $\Sigma$ of triples
$({\mathcal{G}}_{i},{\mathcal{D}}_{i},{\mathcal{N}}_{i})_{i\in\Sigma}$, where
$({\mathcal{G}}_{i},{\mathcal{D}}_{i})$ is a sub-Hodge datum of the generic
Hodge datum $({\mathcal{G}},{\mathcal{D}}_{{\mathcal{G}}})$,
${\mathcal{N}}_{i}$ is a normal subgroup of ${\mathcal{G}}_{i}$ whose
reductive part is semisimple, and the following property holds: for each
maximal monodromically atypical subvariety $Y\subset S$, there exist
$i\in\Sigma$ and $y\in{\mathcal{D}}_{i}$ for which
${\mathcal{D}}_{Y}^{0}={\mathcal{N}}_{i}(\mathbb{R})^{+}{\mathcal{N}}_{i}(\mathbb{C})^{u}\cdot
y$ (up to the action of ${\mathit{\Gamma}}$). If in addition $Y$ has positive
period dimension, then ${\mathcal{N}}_{i}^{ad}$ is a nontrivial algebraic
group.

######  Remark 2.4.

In the previous theorem, the statement that
${\mathcal{D}}_{0}^{Y}={\mathcal{N}}_{i}(\mathbb{R})^{+}{\mathcal{N}}_{i}(\mathbb{C})^{u}\cdot
y$ implies (up to the action of ${\mathit{\Gamma}}$) that ${\mathcal{N}}_{i}$
equals the geometric monodromy group of ${\mathcal{E}}|_{Y}$, and $\Phi|_{Y}$
factors through ${\mathit{\Gamma}}_{i}\backslash{\mathcal{D}}_{i}$,333Here one
can take ${\mathit{\Gamma}}_{i}:={\mathit{\Gamma}}\cap{\mathcal{G}}_{i}$. with
image in the fiber of
${\mathit{\Gamma}}_{i}\backslash{\mathcal{D}}_{i}\overset{\pi_{i}}{\to}\overline{{\mathit{\Gamma}}}_{i}\backslash\overline{{\mathcal{D}}}_{{\mathcal{G}}_{i}/{\mathcal{N}}_{i}}$
over $\pi_{i}(y)$. In particular, ${\mathcal{G}}_{i}$ need _not_ equal the
generic Mumford-Tate group of ${\mathcal{E}}|_{Y}$ — it merely _contains_ it,
and normalizes the geometric monodromy group. Hence if
${\mathcal{N}}_{i}=\\{1\\}$, then ${\mathcal{G}}_{i}$ is unconstrained by $Y$
and no “finiteness” result is achieved for subvarieties of period dimension
zero.

######  Remark 2.5.

In this section we have used italicized letters for (Mumford-Tate and
monodromy) groups associated to mixed variations. We shall denote in the
sequel by roman letters the groups attached to the associated gradeds (which
are direct sums of pure variations), a passage denoted by
${\mathcal{G}}\mapsto\mathrm{Gr}\,{\mathcal{G}}$ in [And92]. So for us $G$
denotes the reductive group $\mathrm{Gr}\,{\mathcal{G}}$, and ${\mathcal{G}}$
is an extension of $G$ by a unipotent group (which records the extension
data).

###  2.2. Brief review of normal functions

An admissible normal function on $S$ with values in a polarizable VHS
${\mathcal{V}}$ (of negative weight) is an extension of
admissible444Admissibility is an asymptotic condition at the boundary
$\overline{S}\backslash S$ which is always satisfied for VMHS arising from
algebraic geometry. variations of MHS on $S$ of the form

$0\to{\mathcal{V}}\to{\mathcal{E}}\to\mathbb{Z}(0)\to 0\,,$ (1)

where $\mathbb{Z}(0)$ is the constant VHS of rank 1 and weight 0. They
constitute a group $\mathrm{ANF}_{S}({\mathcal{V}})$ under Baer sum.

They may be viewed as functions in the following way. There exist local lifts
of $1\in\mathbb{Z}(0)$: to a holomorphic section $e_{F}$ of
${\mathcal{F}}^{0}{\mathcal{E}}$; and to a section $e_{\mathbb{Z}}$ of
${\mathbb{E}}_{\mathbb{Z}}$. Their difference $e_{\mathbb{Z}}-e_{F}$ lifts to
a local section of ${\mathcal{V}}$, with ambiguity (arising from choices of
$e_{F},e_{\mathbb{Z}}$) in
${\mathcal{F}}^{0}{\mathcal{V}}+{\mathbb{V}}_{\mathbb{Z}}$. Passing to the
generalized intermediate Jacobian bundle
${\mathcal{J}}({\mathcal{V}})={\mathcal{V}}/({\mathcal{F}}^{0}{\mathcal{V}}+{\mathbb{V}}_{\mathbb{Z}})$,
this construction produces a well-defined global holomorphic section
$\nu\colon S\to{\mathcal{J}}({\mathcal{V}})$. (The group structure is given by
adding sections.) We write $\nu\in\mathrm{ANF}_{S}({\mathcal{V}})$ and
${\mathcal{E}}_{\nu}$ for the corresponding AVMHS.

By Griffiths transversality for ${\mathcal{E}}_{\nu}$, a local lift
$\tilde{\nu}$ of $\nu$ (to a section of ${\mathcal{V}}$ itself) has derivative
$\nabla\tilde{\nu}$ a local section of
${\mathcal{F}}^{-1}{\mathcal{V}}\otimes\Omega^{1}_{S}$ (a property of $\nu$
known as _quasi-horizontality_). We can also analytically continue
$\tilde{\nu}$ along a loop $\gamma\in\pi_{1}(S,s_{0})$ and consider its
monodromy $(\gamma-I)\nu\in V$ (the fiber of ${\mathcal{V}}$ over $s_{0}$)
which becomes well-defined in $V/(T_{\gamma}-I)V$ (with $T_{\gamma}$ given by
the monodromy of the local system). These notions of derivative and monodromy
of $\nu$ are formalized by the _infinitesimal_ and _topological invariants_
$\delta\nu$ and $[\nu]$, defined as follows.

Consider the exact sequence of complexes of sheaves

$0\to{\mathcal{F}}^{0}\mathscr{C}^{\bullet}\oplus{\mathbb{V}}_{\mathbb{Z}}\to\mathscr{C}^{\bullet}\to\frac{\mathscr{C}^{\bullet}}{{\mathcal{F}}^{0}\mathscr{C}^{\bullet}\oplus{\mathbb{V}}_{\mathbb{Z}}}\to
0$

where
$\mathscr{C}^{\bullet}:=({\mathcal{V}}\otimes\Omega^{\bullet}_{S},\nabla)$,
${\mathcal{F}}^{0}\mathscr{C}^{\bullet}:=({\mathcal{F}}^{-\bullet}{\mathcal{V}}\otimes\Omega^{\bullet}_{S},\nabla)$,
and ${\mathbb{V}}_{\mathbb{Z}}$ is placed in degree zero. Write
$\mathrm{NF}_{S}({\mathcal{V}}):={\mathbb{H}}^{0}(S,\tfrac{\mathscr{C}^{\bullet}}{{\mathcal{F}}^{0}\mathscr{C}^{\bullet}\oplus{\mathbb{V}}_{\mathbb{Z}}})$
for the group of holomorphic quasi-horizontal sections of
${\mathcal{J}}({\mathcal{V}})$.

######  Definition 2.6.

(i) In the composition

$\textstyle{\mathrm{ANF}_{S}({\mathcal{V}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=:(\delta,[\cdot])}$$\textstyle{\mathrm{NF}_{S}({\mathcal{V}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$conn.
hom.
$\textstyle{{\mathbb{H}}^{1}({\mathcal{F}}^{0}\mathscr{C}^{\bullet}\oplus{\mathbb{V}}_{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$edge
hom.$\textstyle{\Gamma(S,{\mathcal{H}}^{1}_{\nabla}({\mathcal{F}}^{0}\mathscr{C}^{\bullet}))\oplus
H^{1}(S,{\mathbb{V}}_{\mathbb{Z}})}$

$\delta\nu$ is the _infinitesimal invariant_ of
$\nu\in\mathrm{ANF}_{S}({\mathcal{V}})$, and $[\nu]$ the _topological
invariant_ of $\nu$.

(ii) $\nu$ is said to be _flat_ if $\delta\nu=0$, and _monodromically trivial_
(resp.  _torsion_) if $[\nu]$ is trivial (resp. torsion).

######  Proposition 2.7.

Consider the following conditions on $\nu$:

  * (a)

$\nu$ is torsion;

  * (b)

$\nu$ is flat;

  * (c)

$\nu$ is monodromically torsion;

  * (d)

${\mathbb{E}}_{\nu}$ is split (there is a lift of $1\in\mathbb{Z}(0)$ to
${\mathbb{E}}_{\nu}$) after multiplying $\nu$ by a nonzero integer or passing
to a finite subgroup of $\pi_{1}(S)$;

  * (e)

the geometric monodromy group ${\mathcal{H}}$ of ${\mathcal{E}}_{\nu}$ is
semisimple (has trivial unipotent radical), i.e., equals the geometric
monodromy group $H$ of ${\mathcal{V}}$.

In general, (a) $\implies$ (b,c,d,e), and (c,d,e) are all equivalent. If the
“fixed part” $H^{0}(S,{\mathbb{V}})=\\{0\\}$, then (a) and (c,d,e) are
equivalent. If, in addition, the sheaf map
$\nabla\colon{\mathcal{F}}^{0}{\mathcal{V}}\to{\mathcal{F}}^{-1}{\mathcal{V}}\otimes\Omega^{1}_{S}$
is everywhere injective, then all conditions are equivalent.

###### Sketch.

Working $\otimes\mathbb{Q}$, if $[\nu]=0$ then
${\mathbb{E}}_{\nu}={\mathbb{V}}\oplus\mathbb{Q}$. If also
$H^{0}(S,{\mathbb{V}})=\\{0\\}$, then $H^{0}(S,{\mathbb{E}}_{\nu})=\mathbb{Q}$
and the Theorem of the Fixed Part [SZ85, (4.19)] says the latter underlies a
(constant) sub-AVMHS of ${\mathcal{E}}_{\nu}$. Since it is of rank one, it can
only be of type $(0,0)$, splitting (1) and rendering $\nu=0$. Thus $[\cdot]$
is injective, whence (c)$\implies$(a), when ${\mathbb{V}}$ has trivial fixed
part. The additional condition required for (b)$\implies$(a) is to make the
edge homomorphism injective.

For the equivalence of (c) and (d), tracing through the definition shows that
$[\nu]$ is the image of $1\in H^{0}(S,\mathbb{Z}(0))$ under the connecting
homomorphism for the extension

$0\to{\mathbb{V}}_{\mathbb{Z}}\to{\mathbb{E}}_{\nu,\mathbb{Z}}\to\mathbb{Z}(0)\to
0$ (2)

of local systems. Finally, ${\mathcal{H}}$ is an extension of
$\mathrm{Gr}\,{\mathcal{H}}$ by its unipotent radical $U$; (e) says that
$U=\\{1\\}$, and (d) is the same as
${\mathcal{H}}=\mathrm{Gr}\,{\mathcal{H}}$, so (d)$\iff$(e). ∎

######  Definition 2.8.

A _projection_ of $\nu$ is its image under
$\mathrm{ANF}_{S}({\mathcal{V}})\to\mathrm{ANF}_{S}({\mathcal{V}}^{\prime})$
for any _nontrivial_ surjection
${\mathcal{V}}\twoheadrightarrow{\mathcal{V}}^{\prime}$ of VHS. We say that
$\nu$ is _generic_ if no projection of $\nu$ has torsion topological
invariant.

######  Corollary 2.9.

(i) $\nu$ is generic if and only if the unipotent radical of ${\mathcal{H}}$
(the geometric monodromy group of ${\mathcal{E}}_{\nu}$) is isomorphic to $V$.

(ii) If ${\mathcal{V}}$ has no constant factors, $\nu$ is generic if, and only
if, every projection of $\nu$ is nontorsion.

(iii) If ${\mathcal{V}}$ is irreducible and non-constant, $\nu$ is generic if,
and only if, it is nontorsion.

###### Proof.

If ${\mathcal{V}}$ is simple, then non-semisimplicity of ${\mathcal{H}}$ is
the same as having $U=V$. More generally, applying this to projections with
${\mathcal{V}}^{\prime}$ simple and using the equivalence of (c) and (e) in
the Proposition shows that genericity is equivalent to $U$ projecting to each
$V^{\prime}$, i.e., $U=V$. For (ii-iii), the fixed part is trivial and we use
the resulting equivalence of (a) and (c). ∎

######  Remark 2.10.

In the geometric context, normal functions graph Abel-Jacobi mappings in
families of cycles and higher cycles. Given a smooth projective family
${\mathcal{X}}\to S$, with
${\mathcal{V}}\subset{\mathcal{H}}^{n}({\mathcal{X}}/S)(m)$ and $r=2m-n-1\geq
0$, a primitive555This means that the restrictions $\mathfrak{Z}|_{X_{s}}$ to
fibers of ${\mathcal{X}}\to S$ are homologically trivial, which is a
nontrivial condition only for $r=0$. Also note that ${\mathcal{V}}$ has
negative weight $n-2m=-r-1$. cycle
$\mathfrak{Z}\in\mathrm{CH}^{m}({\mathcal{X}},r)\cong_{\otimes\mathbb{Q}}\mathrm{Gr}_{\gamma}^{m}K_{r}^{\text{alg}}({\mathcal{X}})$
produces a normal function
$\nu_{\mathfrak{Z}}\in\mathrm{ANF}_{S}({\mathcal{V}})$ via

$\nu(s):=\mathrm{AJ}_{X_{s}}(\mathfrak{Z}|_{X_{s}})\in{\mathcal{J}}(V_{s})=\frac{V_{s,\mathbb{C}}}{F^{0}V_{s,\mathbb{C}}+{\mathbb{V}}_{\mathbb{Z}}}\cong\mathrm{Ext}^{1}_{\text{MHS}}(\mathbb{Z}(0),V_{s})\,.$

When $r=0$, ${\mathcal{J}}(V_{s})$ is a compact complex torus known as an
intermediate Jacobian; it is usually666Exceptions here would include the case
where $V_{s}$ is a CM Hodge structure. not algebraic if the level of $V_{s}$
is larger than one. When $r>0$, it is a noncompact complex torus.

By the main theorem of [BP13], the zero locus of $\nu$ is an algebraic
subvariety of $S$. Since the zero-locus of $m\nu$ is the $m$-torsion locus of
$\nu$, we get that the torsion locus with torsion order bounded (say, by $N$)
is also algebraic. Thus the torsion locus of $\nu$ is a countable union of
algebraic subvarieties. In the remainder of this paper we aim to improve this
statement.

##  3\. Torsion loci of admissible normal functions

###  3.1. The main theorem

Let $\mathcal{M}$ be a smooth quasi-projective complex variety and let
${\mathcal{V}}=\\{{\mathbb{V}}_{\mathbb{Z}},{\mathcal{F}}^{\bullet},\nabla\\}$
be a non-trivial polarized $\mathbb{Z}$-VHS of negative weight on
$\mathcal{M}$. Let $G$ denote the generic Mumford-Tate group of
${\mathcal{V}}$ and $\phi\colon\mathcal{M}_{\mathbb{C}}^{\text{an}}\to
X=\Gamma\backslash D=\Gamma\backslash G_{\mathbb{R}}/K$ be the induced period
map. By construction the image $M:=\phi({\mathcal{M}})$ is Hodge-generic in
$X$; while $X$ is in general not an algebraic variety, we know that $M$ is one
by [BBT23a].

Let $(H,{\mathcal{D}}_{H})$ be the monodromy data of ${\mathcal{V}}$ and let
$\phi_{H}:{\mathcal{M}}\rightarrow
X_{H}:=\Gamma_{H}\backslash{\mathcal{D}}_{H}$ be the monodromy period map.

######  Definition 3.1.

A subvariety $Y\subset{\mathcal{M}}$ is said to be of _factorwise positive
dimension_ with respect to $\phi$, if the projection of $\phi(Y)$ to each
simple factor of $X_{H}$ has positive dimension.777One may need to take a
finite étale cover to define these projections.

Let ${\mathcal{V}}_{0}$ be the largest isotrivial summand of ${\mathcal{V}}$.
Then the variation ${\mathcal{V}}$ admits a decomposition:

$\displaystyle{\mathcal{V}}={\mathcal{V}}_{0}\oplus\bigoplus_{\ell\in
I}{\mathcal{V}}_{\ell}$ (3)

where for $\ell\in I$, ${\mathcal{V}}_{\ell}$ are $\mathbb{Q}$-simple non-
isotrivial variations. Let $\nu\in\mathrm{ANF}_{\mathcal{M}}(\mathcal{V})$ be
a nontorsion admissible normal function. Then $\nu$ admits a decomposition
$\nu=(\nu_{\ell})_{\ell}$ which reflects the decomposition in (3). Let $V$ be
the $\mathbb{Q}$-vector space underlying a fiber of ${\mathcal{V}}$, which is
also equal to
$H^{0}(\widetilde{{\mathcal{M}}},\iota^{*}{\mathbb{V}}_{\mathbb{C}})$, where
$\iota:\widetilde{{\mathcal{M}}}\rightarrow{\mathcal{M}}$ is the universal
cover.

We can now state the main theorem of this paper.

######  Theorem 3.2.

Let $\nu\in\mathrm{ANF}_{\mathcal{M}}({\mathcal{V}})$ be a generic admissible
normal function on $\mathcal{M}$. Let $\mathrm{TL}_{f-pos}(\nu)$ be the
(countable) union of components of the torsion locus of $\nu$ which are of
factorwise positive dimension with respect to $\phi$. Assume that at least one
of the following conditions holds:

  * (i)

the vector bundle ${\mathcal{V}}/{\mathcal{F}}^{-1}{\mathcal{V}}$ is non-
trivial; or

  * (ii)

$\dim(\mathcal{M})\leq\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))$.

Then $\mathcal{T}:=\overline{\mathrm{TL}_{f-pos}(\nu)}^{Zar}$ is a strict
algebraic subvariety of $\mathcal{M}$. Moreover, there exist finitely many
projections $(\nu_{i})_{i}$ of $\nu$ such that each component of $\mathcal{T}$
is either contained in a component of the Hodge locus for $\phi$, or contained
in $\mathrm{TL}_{f-pos}(\nu_{i})$.

######  Remark 3.3.

If ${\mathcal{V}}$ is in weight $-1$ (for “usual” normal functions), then
${\mathcal{V}}$ has odd level and (i) says this level is $>1$; if
${\mathcal{V}}$ has weight $-2$, (i) says that ${\mathcal{V}}$ has level $>0$;
and if ${\mathcal{V}}$ has weight $<-2$, (i) holds automatically. Condition
(ii) is thus of interest only when ${\mathcal{V}}$ has weight $-1$ level $1$
(where
$\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))=\tfrac{1}{2}\dim_{\mathbb{Q}}(V)$),
or weight $-2$ level $0$ (where
$\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))=\dim_{\mathbb{Q}}(V)$).

###### Proof.

We will regard $\nu$ as a variation of mixed Hodge structure
$\mathcal{E}_{\nu}\in\mathrm{Ext}^{1}_{\text{AVMHS}(\mathcal{M})}(\mathbb{Z}(0),{\mathcal{V}})$.
Since $\nu$ is generic, the geometric monodromy group ${\mathcal{H}}$ of
$\mathcal{E}_{\nu}$ is an extension of $H$ by $V$ (by Corollary 2.9(i)); and
thus the Mumford-Tate group ${\mathcal{G}}$ of ${\mathcal{E}}_{\nu}$ is an
extension of $G$ by $V$. Consider the period map for ${\mathcal{E}}_{\nu}$:

$\Phi\colon\mathcal{M}\to{\mathcal{X}}={\mathit{\Gamma}}\backslash\mathcal{D}={\mathit{\Gamma}}\backslash\mathcal{G}(\mathbb{R})/\mathcal{K}~{},$

where the arithmetic group $\tilde{\Gamma}$ is an extension of $\Gamma$ by the
translation group $V_{\mathbb{Z}}$. Consider also the monodromy period map

$\Phi_{\mathcal{H}}\colon\mathcal{M}\to{\mathcal{X}}_{\mathcal{H}}={\mathit{\Gamma}}_{\mathcal{H}}\backslash\mathcal{D}_{\mathcal{H}}={\mathit{\Gamma}}\backslash\mathcal{H}(\mathbb{R})/\mathcal{K}_{{\mathcal{H}}}~{}.$

The $\mathbb{Q}$-vector space $V$ admits an $H$-stable decomposition into
irreducible factors $V=V_{0}\oplus\bigoplus_{\ell\in I}V_{\ell}$ which
reflects (3), and where $V_{0}=V^{H}$.

There is a natural projection $\pi\colon{\mathcal{X}}\to X$ with fibers
generalized intermediate Jacobians $\\{{\mathcal{J}}_{x}\\}$, as well as
natural sections corresponding to torsion points in
$V_{\mathbb{C}}/V_{\mathbb{Z}}$ preserved by $\Gamma$. This projection is
compatible with the decomposition from (3), since the intermediate Jacobian
$\\{{\mathcal{J}}_{x}\\}$ decomposes as the product of the intermediate
Jacobians associated to each summand in (3). This results in a decomposition
$\pi\colon{\mathcal{D}}={\mathcal{D}}_{0}\times_{D}\prod_{\ell\in
I}{\mathcal{D}}_{\ell}\rightarrow D$ and
$\pi\colon{\mathcal{X}}={\mathcal{X}}_{0}\times_{X}\prod_{\ell\in
I}{\mathcal{X}}_{\ell}\rightarrow X$. Here the fiber of
${\mathcal{D}}_{\ell}\to D$ over $\varphi\in D$ is
$V_{\ell,\mathbb{C}}/F^{0}_{\varphi}V_{\ell,\mathbb{C}}$, and that of
${\mathcal{X}}_{\ell}\to X$ is the generalized Jacobian
${\mathcal{J}}(V_{\ell})$. The monodromy period map factors furthermore as:

$\Phi_{\mathcal{H}}:\mathcal{M}\rightarrow{\mathcal{J}}(V_{0})\times\prod_{\ell\in
I}{\mathcal{X}}_{\ell}~{},$

where ${\mathcal{J}}(V_{0})$ is the generalized Jacobian associated to the
isotrivial variation ${\mathcal{V}}_{0}$.888One may need to take a finite
étale cover to define this map, which we assume already done. All products
“$\prod$” are relative (to $D$ or to $X$).

Let $Y\subset\mathcal{M}$ be an irreducible subvariety, and denote by
$(\mathcal{G}_{Y},\mathcal{D}_{Y})$ the mixed Hodge data associated to
$\mathcal{E}_{\nu}|_{Y}$. Further, we write
$\mathcal{H}_{Y}\trianglelefteq\mathcal{G}_{Y}^{\text{der}}$ and
$\mathcal{D}_{Y}^{0}\subseteq\mathcal{D}_{Y}$ for the geometric monodromy
group and the monodromy period domain, which is the orbit of a
basepoint999that is, a point whose image in $\tilde{X}$ is contained in the
image of $\Phi|_{Y}$ in $\mathcal{D}_{Y}$ by
$\mathcal{H}_{Y}(\mathbb{R})^{+}\mathcal{H}^{\text{un}}_{Y}(\mathbb{C})$.
Recall that $Y$ is monodromically atypical for $\Phi$ if $Y$ is weakly special
and
$\text{codim}_{\mathcal{M}}(Y)<\text{codim}_{\mathcal{D}_{\mathcal{H}}}(\mathcal{D}^{0}_{Y})$.

Now let
$\\{({\mathcal{G}}_{i},{\mathcal{D}}_{i},{\mathcal{N}}_{i})\\}_{i\in\Sigma}$
be the finite set of triples guaranteed for ${\mathcal{E}}_{\nu}$ by Theorem
2.3. Take $Y\subset{\mathcal{M}}$ to be a _maximal_ monodromically atypical
subvariety, i.e., monodromically atypical and not contained in a larger
monodromically atypical subvariety. Then up to the action by
${\mathit{\Gamma}}$,101010In fact, we can ignore the action by $\Gamma$ here,
since this doesn’t affect images in ${\mathcal{X}}$. there exist $i\in\Sigma$
and $y\in{\mathcal{D}}_{i}$ such that ${\mathcal{H}}_{Y}={\mathcal{N}}_{i}$
and
$\mathcal{D}_{Y}^{0}=\mathcal{N}_{i}(\mathbb{R})\mathcal{N}_{i}^{u}(\mathbb{C})\cdot
y$.

We consider the various possibilities for this triple. For $i\in\Sigma$
belonging to some subset (say, $\Sigma_{0}\subset\Sigma$),
$G_{i}:=\mathcal{G}_{i}/\mathcal{G}_{i}\cap V$ has $G_{i}^{der}<G^{der}$ a
proper subgroup. In this case, $D_{Y}:=\pi(\mathcal{D}_{Y})$ is a proper
subdomain of $D$ and $Y$ is contained in a component $Z$ of the Hodge locus
for $\phi$ (i.e., for ${\mathcal{V}}$). More precisely, there is a Mumford-
Tate subdomain $D_{i}:=G_{i,\mathbb{R}}/(K\cap G_{i,\mathbb{R}})\subset D$ and
$Z\subset{\mathcal{M}}$ is one of finitely many components of the preimage of
$\Gamma_{i}\backslash D_{i}\subset X$. The union of these $Z$ as $i$ ranges
over $\Sigma_{0}$ is a proper algebraic subvariety
$\mathscr{Z}\subset{\mathcal{M}}$.

If, on the other hand, $G_{i}^{der}=G^{der}$, then we assume furthermore that
$Y$ has factorwise positive dimension for $\phi$. Since
${N}_{i}:={\mathcal{N}}_{i}/({\mathcal{N}}_{i}\cap V)$ is normal in $G^{der}$
and equals $H_{Y}$, and $H_{Y}=H$ by factorwise positivity, we have
$N_{i}=H_{Y}=H$. Since $\mathcal{N}_{i}\cap V$ is $N_{i}$-stable, it is of the
form111111If $V$ has repeated factors, one may need to rearrange the
decomposition of isotypical components. However, such rearrangements are still
subject to the finiteness of $\Sigma$.

$V^{\prime}_{0}\oplus\bigoplus_{\ell\in I^{\prime}}V_{\ell}\subsetneq
V_{0}\oplus\bigoplus_{\ell\in I}V_{\ell}~{},$

for a subset $I^{\prime}\subset I$ and a $\mathbb{Q}$ sub-Hodge structure
$V^{\prime}_{0}\subset V_{0}$. Hence we conclude that ${\mathcal{H}}_{Y}$ is,
up to an irrelevant torus part, an extension of $H$ by
$V^{\prime}_{0}\oplus\bigoplus_{\ell\in I^{\prime}}V_{\ell}$.

The group $\mathcal{G}_{i}$ is on the other hand an extension of $G$ by
$W_{0}\oplus\bigoplus_{\ell\in I_{1}}V_{\ell}$ for some $I_{1}\subseteq I$
$W_{0}\subset V_{0}$ a sub Hodge structure. Since
${\mathcal{H}}_{Y}={\mathcal{N}}_{i}$ is a subgroup of
$\mathcal{G}_{i}^{der}$, $I^{\prime}\subseteq I_{1}$ and
$V^{\prime}_{0}\subseteq W_{0}$. Let
$U=W_{0}/V^{\prime}_{0}\oplus\bigoplus_{\ell\in I_{1}\backslash
I^{\prime}}V_{\ell}$. Then $U$ and ${\mathcal{H}}_{Y}$ are both normal
subgroups of $\mathcal{G}_{i}$ with
$\mathcal{G}_{i}^{der}/{\mathcal{H}}_{Y}\simeq U$, whence
$\mathcal{G}_{i}^{der}$ is an almost direct product of ${\mathcal{H}}_{Y}$ and
$U$. This implies that $H_{Y}=H$ acts trivially on $U$, and hence that
$U\subset V_{0}$ as a sub-Hodge structure. Thus $I_{1}=I^{\prime}$.

Let ${\mathcal{D}}_{W_{0}}\rightarrow D$,
${\mathcal{D}}_{V_{0}/W_{0}}\overset{\pi_{0}}{\to}D$, and
${\mathcal{D}}_{\ell}\overset{\pi_{\ell}}{\to}D$ be the pre-intermediate-
Jacobian fibrations associated to $W_{0}$, $V_{0}/W_{0}$, and $V_{\ell}$. Then
there are torsion sections $\sigma_{i}^{(0)}$ of $\pi_{0}$, and
$\sigma_{i}^{(\ell)}$ of $\pi_{\ell}$ for $\ell\in J:=I_{1}^{c}$, such that

${\mathcal{D}}_{i}={\mathcal{D}}_{W_{0}}\times_{D}\sigma_{i}^{(0)}(D_{V_{0}/W_{0}})\times_{D}\prod_{\ell\in
I_{1}}{\mathcal{D}}_{\ell}\times_{D}\prod_{\ell\in
J}\sigma_{i}^{(\ell)}(D_{\ell})\,.$

Writing ${\mathcal{X}}_{i}$ for the image of ${\mathcal{D}}_{i}$ in
${\mathcal{X}}$, $\Phi^{-1}({\mathcal{X}}_{i})\subset{\mathcal{M}}$ is a union
of (finitely many) components of the torsion locus of the projection
$\nu_{i}=\nu_{V_{0}/W_{0}}+\sum_{\ell\in J}\nu_{\ell}$ of our normal function
$\nu$. By the finiteness result of Theorem 2.3, only finitely many such
projections $\nu_{i}$ and sections
$\sigma_{i}=\\{\sigma_{i}^{(0)},\\{\sigma_{i}^{(\ell)}\\}\\}$ appear (up to
translation by $V_{\mathbb{Z}}$). In particular, we can bound the torsion
order of $\nu_{i}$ uniformly in $i$, say by $m$.

As for ${\mathcal{D}}_{Y}^{0}$, it is determined up to the choice of a point
$y\in{\mathcal{J}}(U)$, the intermediate Jacobian of the constant Hodge
structure $U$.121212Here $U$ is constant possibly after passing to a finite
étale cover of ${\mathcal{M}}$. As we will be primarily interested in the
situation where $y$ is torsion, we make that assumption from now on. In other
words, we suppose that $Y$ is special. Notice that Theorem 2.3 does not say
anything about the boundedness of the torsion orders of such points, but we
are going to prove that this torsion order is indeed bounded.

Notice first that if either $J$ is nonempty or $V_{0}/W_{0}\neq\\{0\\}$, then
$Y$ is contained in the strict subvariety where $m\nu_{i}$ vanishes and we can
skip the next discussion. Otherwise, assume that $V_{0}=W_{0}$ and
$J=\emptyset$, and thus that ${\mathcal{G}}_{i}={\mathcal{G}}$. Denote the
projection of the monodromy period map to ${\mathcal{J}}(U)$ by
$\Phi_{0}:\mathcal{M}\rightarrow{\mathcal{J}}(U)$. Since $Y$ is assumed to be
(monodromically) atypical, and the only “drop” from ${\mathcal{G}}$ to
${\mathcal{H}}_{Y}={\mathcal{N}}_{i}$ is $U$, $Y$ must be an atypical fiber of
$\Phi_{0}$ over a torsion point.

Suppose $U$ has an irreducible sub-Hodge structure $U^{\prime}$ which is not
of weight -1 level 1 or weight -2 level 0. The topological invariant
associated to the projection $\nu_{U^{\prime}}$ is computed by the composition
$\pi_{1}({\mathcal{M}})\twoheadrightarrow H_{1}({\mathcal{M}},\mathbb{Z})\to
H_{1}({\mathcal{J}}(U^{\prime}),\mathbb{Z})\cong U^{\prime}_{\mathbb{Z}}$
induced by $\Phi_{0}^{\prime}$, the Zariski closure of whose image is (the
underlying $\mathbb{Q}$-vector subspace of) a sub-HS $U^{\prime\prime}\subset
U^{\prime}$. On the other hand, we have
$F^{-1}U_{\mathbb{C}}^{\prime}\subsetneq U_{\mathbb{C}}^{\prime}$; and the
differential $d\Phi_{0}^{\prime}\colon T{\mathcal{M}}\to
T{\mathcal{J}}(U^{\prime})\cong
U_{\mathbb{C}}^{\prime}/F^{0}U_{\mathbb{C}}^{\prime}$ factors through
$F^{-1}U_{\mathbb{C}}^{\prime}/F^{0}U_{\mathbb{C}}^{\prime}$ by Griffiths
transversality, which forces $U^{\prime\prime}_{\mathbb{C}}\subset
F^{-1}U^{\prime}_{\mathbb{C}}$. Since $U^{\prime}$ is irreducible, this forces
$U^{\prime\prime}=0$, in contradiction to the assumed genericity of $\nu$.

Hence we may assume that all the Hodge structures appearing in $U$ have either
weight -1 level 1 or weight -2 level 0. Then ${\mathcal{J}}(U)$ is a semi-
abelian scheme, and it is an algebraic variety. Assume for the sake of
contradiction that the image of $\Phi_{0}$ contains infinitely many torsion
points, whose preimages are atypical. By [BBT23b], the image of $\Phi_{0}$ is
an algebraic subvariety and by the solution to the Manin-Mumford conjecture
[Ray83, PZ08], the closure of the torsion locus in ${\Phi}_{0}(\mathcal{M})$
is a translate of a subgroup scheme $B$. If $B$ is strict, then
${\Phi}^{-1}_{0}(B)$ is a (monodromically) atypical subvariety of
$\mathcal{M}$ that strictly contains $Y$, which contradicts the maximality of
$Y$. Hence $B={\mathcal{J}}(U)$ and $\Phi_{0}$ is dominant. But since (by
atypicality of $Y$) the fibers of $\Phi_{0}$ have dimension
$>\dim(\mathcal{M})-\dim{\mathcal{J}}(U)$ over a Zariski dense locus in $B$,
this implies that $\Phi_{0}$ has relative dimension
$>\dim(\mathcal{M})-\dim{\mathcal{J}}(U)$, and as it is dominant, we get a
contradiction. Thus $\Phi_{0}({\mathcal{M}})$ contains only finitely many
torsion points and we can bound the torsion order in the factor
${\mathcal{J}}(U)$. Conclude that $Y$ is a component of the torsion locus of a
normal function $\nu_{i}=\nu_{W_{0}/V^{\prime}_{0}}$ and where the possible
torsion orders are bounded (say, by $m$ again).

We have now produced the finite collection $(\nu_{i})_{i}$ of projections of
$\nu$ in the statement of the Theorem. Let $\mathscr{T}\subset{\mathcal{M}}$
denote the union of the zero loci of $m\nu_{i}$. By [BP13] and our genericity
assumption on $\nu$, this is a proper algebraic subvariety.

Finally, let $T$ be a factorwise positive-dimensional component of
$\mathrm{TL}(\nu)$. On $T$, the monodromy period map factors through some
torsion section $\sigma\colon
D_{{\mathcal{H}}}\to{\mathcal{D}}_{{\mathcal{H}}}$ of codimension
$\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))$. Now if (i) holds, then (invoking
Griffiths transversality once more) the rank of $\delta\tilde{\nu}|_{p}\colon
T_{p}{\mathcal{M}}\to{\mathcal{F}}^{-1}V_{p}/{\mathcal{F}}^{0}V_{p}$ at a
point $p\in T$, and hence $\text{codim}_{{\mathcal{M}}}(T)$, is strictly less
than $\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))$. If (ii) holds, then
$\text{codim}_{{\mathcal{M}}}(T)<\dim({\mathcal{M}})\leq\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))$.
So in either case
$\text{codim}_{{\mathcal{M}}}(T)<\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))\leq$
monodromy codimension of $T$, which makes $T$ monodromically atypical for
${\mathcal{E}}_{\nu}$. Let $Y$ be a maximal monodromically atypical subvariety
containing $T$, and assume $Y$ is not contained in $\mathscr{Z}$. Then the
analysis above shows that ${\mathcal{D}}_{Y}^{0}$ is determined by a torsion
point, since it contains $T$, and that there exists $i$ such that $Y$ is a
component of $\mathrm{TL}(\nu_{i})$ with torsion order bounded by $m$. That
is, $Y\subset\mathscr{T}$. ∎

We now state some corollaries that follow from the previous theorem.

######  Corollary 3.4.

If $\overline{\mathrm{TL}_{f-pos}(\nu)}^{Zar}$ is Zariski dense in
$\mathcal{M}$, then ${\mathcal{V}}/{\mathcal{F}}^{-1}{\mathcal{V}}$ is trivial
and ${\mathcal{V}}$ has rank less than $2\dim(\mathcal{M})$.

If the variation ${\mathcal{V}}$ is $\mathbb{Q}$-simple and non-isotrivial,
then notice that in the proof of Theorem 3.2, the only possibility for
$\nu_{i}$ is $\nu$, and if $\nu$ is generically non-torsion, then it is
generic by Corollary 2.9(iii). Hence we get the following result.

######  Corollary 3.5.

Assume that ${\mathcal{V}}$ is $\mathbb{Q}$-simple and non-isotrivial of
negative weight with quasi-finite period map. Let
$\nu\in\mathrm{ANF}_{\mathcal{M}}({\mathcal{V}})$ be a non-torsion admissible
normal function. Assume that either the vector bundle
${\mathcal{V}}/{\mathcal{F}}^{-1}{\mathcal{V}}$ is non-trivial or the rank of
$\mathcal{V}/F^{0}\mathcal{V}$ is at least131313It is worth remarking here
that if this rank _exceeds_ $\dim({\mathcal{M}})$, then one _expects_ the non-
special zero-dimensional part of the torsion locus to be finite.
$\dim({\mathcal{M}})$.

Then
$\mathcal{T}:=\overline{\mathrm{TL}_{\text{pos}}(\nu)}^{\text{Zar}}\subset\mathcal{M}$
is a proper algebraic subvariety. Moreover, any component of $\mathcal{T}$ is
either contained in the preimage under $\phi$ of a proper locally symmetric
subvariety of $X$, or is a component of $\mathrm{TL}_{\text{pos}}(\nu)$.

######  Remark 3.6.

The previous theorem shows there are at most finitely many Shimura
subvarieties $X_{i}\subsetneq X$ such that the normal function is generically
non-torsion on $\phi^{-1}(X_{i})$ but is torsion on a (possibly dense)
countable union of subvarieties thereof. As an immediate consequence, at most
finitely many Hecke translates of such an $X_{i}$ will support dense positive-
dimensional torsion locus of $\nu$.

###  3.2. Analytic density and equidistribution of the torsion locus

We assume in this section that ${\mathcal{V}}$ has weight $-1$ and level $1$,
that all its $\mathbb{Q}$-factors are non-isotrivial, and $H=G^{der}$. We
continue to assume that the period map for ${\mathcal{V}}$ is quasi-finite.
Let $\nu\in\mathrm{ANF}_{\mathcal{M}}({\mathcal{V}})$ be a generic normal
function in the sense of Definition 2.8.

To analyze the density properties of the torsion locus of $\nu$, we need to
understand better the local properties of the normal function $\nu$, namely
its _generic rank_ which we will now define.

Let ${\mathcal{J}}({\mathcal{V}})\rightarrow\mathcal{M}$ be the family of
intermediate Jacobians of which $\nu$ is a section. For $U\subset\mathcal{M}$
an open simply connected subset, the family of Jacobians
${\mathcal{J}}({\mathcal{V}})\rightarrow U$ admits a
$C^{\infty}$-trivialization ${\mathcal{J}}(V_{\mathbb{R}})\times U\rightarrow
U$ where ${\mathcal{J}}(V_{\mathbb{R}})$ is a compact torus. Projecting to the
first factor and lifting to $V_{\mathbb{R}}$ yields the _Betti map_
$\nu_{\mathbb{R}}:U\rightarrow V_{\mathbb{R}}$.

For $s\in U$, we can define the rank of $\nu$ at $s$ as the rank of the
differential of $\nu_{\mathbb{R}}$ at $s$. Denoting by
$\widetilde{\mathcal{M}}$ the universal cover of $\mathcal{M}$, the previous
discussion produces a map $\nu_{\mathbb{R}}:\widetilde{\mathcal{M}}\rightarrow
V_{\mathbb{R}}$. The fibers of $\nu_{\mathbb{R}}$ are complex analytic
subvarieties of $\widetilde{{\mathcal{M}}}$.

######  Definition 3.7.

The generic rank of $\nu$ is
$\max_{s\in\widetilde{\mathcal{M}}}\mathrm{rank}_{s}(d\nu_{\mathbb{R}})$. It
is an even integer.

The following theorem is inspired by [KU23]; see also [Gao20] for earlier
results.

######  Theorem 3.8.

Assume that $G^{ad}$ is simple and ${\mathcal{V}}$ has no isotrivial factors.
Then the generic rank of $\nu$ is equal to
$\mu:=\min(\dim_{\mathbb{R}}{\mathcal{M}},\dim_{\mathbb{Q}}V)$.

###### Proof.

Let $d$ be the complex dimension of $\mathcal{M}$. Obviously the generic rank
$2k$ of $\nu$ cannot be larger than $\mu$. By way of contradiction, assume
that this rank is smaller than $\mu$ (hence that $k<d$ and
$k<\tfrac{1}{2}\dim_{\mathbb{Q}}(V)$). Let
$\widetilde{\Phi}:\widetilde{\mathcal{M}}\rightarrow{\mathcal{D}}$ be the lift
of the period map associated to the variation of mixed Hodge structure
${\mathcal{E}}_{\nu}$. Every point $x\in{\mathcal{J}}(V_{\mathbb{R}})$ defines
a holomorphic section $\sigma_{x}:D\rightarrow{\mathcal{D}}$ and it is clear
that
$\tilde{\Phi}^{-1}(\sigma_{x}(D)\cap\widetilde{\Phi}(\widetilde{{\mathcal{M}}}))=\nu_{\mathbb{R}}^{-1}(\\{x\\})$.

Since $\nu_{\mathbb{R}}$ has generic rank $2k$, we can find an analytic open
subset $U\subset\widetilde{\mathcal{M}}$ such that the fibers of
$\nu_{\mathbb{R}}$ above $\nu_{\mathbb{R}}(U)$ are complex analytic of complex
dimension $d-k$. Furthermore, since the Hodge locus of
$\widetilde{\mathcal{M}}$ is a countable union of strict analytic
subvarieties, we can find a point $y\in U$ which is Hodge generic by Baire’s
theorem. Let $x=\nu_{\mathbb{R}}(y)\in V_{\mathbb{R}}$; then it satisfies the
following properties:

  1. (1)

the fiber $Y_{0}=\nu_{\mathbb{R}}^{-1}(x)$ is complex analytic of complex
dimension $d-k$ ($>0$ by the assumption on $k$);

  2. (2)

$Y_{0}$ is Hodge generic (for ${\mathcal{E}}_{\nu}$) inside
$\widetilde{\mathcal{M}}$.

Let $\Delta={\mathcal{D}}\times_{\Gamma\backslash{\mathcal{D}}}\mathcal{M}$
and $W=D_{x}\times\mathcal{M}$. Then $Y_{0}$ is a complex analytic component
of the intersection $W\cap\Delta$ in ${\mathcal{D}}\times\mathcal{M}$ and we
have:

$\mathrm{codim}_{{\mathcal{D}}\times\mathcal{{M}}}(Y_{0})=\dim({\mathcal{D}})+k,\mspace{20.0mu}\mathrm{codim}_{{\mathcal{D}}\times\mathcal{{M}}}(W)=\frac{1}{2}\dim_{\mathbb{Q}}V,\mspace{20.0mu}\mathrm{codim}_{{\mathcal{D}}\times\mathcal{{M}}}(\Delta)=\dim({\mathcal{D}})\,.$

Hence (by our assumption on $k$) the following inequality is satisfied:

$\mathrm{codim}_{{\mathcal{D}}\times\mathcal{{M}}}(Y_{0})<\mathrm{codim}_{{\mathcal{D}}\times\mathcal{{M}}}(W)+\mathrm{codim}_{{\mathcal{D}}\times\mathcal{{M}}}(\Delta)~{}.$

By the Ax–Schanuel theorem for variations of mixed Hodge structures [GK24], we
conclude that the projection of $Y_{0}$ to ${\mathcal{M}}$ is contained in a
_strict_ weakly special subvariety ${\mathcal{Z}}$. That is, the period map
sends ${\mathcal{Z}}$ into (a quotient of) a _proper_ weak Mumford-Tate
subdomain of ${\mathcal{D}}$, and so the geometric monodromy group
${\mathcal{H}}_{{\mathcal{Z}}}$ of ${\mathcal{E}}_{\nu}|_{{\mathcal{Z}}}$ is a
_proper_ subgroup of ${\mathcal{G}}^{der}$. But since $Y_{0}$ is Hodge
generic, the generic Mumford-Tate group of
${\mathcal{E}}_{\nu}|_{{\mathcal{Z}}}$ is equal to ${\mathcal{G}}$; and
because $Y_{0}$ is positive-dimensional (and $G^{ad}$ is simple), we have that
$H_{\mathcal{Z}}=G^{der}$. As $\mathcal{H}_{\mathcal{Z}}$ is a normal subgroup
of ${\mathcal{G}}$ and $G^{der}$ acts non-trivially on every factor of $V$, we
conclude that ${\mathcal{H}}_{\mathcal{Z}}={\mathcal{H}}={\mathcal{G}}^{der}$,
which is a contradiction. This proves the desired result. ∎

Let $\psi:V_{\mathbb{Z}}\times V_{\mathbb{Z}}\rightarrow\mathbb{Z}$ be the
polarization form and let $\beta$ be the corresponding harmonic $(1,1)$ form
on ${\mathcal{J}}(V_{\mathbb{R}})$. Then $\beta^{r}$ determines a volume form
on ${\mathcal{J}}(V_{\mathbb{R}})$, where $r=\frac{1}{2}\dim_{\mathbb{Q}}V$
and let $\mathrm{vol}({\mathcal{J}}(V_{\mathbb{R}}))$ be the volume of
${\mathcal{J}}(V_{\mathbb{R}})$ with respect to $\beta^{r}$, which is also the
covolume of the lattice $V_{\mathbb{Z}}$ in $V_{\mathbb{R}}$141414The
quadratic form on $V_{\mathbb{R}}$ is $\psi(\cdot,J\cdot)$ where $J$ is the
almost complex structure on $V_{\mathbb{R}}$.. It is well known that the
torsion points on ${\mathcal{J}}(V_{\mathbb{R}})$ are equidistributed with
respect to $\beta^{r}$; that is, for every open subset
$U\subset{\mathcal{J}}(V_{\mathbb{R}})$ with measure-zero boundary, we have

$\left|\\{z\in U|\,n\cdot
z=0\\}\right|\underset{n\rightarrow\infty}{\sim}\frac{n^{2r}}{\mathrm{vol}({\mathcal{J}}(V_{\mathbb{R}}))}\cdot\int_{U}\beta^{r}~{}.$

The pullback form $\omega_{Betti}=\nu^{*}\beta$ to $\widetilde{\mathcal{M}}$
is translation invariant and descends to a form on $\mathcal{M}$, also known
as the _Betti form_.

######  Theorem 3.9.

We keep the same assumptions as Theorem 3.8 and we assume furthermore that
${\mathcal{V}}$ has rank $2r\leq 2\dim(\mathcal{M})=2d$. Then
$\mathrm{TL}(\nu)$ is analytically dense in $\mathcal{M}$ and equidistributed
with respect to the Betti form $\omega_{Betti}$, i.e., for every
$U\subset\mathcal{M}$ relatively compact subset with zero measure boundary and
differential form $\alpha$, we have:

$\int_{\\{x\in
U|\,n\cdot\nu(x)=0\\}}\alpha\underset{n\rightarrow\infty}{\sim}\frac{n^{2r}}{\mathrm{vol}({\mathcal{J}}(V_{\mathbb{R}}))}\cdot\int_{U}\alpha\wedge\omega_{Betti}^{r}~{}.$
(4)

###### Proof.

Let $U\subset\mathcal{M}$ be a simply connected open subset as in the theorem
and let $\nu_{\mathbb{R}}:U\rightarrow V_{\mathbb{R}}$ be the restriction of
the Betti map to $U$. By Theorem 3.8, the generic rank of $\nu_{\mathbb{R}}$
is equal to $\dim_{\mathbb{R}}V$, hence $\nu_{\mathbb{R}}$ is submersive,
outside a measure zero subset. Notice also that the preimages of
$V_{\mathbb{Q}}$ are exactly the components in $U$ of the torsion locus of
$\nu$, which shows the analytic density of $\mathrm{TL}(\nu)$.

As for the equidistribution, one can argue exactly as in the proof of [TT23,
Theorem 3.6] using the submersivity of $\nu_{\mathbb{R}}$, to prove that
$\nu_{\mathbb{R}}^{-1}(V_{\mathbb{Q}})$ is equidistributed with respect to
$\nu_{\mathbb{R}}^{*}\beta_{Betti}^{r}=\omega_{Betti}^{r}$. That is, (4) holds
for every differential form $\alpha$, which is the desired result. ∎

######  Remark 3.10.

The conclusions of Theorem 3.9 hold as long as
$\dim({\mathcal{M}})\geq\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))$ and
$\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))=k$ (i.e.,
$\mathrm{rk}(d\nu_{\mathbb{R}})=\dim_{\mathbb{Q}}(V)$). If ${\mathcal{V}}$ is
isotrivial, one may be able to check the latter condition directly; for
example, it obviously holds if $\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))=1$
and $\nu$ is not flat.

##  4\. Applications

###  4.1. The torsion locus for the Ceresa cycle

For a curve $C$ of genus $g$ and $s\in C$ a complex base point, we have the
Ceresa cycle $Z_{C}$ and associated normal function $\nu$ as described in the
introduction. In the following we work with a level $\ell\geq 3$ structure (in
the automorphic sense), so that the period map is 2:1 and stacky issues do not
enter. By the “Main Theorem” below we shall mean Theorem 3.2, though the
reader can also take it to mean Theorem 1.3 (which also applies in each
instance).

######  Corollary 4.1.

Let $\phi\colon\mathcal{M}_{g}[\ell]\to\mathcal{A}_{g}[\ell]=X$ be the usual
period map, $\mathcal{V}$ be the VHS whose fiber over
$[C]\in\mathcal{M}_{g}[\ell]$ is
$H^{3}_{\text{pr}}(J(C))(2)\cong\textstyle(\bigwedge^{3}H^{1}(C))(2)/(H^{1}(C)(1))$,
and $\nu\in\mathrm{ANF}_{{\mathcal{M}}_{g}[\ell]}({\mathcal{V}})$ be the
Ceresa normal function. Write $H_{g}\subset\mathcal{M}_{g}[\ell]$ for the
hyperelliptic locus (along which $\Phi$ is ramified). Then there is a finite
set of Shimura subvarieties $Y_{i}\subset X$ of codimension $\geq 2$, and a
finite set of subvarieties $T_{i}\subset\mathcal{M}_{g}[\ell]$ of dimension
$\leq 2g-2$ (or $3$, if $g=3$), such that if $U\subset\mathcal{M}_{g}[\ell]$
denotes the complement of $H_{g}$, $\cup T_{i}$, and $\cup\phi^{-1}(Y_{i})$,
then $\mathrm{TL}(\nu)\cap U(\mathbb{C})$ is a countable set of points.

If one is interested in the locus where the _cycle itself_ is torsion, then in
the above one can remove any points not in $U(\bar{\mathbb{Q}})$. (One can
make sense of this over $\mathcal{M}_{g}[\ell]$ by choosing the canonical
class as “base point”.)

###### Proof.

The first point is that ${\mathcal{F}}^{-1}{\mathcal{V}}\neq{\mathcal{V}}$
because ${\mathcal{V}}$ is irreducible of (Hodge-theoretic) level three. The
remaining assumptions of the Main Theorem are satisfied by the Torelli theorem
and Ceresa’s theorem (that the Ceresa normal function is nontorsion). The
(co)dimension bounds are from a result of [CP95]. Finally, if a
non-$\bar{\mathbb{Q}}$-point is contained in the torsion locus of the family
of cycles themselves, then so is its $\bar{\mathbb{Q}}$-spread; this does not
apply to the normal function. ∎

######  Remark 4.2.

The rank of the intermediate Jacobian bundle ${\mathcal{J}}(\mathcal{V})$ is
$\tfrac{1}{2}(\binom{2g}{3}-2g)$, which exceeds $3g-3$ for every $g\geq 3$. So
in fact both (i) and (ii) hold in Theorem 3.2. While we don’t need (ii), the
strict inequality means that zero-dimensional components of the torsion locus
are atypical so that one _expects_ only finitely many.

######  Example 4.3.

We know from [LS23] that in genus 3, there is a ball quotient
$Y\subset\mathcal{A}_{3}$ ruled by rational curves, countably many of which
are contained in $\Phi(\mathrm{TL}_{\text{pos}}(\nu))$ for the Ceresa normal
function. The torsion orders here are unbounded and the torsion locus is
Zariski dense, as predicted by Theorem 3.9 and Remark 3.10 (as $\nu|_{Y}$
takes values in the Jacobian of an isotrivial VHS of rank 2); while the fact
that this dense union is contained in (the preimage of) a Shimura subvariety
of ${\mathcal{A}}_{3}$ illustrates the main theorem.

###  4.2. Normal functions over locally symmetric spaces

In general there are no nontrivial admissible normal functions on (most)
locally symmetric varieties as a consequence of Raghunathan’s theorem, cf.
[KK18]. Instead, one passes to a finite branched cover (or, deleting the
branch locus, an étale neighborhood) and considers normal functions there with
values in the basechange of a Hermitian homogeneous VHS
${\mathcal{V}}=\widetilde{{\mathcal{V}}}^{\lambda}\\{\tfrac{a}{2}\\}$ over
$X=\Gamma\backslash D=\Gamma\backslash G_{\mathbb{R}}/K$. Such variations
arise from an irreducible finite-dimensional representation of
$G_{\mathbb{R}}$ of highest weight $\lambda$ (which we assume has descent to
$\mathbb{Z}$); for $G$ of Cartan type $A_{n}$, $D_{\text{odd}}$, or $E_{6}$,
the generalized half-twist $\\{\tfrac{a}{2}\\}$ is needed to make the Hodge
numbers $(p,q)$ integral. See [op. cit.] for details. Of course, we can always
apply a Tate twist to put ${\mathcal{V}}$ into negative weight.

######  Corollary 4.4.

Let $\phi\colon{\mathcal{M}}\to X$ be an étale neighborhood of an irreducible
locally symmetric variety of dimension $>1$. The basechange by $\phi$ of an
irreducible Hermitian homogeneous $\mathbb{Z}$-VHS ${\mathcal{V}}$ (as above)
tautologically has $\phi$ as its period map. Assume that ${\mathcal{V}}$ has
negative weight and (Hodge-theoretic) level $>1$, and consider a nontorsion
admissible normal function $\nu\in\mathrm{ANF}_{\mathcal{M}}(\mathcal{V})$.

Then
$\mathcal{T}:=\overline{\mathrm{TL}_{\text{pos}}(\nu)}^{\text{Zar}}\subset\mathcal{M}$
is a proper algebraic subvariety. Moreover, any component of $\mathcal{T}$ is
either contained in the preimage of a proper locally symmetric subvariety of
$X$, or is a component of $\mathrm{TL}_{\text{pos}}(\nu)$.

###### Proof.

Obviously ${\mathcal{V}}$ is non-isotrivial and $\phi$ is quasi-finite, and
level $>1$ forces ${\mathcal{F}}^{-1}{\mathcal{V}}\neq{\mathcal{V}}$. So this
is an immediate consequence of the Main Theorem. ∎

######  Remark 4.5.

The $\tilde{{\mathcal{V}}}^{\lambda}\\{\tfrac{a}{2}\\}$ admitting
infinitesimal normal functions (a necessary condition for the existence of
nontorsion $\nu$) have been classified in [KK18, Ker24]. For $\dim(X)>1$, only
weights $-1$ and $-2$ occur, i.e., $r=0$ (“usual” $K_{0}$-cycles) or $1$
($K_{1}$-cycles) in the sense of Remark 2.10. At least for $D$ a tube domain,
one can do a case-by-case check that when ${\mathcal{V}}$ has level $>1$, we
always have $\mathrm{rk}({\mathcal{J}}({\mathcal{V}}))>\dim(D)$, so that both
(i) and (ii) hold in Theorem 3.2.

######  Example 4.6.

(i) The Ceresa normal function over ${\mathcal{M}}_{3}[\ell]$ is the obvious
example, but the point of Corollary 4.4 is to go beyond Ceresa. So here are
two more:

(ii) The Prym-Ceresa normal function over Faber’s moduli space
$\mathcal{R}^{3}_{4}$ of 3:1 étale covers $\tilde{C}\overset{\Phi}{\to}C$ of
genus $4$ curves. Here the period map is to the moduli space of Weil abelian
6-folds, given by $A_{\Phi}:=J(\tilde{C})/J(C)$; and the cycle is the
projection of Ceresa for $\tilde{C}$ to $A$. See [KK18, §3.5] for details.

(iii) The Collino normal function [Col97] over an étale neighborhood of
${\mathcal{M}}_{2}$ (namely, the 15-to-1 cover of the locus where
$C\in{\mathcal{M}}_{2}$ has only one involution, parametrized by triples
$(C,p,q)$ with $p,q\in C$ distinct Weierstrass points.) Here $\mathcal{V}$ has
weight $-2$ and corresponds to a $K_{1}$/‘higher’ cycle in the hyperelliptic
Jacobian.

###  4.3. The Fano normal function

Similar in spirit to the Ceresa normal function is the _Fano normal function_
$\nu$ over the (10-dimensional) moduli space ${\mathcal{M}}$ of cubic
threefolds. Given a threefold ${\mathbb{T}}$ and fixing a line
$L_{0}\subset{\mathbb{T}}$, the Abel-Jacobi images
$\mathrm{AJ}_{{\mathbb{T}}}(L-L_{0})\in{\mathcal{J}}(H^{3}({\mathbb{T}})(2))=:J({\mathbb{T}})$
(over all lines $L\subset{\mathbb{T}}$) sweep out a surface
$F\subset{\mathbb{T}}$. Setting $F^{-}:=\imath(F)$, the cycle
$Z_{{\mathbb{T}}}:=F-F^{-}$ is homologous to zero, and its Abel-Jacobi class
$\mathrm{AJ}_{J({\mathbb{T}})}(Z_{{\mathbb{T}}})\in{\mathcal{J}}(H^{5}_{pr}(J({\mathbb{T}}))(3))$
(resp. the HS $H^{5}_{pr}(J({\mathbb{T}})(3))$) defines $\nu$ (resp.
${\mathcal{V}}$) at $[{\mathbb{T}}]\in{\mathcal{M}}$. Then
$\phi\colon{\mathcal{M}}\to{\mathcal{A}}_{5}$ is an embedding, ${\mathcal{V}}$
is irreducible level 5 (whence
${\mathcal{F}}^{-1}{\mathcal{V}}\neq{\mathcal{V}}$), and it is shown in
[CNP12] that $\nu\in\mathrm{ANF}_{{\mathcal{M}}}({\mathcal{V}})$ is
nontorsion. So the conclusion of Theorem 1.3 holds.

Even less is known about the torsion locus for the Fano normal function than
for Ceresa. However, if we extend $\nu$ to the closure
$\overline{{\mathcal{M}}}$ of $\phi({\mathcal{M}})$, then
$\overline{{\mathcal{M}}}$ contains the hyperelliptic Jacobian locus $H_{5}$
and the extended normal function is torsion (in fact zero) on this locus [op.
cit.].

###  4.4. Fields of definition of the torsion locus

In this section, we would like to determine the field of definition of the
“isolated” components of the torsion locus of a normal function. We first
recall a general result about local systems on algebraic varieties.

Let $S$ be a smooth complex quasi-projective algebraic variety and let
${\mathbb{V}}$ be a complex local system on $S$. For any subvariety $Y\subset
S$, we denote by ${\mathcal{H}}_{Y}$ the monodromy group of $Y$.151515Defined
as the neutral component of the monodromy representation of ($\pi_{1}$ of) the
normalization of $Y$.

######  Definition 4.7.

Let $Y\subseteq S$ be a closed algebraic subvariety. We say that $Y$ is weakly
non-factor for ${\mathbb{V}}$, if there is no $Z\supset Y$ for which
${\mathcal{H}}_{Y}$ is a strict normal subgroup of ${\mathcal{H}}_{Z}$.

######  Theorem 4.8 ([KOU23, Theorem 2.6]).

Let $S$ be a smooth quasi-projective variety defined over a number field
$K\subset\mathbb{C}$ and let ${\mathbb{L}}$ be a complex local system on
$S(\mathbb{C})^{an}$, whose associated flat bundle is also defined over $K$.
Then any weakly special, weakly non-factor subvariety for ${\mathbb{L}}$ is
defined over $\overline{\mathbb{Q}}$ and its Galois conjugates are also weakly
non-factor, weakly special subvarieties.

This leads at once to the main result of this subsection.

######  Theorem 4.9.

Suppose we have:

  * •

$\mathcal{M}$ a smooth quasi-projective variety defined over a number field
$K$;

  * •

${\mathcal{V}}$ a non-isotrivial $\mathbb{Q}$-simple polarizable variation of
Hodge structure of negative weight on ${\mathcal{M}}$, with associated flat
bundle defined over $K$, $\mathbb{Q}$-simple derived generic Mumford-Tate
group $G$, and associated period map quasi-finite;

  * •

$\nu\in\mathrm{ANF}_{{\mathcal{M}}}({\mathcal{V}})$ a nontorsion admissible
normal function arising from a family of cycles defined over $K$; and

  * •

$Y$ a positive-dimensional component of the torsion locus of $\nu$.

Then either $Y$ is contained in a component of the Hodge locus for
${\mathcal{V}}$ or it is defined over $\overline{\mathbb{Q}}$. In general, the
components of the torsion locus which are weakly non-factor are defined over
$\overline{\mathbb{Q}}$ and their Galois conjugates are again components of
the torsion locus.

###### Proof.

The hypothesis on the normal function implies that the flat bundle associated
to the extension $0\to{\mathcal{V}}\to{\mathcal{E}}_{\nu}\to\mathbb{Q}\to 0$
is defined over $K$. Thus the associated local system
${\mathbb{E}}_{\nu,\mathbb{C}}$ satisfies the hypothesis of Theorem 4.8. By
[And92], the geometric monodromy group of ${\mathbb{V}}$ is normal in $G$; as
${\mathbb{V}}$ is non-isotrivial, they are therefore equal. Since $\nu$ is
nontorsion, we conclude that the geometric monodromy group ${\mathcal{H}}$ of
${\mathbb{E}}_{\nu}$ is an extension of $G$ by $V$ ($=$ a fiber of
${\mathbb{V}}$).

A component $Y$ of the torsion locus has geometric monodromy group
${\mathcal{H}}_{Y}\leq G$, hence (as ${\mathcal{H}}_{Y}<{\mathcal{H}}$) it is
weakly special. Furthermore, if $Y$ is not contained in a component of the
Hodge locus for ${\mathcal{V}}$, its derived generic mixed Mumford-Tate group
is isomorphic to $G$. So by [And92] we must have
${\mathcal{H}}_{Y}\trianglelefteq G$, and as $G$ is simple and the hypotheses
force ${\mathcal{H}}_{Y}$ to be nontrivial, ${\mathcal{H}}_{Y}=G$. By non-
isotriviality and irreducibility of ${\mathbb{V}}$ on ${\mathcal{M}}$, $G$ is
its own normalizer in ${\mathcal{H}}$. Thus $Y$ is weakly non-factor and we
conclude by Theorem 4.8. ∎

######  Corollary 4.10.

The (finitely many) positive dimensional components of the torsion locus of
the Ceresa normal function on $\mathcal{M}_{g}$ with Mumford-Tate group equal
to $\mathrm{GSp}_{2g}$ are defined over $\overline{\mathbb{Q}}$.

## References

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titleAdopting Model Driven Software Development in Industry – A
Case Study at Two Companies

booktitle2019 ACM/IEEE 22nd International Conference on Model
Driven Engineering Languages and Systems Companion (MODELS-C)
titleInsights for Improving Diagram Editing Gained from an
Empirical Study

Pearson Education

titleEMF: eclipse modeling framework

Association for Computing Machinery

guidelines, grounded theory, software engineering, review
Grounded Theory (GT) has proved an extremely useful research approach in
several fields including medical sociology, nursing, education and management
theory. However, GT is a complex method based on an inductive paradigm that
is fundamentally different from the traditional hypothetico-deductive
research model. As there are at least three variants of GT, some ostensibly
GT research suffers from method slurring, where researchers adopt an
arbitrary subset of GT practices that are not recognizable as GT. In this
paper, we describe the variants of GT and identify the core set of GT
practices. We then analyze the use of grounded theory in software
engineering. We carefully and systematically selected 98 articles that
mention GT, of which 52 explicitly claim to use GT, with the other 46 using
GT techniques only. Only 16 articles provide detailed accounts of their
research procedures. We offer guidelines to improve the quality of both
conducting and reporting GT studies. The latter is an important extension
since current GT guidelines in software engineering do not cover the
reporting process, despite good reporting being necessary for evaluating a
study and informing subsequent research.

booktitleProceedings of the 38th International Conference on
Software Engineering
seriesICSE '16
titleGrounded Theory in Software Engineering Research: A Critical
Review and Guidelines
Austin, Texas

* Can't use 'location' + 'address'

booktitleHuFaMo@ MoDELS
titleComparing Reuse Mechanisms for Model Transformation
Languages: Design for an Empirical Study.

booktitleInternational Conference on Theory and Practice of Model
titleRequirements engineering in model-transformation development:
An interview-based study

ACM New York, NY

titleModel Transformation Testing and Debugging: A Survey
journaltitleACM Computing Surveys (CSUR)

titleDomain-specific language design requires feature
journaltitleJournal of Computing and Information Technology

Springer International Publishing

In this chapter we introduce grounded theory methodology and methods. In
particular we clarify which research questions are appropriate for a grounded
theory study and give an overview of the main techniques and procedures, such
as the coding procedures, theoretical sensitivity, theoretical sampling, and
theoretical saturation. We further discuss the role of theory within grounded
theory and provide examples of studies in which the coding paradigm of
grounded theory has been altered in order to be better suitable for
applications in mathematics education. In our exposition we mainly refer to
grounded theory techniques and procedures according to Strauss and Corbin
(Basics of qualitative research: Grounded theory procedures and techniques,
Sage Publications, Thousand Oaks, 1990), but also include other approaches in
the discussion in order to point out the particularities of the approach by
Strauss and Corbin.

booktitleCompendium for Early Career Researchers in Mathematics
titleAn Introduction to Grounded Theory with a Special Focus on
Axial Coding and the Coding Paradigm

titleStrukturgleichungsmodellierung: Eine anwendungsorientierte
Einführung in die Kausalanalyse mit Hilfe von AMOS, SmartPLS und SPSS

* Can't use 'date' + 'year'

An oft-cited reason for lack of adoption of model-driven engineering (MDE)
is poor tool support. However, studies have shown that adoption problems are
as much to do with social and organizational factors as with tooling issues.
This paper discusses the impact of tools on MDE adoption and places tooling
within a broader organizational context. The paper revisits previous data on
MDE adoption (19 in-depth interviews with MDE practitioners) and re-analyzes
the data through the specific lens of MDE tools. In addition, the paper
presents new data (20 new interviews in two specific companies) and analyzes
it through the same lens. The key contribution of the paper is a taxonomy of
tool-related considerations, based on industry data, which can be used to
reflect on the tooling landscape as well as inform future research on MDE

booktitleModel-Driven Engineering Languages and Systems
seriesMODELS 2013
titleIndustrial Adoption of Model-Driven Engineering: Are the
Tools Really the Problem?

Springer Berlin Heidelberg

The standardized QVT Relations language, one cornerstone of Model-Driven
Architecture (MDA), has not yet gained widespread use in practice, not least
due to missing tool support in general and inadequate debugging support in
particular. Transformation engines interpreting QVT Relations operate on a
low level of abstraction, hide the operational semantics of a transformation
and scatter metamodels, models, QVT code, and traces across different
artifacts. We propose a model-based debugger representing QVT Relations on
bases of TROPIC, a model transformation framework which utilizes a variant of
Colored Petri Nets (CPNs) providing an explicit runtime model and a
homogenous view on all artifacts of a transformation.

booktitleModel Driven Engineering Languages and Systems
titleReviving QVT Relations: Model-Based Debugging Using Colored
Petri Nets
Berlin, Heidelberg