# Slow Light Frequency Reference Cavities – Proof of Concept for Reducing the
Frequency Sensitivity Due to Length Fluctuations

Sebastian P. Horvath  Chunyan Shi  David Gustavsson  Andreas Walther  Adam
Kinos  Stefan Kröll  Lars Rippe<EMAIL_ADDRESS>Department of Physics, Lund
University, P.O. Box 118, SE-22100 Lund, Sweden.

(August 27, 2024)

###### Abstract

A key limiting factor of present day state-of-the-art laser frequency
stabilization using Fabry-Pérot cavities are length changes due to thermo-
mechanical noise, originating from, for example, Brownian motion. We present a
laser-frequency stabilization concept using an optical cavity with a strong
slow-light effect to reduce the impact of cavity length changes on the
frequency stability. The resulting noise-reduction factor is proportional to
the ratio between the light phase and group velocities in the highly
dispersive cavity spacer. We experimentally demonstrate a proof-of-principle
implementation of this laser-frequency stabilization technique using a rare-
earth doped crystalline cavity spacer in conjunction with semi-permanent
spectral tailoring to achieve precise control of the dispersive properties of
the cavity. A reduction in frequency sensitivity of four orders of magnitude
was achieved compared to the same setup in the absence of the slow-light
effect.

##  I Introduction

Precision frequency stabilized lasers are a fundamental component of modern
metrology. They serve as a local oscillator in optical atomic clocks Ludlow
_et al._ (2015); Nicholson  _et al._ (2015); Ushijima  _et al._ (2015), and
are at the heart of interferometer based tests of fundamental physics, such as
gravitational wave detection LIGO Scientific Collaboration and Virgo
Collaboration (2016) and tests of relativity Chou  _et al._ (2010); Eisele
_et al._ (2009). State-of-the-art frequency stabilization is performed using
Fabry-Pérot (FP) cavities in conjunction with the Pound-Drever-Hall (PDH)
locking technique. In order to achieve good performance, thermo-mechanical
isolation of the cavity, along with a careful material selection, is
imperative. Typical cavities, with a length on the order of $10\;\mathrm{cm}$,
have a fractional stability of $10^{-15}$ Ludlow  _et al._ (2007); Webster
_et al._ (2008); Chen  _et al._ (2014).

Present day leading stabilities are achieved by utilizing long cavities ($\sim
50\;\mathrm{cm}$), with a fractional stability of $8\times 10^{-17}$ Häfner
_et al._ (2015), or cryogenically cooling single-crystal silicon cavities,
with a fractional stability of $4\times 10^{-17}$ Matei  _et al._ (2017). Such
designs have been shown to operate at the frequency stability limit imposed by
the thermo-mechanical noise in the mirror coatings Matei  _et al._ (2017);
Numata  _et al._ (2004). In addition to utilizing long cavities Häfner  _et
al._ (2015); Jiang  _et al._ (2011), or operating single-crystal cavities at
cryogenic temperatures Seel  _et al._ (1997); Matei  _et al._ (2017); Kessler
_et al._ (2012), investigations to reduce the thermo-mechanical noise
originating from the mirror coatings has focused on increasing the size of the
optical mode Amairi  _et al._ (2013), as well as utilizing single-crystal
mirrors with low dielectric losses Cole  _et al._ (2013); Brückner  _et al._
(2010).

While the fractional stability has been considerably improved over the last
decade by refining cavity designs, the thermo-mechanical noise limit of FP
cavities has prompted investigations into alternative stabilization techniques
including cavity quantum-electrodynamic systems Bishof  _et al._ (2013);
Christensen  _et al._ (2015); Norcia and Thompson (2016) and locking to
spectral holes in rare-earth ion doped crystals Sellin  _et al._ (1999);
Julsgaard  _et al._ (2007); Thorpe  _et al._ (2011); Chen  _et al._ (2011);
Leibrandt  _et al._ (2013); Cook  _et al._ (2015); Gobron  _et al._ (2017);
Galland  _et al._ (2020). In this work we demonstrate a new locking technique
that utilizes the slow-light effect to reduce the sensitivity to frequency
changes originating from thermo-mechanical noise.

The cavity is constructed by using a rare-earth ion doped crystal with mirror
coatings directly deposited on the crystal surfaces. By spectral tailoring, a
narrow semi-permanent frequency transmission window is burned into the
inhomogeneously broadened rare-earth ion ensemble. The narrow transmission
window leads to a gradient of the index of refraction across the spectral
window, which reduces the speed of light in the cavity by up to four orders of
magnitude (see Fig. 1(a)). Such an effect has been previously demonstrated to
lead to a narrowing of the cavity-linewidth by a factor comparable to the
reduction of the speed of light Sabooni  _et al._ (2013).

##  II Results

Figure 1: (a) Schematic of a spectral hole and the resulting rapidly varying
dispersion across this frequency region. (b) Schematic of the crystal
geometry, showing the burning beam perpendicular to the locking beam. (c) The
experimental layout of the core laser-locking setup, with representative
signals observed at key locations in the locking loop. The EOM modulation was
disabled to record the cavity transmission and reflection signals. The steep
gradient visible in the error signal at frequencies where there is no peak in
the reflection signal originates from the large modulation used
($1.5\;\mathrm{MHz}$, which is almost half of the cavity mode spacing). For a
detailed description of these signals in the context of PDH locking, see Ref.
Black (2000).

The slow-light effect increases the cavity round-trip time, which,
heuristically, can be thought of as an increase of the effective cavity length
from $6\;\mathrm{mm}$ to $60\;\mathrm{m}$. Therefore, this technique enables
designs with a very large effective length without the mechanical degradation
associated with a physically longer cavity. More concretely, assuming an
inhomogeneously broadened ensemble with an absorption $\alpha_{0}$ and a
square spectral transmission window, as indicated in Fig. 1, the cavity free-
spectral range (FSR) has the form

$\nu_{\text{FSR}}=\frac{v_{g}}{2L}=\frac{1}{2L}\frac{\pi^{2}\Gamma}{\alpha_{0}}=\frac{c}{2L}\frac{1}{n+\nu\frac{dn}{d\nu}}.$
(1)

Here, $v_{g}$ is the group velocity of light, $L$ is the cavity length,
$\Gamma$ is the transparency window width, $c$ is the speed of light, $n$ is
the refractive index, and $\nu$ the frequency of the light; a derivation of
this relation is provided in the supplemental material (SM). Note that the
above equation is only valid at the center of the transparency window;
nevertheless, since the dispersive gradient is close to constant across a
large portion of the transparency window, Eq. (1) is a good approximation for
all cavity modes not adjacent to the edges.

The slow-light effect not only reduces the cavity FSR, but equally scales the
cavity linewidth $\delta\nu_{\text{cav}}$. Assuming a high-finesse cavity with
only small mirror losses Siegman (1986), one finds

$\delta\nu_{\text{cav}}=\frac{\nu_{\text{FSR}}}{\mathcal{F}}=\frac{v_{g}}{2L}\left(\frac{T_{1}+T_{2}+2\alpha_{c}L}{2\pi}\right).$
(2)

Here, $\mathcal{F}$ is the cavity finesse, $\alpha_{c}$ is the residual
absorption at the center of the transparency window due to off-resonant
excitation, while $T_{1}$ and $T_{2}$ are the transmission of the first and
second mirror. From Eqs. (1) and (2) it is apparent that the finesse
$\mathcal{F}$ is independent of the slow-light effect.

In addition to a narrowing of the cavity modes, the slow-light effect also
results in an insensitivity to cavity-length changes. By considering the
relative frequency shift due to a cavity length change, one obtains

$\frac{d\nu}{\nu}=-\frac{dL}{L}\frac{v_{g}}{c}.$ (3)

The same scaling with respect to $v_{g}$ is true for the frequency noise
amplitude $\sqrt{S_{\nu}}$ resulting from a corresponding displacement noise
amplitude. As a consequence, all optical length-change noise sources,
including from the mirror coatings, are reduced by a factor corresponding to
the group velocity divided by the speed of light.

The experimental proof-of-principle implementation utilized a Pr3+:Y2SiO5
crystal as a spacer material for an FP cavity with mirror coatings applied
directly to two, almost parallel, flat crystal surfaces. While a better
ultimate stability could be achieved by utilizing a material with a narrower
optical homogeneous linewdith and slower hyperfine relaxation rate, the
present demonstration of this locking technique was implemented in Pr3+:Y2SiO5
due to the well established spectral-tailoring routines available for this
material Nilsson  _et al._ (2004). An optical transmission window was burned
at $494.723\;\mathrm{THz}$, on resonance with the 3H${}_{4}\to^{1}$D2
transition of Pr3+ ions. The geometry utilized for spectral holeburning with
respect to the crystal orientation and the mirror coatings is shown in Fig.
1(b).

Figure 2: (a) Representative cavity modes observed at three distinct
frequencies relative to the inhomogeneously broadened rare-earth ion
absorption profile. Colors designate the cavity frequency: blue is at maximum
absorption, orange is detuned from the maximum absorption by 3 GHz, and green
is far away from the absorption peak, detuned by $\simeq 5.3\;\mathrm{THz}$,
which is sufficient to ensure that the slow-light effect is negligible. The
inhomogeneous line, shown in grey, has an FWHM of $\simeq 5\;\mathrm{GHz}$,
determined by an absorption measurement in a portion of the crystal that was
not mirror coated. The frequencies of the two insets relative to the
inhomogeneous profile are indicated using the blue and orange vertical bars.
(b) The frequency sensitivity to cavity length changes at the previously
enumerated cavity frequencies (see Fig. 1 for the experimental configuration).
The blue and orange measurements were performed with laser locked to the slow-
light cavity, while the green measurements were performed by directly
measuring the shift of the cavity modes due to the induced length changes. See
text for further details. The indicated group velocities were determined from
the change of the cavity FSR. The frequency sensitivity can be seen to scale
with $v_{g}$; any deviation from an exact correspondence is attributed to a
deviation from an optimally square spectral transparency window.

Figure 1(a) schematically shows the spectral transmission window and the
corresponding change in refractive index. The gradient of the dispersion leads
to, in accordance with Eqs. (1) and (2), a reduction in both the cavity mode
spacing and the linewidth $\delta\nu_{\text{cav}}$. The group velocity of
light propagating through the transparency window scales proportionally with
the absorption, $\alpha_{0}$, outside the transparency window; therefore, the
cavity mode linewidth is strongly dependent on the frequency of the spectral
hole with respect to the Pr3+ inhomogeneous line. This is illustrated in Fig.
2(a), which shows the transmission cavity modes for three scenarios with
different values for $\alpha_{0}$. Specifically, a spectral transmission
window with a width of $18\;\mathrm{MHz}$ was burned at the center of the Pr3+
ensemble, as well as detuned from the center by $3\;\mathrm{GHz}$. This
yielded a full width at half maximum (FWHM) cavity mode linewidth of $275\pm
2\;\mathrm{kHz}$ and $745\pm 14\;\mathrm{kHz}$, respectively. In contrast, the
cavity linewidth obtained when the laser was detuned by $\simeq
5.3\;\mathrm{THz}$, where we expect any absorption from Pr3+ ions to be
negligible, was $1317\pm 2\;\mathrm{MHz}$.

In order to test the reduction in frequency sensitivity to cavity length
changes, and consequently corroborate Eq. (3), a small relative angle between
the mirror coated sides of the Pr3+:Y2SiO5 crystal was exploited. This angle
is schematically shown in Fig. 1(b), and amounted to a relative slope of
$41.0\pm 0.4\;\mathrm{\mu rad}$. The cavity input beam was equipped with a
motorized mirror mount, such that, if the input beam was translated along the
slope, this corresponded to a calibrated cavity length change. Consequently,
by translating the input beam while monitoring the resulting cavity frequency
shift for the three distinct slow-light regimes described above (see Fig.
2(a)), it was possible to probe the impact of the slow-light effect on the
frequency shift as a function of length change. More specifically, in order to
determine the cavity mode frequency shift when the laser was on resonance with
the Pr3+:Y2SiO5 inhomogeneous profile, as well as when it was detuned by
$3\;\mathrm{GHz}$, the PDH technique was employed Black (2000). A pre-
stabilized laser was passed through an acousto-optic modulator (AOM) followed
by a PDH lock to the slow-light cavity, as shown in Fig. 1(c). The PDH
feedback signal was applied to the AOM to create a closed loop. Monitoring the
feedback signal thus provided a high-resolution measurement of the stability
and drift attainable by the slow-light cavity. To ensure the locking beam
propagated within a spectral hole throughout the translation, a spatial region
of the crystal surrounding the locking beam was spectrally tailored using a
large burning beam perpendicular to the locking beam.

This measurement was repeated for three different translation step sizes. For
each step size, the experiment was cycled 10 times, and the resulting
frequency shift averaged. By performing this measurement both at the center of
the inhomogeneous Pr3+ ensemble, as well as detuned by $3\;\mathrm{GHz}$, the
dependence of the induced frequency change could be determined with respect to
the slow-light effect. These measurements are shown in Fig. 2(b) for,
respectively, the group velocities of $v_{g}\simeq c/7500$ and $v_{g}\simeq
c/2790$. The frequency shift for each step size was also determined in the
absence of the slow-light effect by detuning the laser by $\simeq
5.3\;\mathrm{THz}$. However, due to the large increase in the cavity FSR,
bandwidth limitations precluded the use of the PDH method outlined above. The
frequency change was instead determined by directly measuring the cavity modes
in transmission before and after each translation step (see SM for additional
details). These measurements are also included Fig. 2(b), denoted by the speed
of light in Y2SiO5 of $v_{g}=c/1.8$. We note that this last measurement was
also used to calibrate the angle of the crystal.

This measurement shows that the frequency shift due to length changes scales
inversely to the group velocity, and achieves a reduction in length-change
sensitivity of almost four orders of magnitude at the center of the Pr3+
ensemble. From this measurement, the minimal drift rate of our locking setup
was found to be $0.55\pm 0.06\;\mathrm{kHz}/s$. The principal limiting factors
for these experiments, as will be discussed in detail below, originated from
two distinct sources of atomic drift. Furthermore, the locking loop included a
single-mode fiber which did not include any Doppler noise cancellation. Figure
3 shows the overlapped Allan deviation after linear drift subtraction, with a
noise floor of $\sigma_{y}(\tau)=4.6\times 10^{-13}$. One possible explanation
for this limit could be the polarization-maintaining fiber that is part of the
locking loop (see Fig. 1) which for a 25 m fiber has been shown to lead to
frequency broadening up to $\sim 1.2$ kHz Ma  _et al._ (1994).

To further analyze the drift, the laser was locked to different cavity modes
across the transmission window. This revealed two unique sources for the
frequency drift: off-resonant excitation, and hyperfine cross-relaxation.
Specifically, the circulating light in the cavity will off-resonantly excite
ions at the high and low frequency edges of the spectral transmission window,
which then have a non-negligible probability of relaxing to a different
hyperfine level. Ions excited from the high-energy edge in Fig. 4(a)
predominantly reside in the $\ket{\pm 5/2g}$ state. When excited they can
decay to the $\ket{\pm 1/2g}$ or $\ket{\pm 3/2g}$ states. This transfer can,
effectively, be considered as a shift of the spectral transmission window. As
a consequence, the dispersion curve within the spectral transmission window
translates to higher frequencies, which also shifts the cavity modes towards
higher frequencies. This process is counteracted by the off-resonant
excitation of ions at the low energy edge of the transmission window in Fig.
4(a). Here, ions occupy the $\ket{\pm 1/2g}$ state (see Fig. 4(b)), but may,
when they are excited, fall down in either the $\ket{\pm 3/2g}$ or $\ket{\pm
5/2g}$ states, effectively leading to a negative shift of the cavity modes.
These two cases of positive and negative frequency drift are indicated by the
blue and orange shaded regions in Fig. 4(c).

The second source of drift stems from the hyperfine relaxation between the
$\ket{\pm 1/2g}$ state and the $\ket{\pm 3/2g}$ state. Relaxation from
$\ket{\pm 1/2g}$ to $\ket{\pm 3/2g}$ results in absorption at low frequencies
of the spectral transmission window; consequently, the cavity modes drift to
higher frequencies. This is indicated by the shaded green region in Fig. 4(a).
Figure 4(c) shows the drift with respect to different locking beam
intensities. The hyperfine relaxation rate is independent of the locking beam
intensity, and therefore leads to a constant positive offset in the drift
rate. A dependence between the locking intensity and the off-resonant
excitation induced drift is clearly established.

Figure 3: Overlapped Allan deviation with linear drift subtracted. The noise
floor was found to be $\sigma_{y}(\tau)=4.6\times 10^{-13}$.

At high locking intensities, the drift becomes non-linear due to a re-burning
of the spectral transmission window by the locking beam. From the locking
point at $5.89\;\mathrm{MHz}$ in Fig. 4(c) we can conclude that, provided the
hyperfine relaxation can be mitigated, there exists a zero drift point close
to the center of the spectral window for which the effects of the off-resonant
excitation cancel. This can be achieved by utilizing Eu3+:Y2SiO5, for which
persistent spectral holes have been observed for up to $50\;\mathrm{days}$
after creation Oswald  _et al._ (2018), such that the drift contribution from
hyperfine relaxation would be negligible for typical locking timescales.

Even when locked to a cavity mode that optimally cancels drift due to off-
resonant excitation, any absolute drift in the resonance frequency of the ions
is nevertheless a source of noise for a locked laser. In particular, ambient
pressure and temperature changes have been determined to impact spectral-hole
locking techniques Thorpe  _et al._ (2013), and are likely also key limiting
sources of drift for the presented locking technique. However, to our
knowledge, this ultimate atomic limit has yet to be achieved by any of the
rare-earth ion based laser frequency stabilization techniques presently under
investigation Leibrandt  _et al._ (2013); Galland  _et al._ (2020).

Figure 4: (a) Changes in absorption in the spectral transmission window during
locking and the resulting drift. Blue and orange shaded regions indicate off-
resonant excitation and subsequent relaxation into the transmission window.
The green shaded region is the result of direct hyperfine level-cross
relaxation and occurs primarily at the low frequency side of the transmission
window. (b) The energy level diagram of the Pr3+:Y2SiO5 ground and excited
state; the green arrow indicates the most likely hyperfine cross-relaxation
pathway. (c) The locked laser drift rate, relative to the stabilized reference
laser, when locked at different frequencies with respect to the center of the
spectral transmission window. The non-linear effect for the higher locking
intensity data is attributed to a re-burning of the spectral transmission
window by the locking beam.

##  III Discussion

We now discuss several factors which impact the performance of a slow-light
cavity. In order to describe these factors, it is necessary develop several
prerequisites which are interesting in their own right. The integrated off-
resonant absorption at the center of a spectral transmission window can be
approximately expressed as

$\alpha_{c}\simeq\frac{2}{\pi}\frac{\Gamma_{h}}{\Gamma}\alpha_{0}.$ (4)

Here, $\alpha_{0}$ is the absorption outside the transmission window,
$\Gamma_{h}$ is the homogeneous linewidth of the ions, and $\Gamma$ is the
transmission window width Nilsson (2005). Substituting Eq. (4) into Eq. (2),
and using the expression for $v_{g}$ in terms of $\alpha_{0}$ and $\Gamma$
given by Eq. (1), it is straightforward to show that the cavity linewidth
takes the form

$\delta\nu_{\text{cav}}\simeq\Gamma_{h}+\frac{\pi\Gamma(T_{1}+T_{2})}{4\alpha_{0}L}.$
(5)

Considering, for simplicity, the case of an impedance matched cavity, that is,
imposing the requirement that $T_{1}=2\alpha_{c}L$, and an output mirror
transmission approaching zero, the above equation reduces to

$\delta\nu_{\text{cav}}\simeq 2\Gamma_{h}.$ (6)

Interestingly, in this scenario the linewidth is, to first order, independent
of the cavity length, absorption coefficient, and the transmission window
width (provided the cavity mirrors are adjusted to maintain impedance
matching); the linewidth is entirely determined by the homogeneous linewidth
of the active resonant absorber ions.

Equation (5) very directly shows the importance of utilizing a material with a
narrow homogeneous linewidth $\Gamma_{h}$. In addition to resolving the issue
of hyperfine cross-relaxation, Eu3+:Y2SiO5 would reduce the homogeneous
linewidth to $200\;\mathrm{Hz}$ Equall  _et al._ (1994) compared to
$2.5\;\mathrm{kHz}$ in Pr3+:Y2SiO5 Equall  _et al._ (1995). This would allow
for cavity modes that are between 1-2 orders of magnitude narrower, which, as
the shot noise limited resolution for a slow-light cavity is the same as for a
traditional cavity, would in turn enable higher stability with significantly
lower locking power Black (2000). Moreover, such a narrowing of the
homogeneous linewidth would yield an order of magnitude improvement of the
slow-light effect compared to the here presented experiments.

A corollary of Eq. (6) is that for an impedance matched cavity the narrowest
achievable cavity linewidth is not affected by the slow-light effect.
Intuitively this can be understood as follows: by narrowing the transparency
window width $\Gamma$ the group velocity can be reduced, and in accordance
with Eq. (2), the cavity linewidth narrows. However, this leads to an increase
in the off-resonant absorption $\alpha_{c}$ at the center of the transparency
window, which, in order to maintain impedance matching, requires an increase
of the input mirror transmission $T_{1}$. While this might suggest that the
narrowest transparency window achievable for a given material $\Gamma_{h}$ is
optimal in order to maximize the reduction in length change sensitivity, such
an optimization impairs the longest attainable locking period. In particular,
the off-resonant excitation width extending into the ions adjacent in
frequency to the spectral transparency window scales with the width $\Gamma$.
Specifically, 50% of the absorption at the center of the spectral window comes
from the atoms in the intervals $[-\Gamma,-\Gamma/2]$ and $[\Gamma/2,\Gamma]$
(see the SM for further details). Thus, the width $\Gamma$ determines the
number of ions interrogated by the circulating light of the cavity, and
consequently, the number of ions over which any deterioration of the spectral
structure is averaged. Broadening the transparency window width increases the
time that the spectral structure can be maintained, and, consequently, the
amount of time that the slow-light cavity can be locked.

At its core, this trade-off between the optical power used to interrogate the
atomic ensemble, spread over the finite spectral width of the ensemble, is
true for all rare-earth ion based laser stabilization techniques. For
spectral-hole locking, the equivalent averaging takes place when one performs
interleaved locking to a large number of individual spectral holes Thorpe  _et
al._ (2011); however, the total number of interrogated ions for
implementations realized to date is considerably fewer than for the here
presented technique.

An additional potential noise source for the presented locking technique that
is not an issue for conventional FP cavities are changes of the refractive
index of the cavity spacer material due to environmental variations in
temperature and pressure. Noise of this type is also reduced by the slow-light
effect according to

$\frac{d\nu}{\nu}=-\frac{dn}{n}\frac{v_{g}}{c},$ (7)

with $n$ the refractive index of the spacer. A detailed analysis of this is
beyond the scope of the present paper, and the order of magnitude where this
will become significant in a material such as Eu3+:Y2SiO5 is yet to be
determined. Ultimately, like with any locking setup, there will be a wide
range of environmental factors that impair the stability, such as temperature,
pressure, and vibration. The impact of some of these will be reduced by the
slow-light effect similar to what has been demonstrated here for cavity length
changes, although this will not hold true universally for all noise sources. A
detailed noise analysis of this locking scheme will be reported in a
subsequent work.

To conclude, we have presented a laser frequency stabilization technique that
utilizes the slow-light effect within an FP cavity to reduce the sensitivity
of the resonance frequencies to cavity length changes. Implementing such a
locking system using a single-crystal Pr3+:Y2SiO5 cavity spacer achieved a
reduction in length change sensitivity by four orders of magnitude. The
concept validation of this system achieved a drift rate of $0.55\pm
0.06\;\mathrm{kHz/s}$ with a noise floor of $\sigma_{y}(\tau)=4.6\times
10^{-13}$ for a $6\;\mathrm{mm}$ long cavity. We anticipate that utilization
of a different material with narrower homogeneous linewidth and lower
hyperfine cross-relaxation would significantly improve these values.

##  IV Methods

The laser used was a Coherent 699-21 ring-dye laser, with a custom intra-
cavity EOM, stabilized to an external ULE cavity. The linewidth is estimated
to be on the order of a few tens of $\mathrm{Hz}$, with a drift rate below
$1\;\mathrm{Hz/s}$ Alnis  _et al._ (2008). Prior to injection of the light
into the locking loop, arbitrary pulse shaping in amplitude, frequency, and
phase, was performed with a double-pass AOM. Locking was performed using the
PDH technique, with a modulation frequency of $1.5\;\mathrm{MHz}$. To verify
that a tight lock was achieved, the pre-stabilized laser was stepped through a
series of frequency changes while the feedback loop was monitored. The sample
was cooled to $2.1\;\mathrm{K}$ in an Oxford Instruments bath cryostat. The
Y2SiO5 crystal has yttrium atoms replaced with praseodymium at a
$500\;\mathrm{ppm}$ level; the sample is a cylinder of diameter
$10\;\mathrm{mm}$ and a height of $6\;\mathrm{mm}$ (purchased from Scientific
Materials). One side of the cylinder has been cut and polished to allow for
the perpendicular burn geometry indicated in Fig. 1(c). The polarization of
both the burn and probe light was rotated to be parallel to the $D_{2}$
optical extinction axis of Y2SiO5. The width of the spectral transmission
window used throughout was $18\;\mathrm{MHz}$.

All measurements, with the exception of the drift rate with respect to locking
point, were performed with a magnetic field of $10\;\mathrm{mT}$ along the
crystal $D_{2}$ axis, which enhanced the lifetime of the ground-state
hyperfine levels. Moreover the drift rate data was taken in a configuration
without the perpendicular burn-beam geometry; consequently burning was
performed through the mirror coating. While this impaired the absolute drift
performance, it still demonstrated the trends of the drift vs intensity when
locking to different cavity modes.

In order to perform the beam-translation experiment, and to achieve optimal
burning, the perpendicular burn geometry was utilized. The burn beam was
shaped with cylindrical lenses and blocked with a mechanical shutter during
locking. To avoid standing waves during burning, a second shutter blocked the
locking beam during the burn operation. Additionally, for the measurements of
the smallest drift rate, as well as the noise floor, the burning beam was
dithered with a piezo driven mirror mount with a frequency of
$10\;\mathrm{Hz}$, and a lens tissue paper inserted into the beam path before
the cryostat window. By making the burning beam more diffuse, this ensured
optimal burning of the entire length of the crystal (including close to the
mirror coated sides), and mitigated residual drift from unburned spatial
regions of the crystal because of (partial) standing waves arising from
reflections of the parallel crystal surfaces. The error signal was amplified
and $30\;\mathrm{kHz}$ low-pass filtered prior to locking with a Vescent
D2-125.

##  V Acknowledgements

The authors would like to thank Ulf Söderlund and Leif Johansson for their
help with dicing the Pr3+:Y2SiO5 crystal. This research was supported by the
Swedish Research Council (no. 2016-05121, no. 2015-03989, no. 2016-04375, and
2019-04949), the Knut and Alice Wallenberg Foundation (KAW 2016.0081), the
Wallenberg Center for Quantum Technology (WACQT) funded by The Knut and Alice
Wallenberg Foundation (KAW 2017.0449), the Royal Physiographic Society in
Lund, and the European Union FETFLAG program, Grant No.820391 (SQUARE)
(2017.0449).

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# Symbol Length in Brauer Groups of Elliptic Curves

Mateo Attanasio  ,  Caroline Choi  ,  Andrei Mandelshtam  Department of
Mathematics, Stanford University, Stanford, CA 94305<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and  Charlotte Ure  Department of
Mathematics, University of Virginia, Charlottesville, VA 22904
<EMAIL_ADDRESS>
###### Abstract.

Let $\ell$ be an odd prime, and let $K$ be a field of characteristic not
$2,3,$ or $\ell$ containing a primitive $\ell$-th root of unity. For an
elliptic curve $E$ over $K$, we consider the standard Galois representation

$\rho_{E,\ell}:\operatorname{Gal}(\overline{K}/K)\rightarrow\operatorname{GL}_{2}(\operatorname{\mathbb{F}}_{\ell}),$

and denote the fixed field of its kernel by $L$. Recently, the last author
gave an algorithm to compute elements in the Brauer group explicitly, deducing
an upper bound of $2(\ell+1)(\ell-1)$ on the symbol length in
$\mathbin{{}_{\ell}\operatorname{Br}(E)}/\mathbin{{}_{\ell}\operatorname{Br}(K)}$.
More precisely, the symbol length is bounded above by $2[L:K]$. We improve
this bound to $[L:K]-1$ if $\ell\nmid[L:K]$. Under the additional assumption
that $\operatorname{Gal}(L/K)$ contains an element of order $d>1$, we further
reduce it to $(1-\frac{1}{d})[L:K]$. In particular, these bounds hold for all
CM elliptic curves, in which case we deduce a general upper bound of $\ell+1$.
We provide an algorithm implemented in SageMath to compute these symbols
explicitly over number fields.

###### 2020 Mathematics Subject Classification:

Primary 11G05; Secondary 14H52, 14F22

This paper was created as part of the 2021 Number Theory REU at the University
of Virginia. We would like to thank everybody involved with organizing,
lecturing, and mentoring at the REU. In particular, we would like to thank Ken
Ono for managing the REU, his advice, and valuable comments on this project.
We thank Andrew Sutherland and Wei-Lun Tsai for their assistance on SageMath
code. We would like to thank Rachel Pries and the anonymous referee for
helpful comments and suggestions. Finally, we are grateful for the generous
support of the National Science Foundation (Grants DMS 2002265 and DMS
205118), the National Security Agency (Grant H98230-21-1-0059), the Thomas
Jefferson Fund at the University of Virginia, and the Templeton World Charity
Foundation.

##  1\. Introduction

The Brauer group of a variety is an important invariant that sheds light on
the geometric and arithmetic properties of the underlying variety. For
instance, Artin and Mumford used Brauer groups of varieties to provide a
counterexample to the Lüroth problem in [AM72], and Manin showed that for a
global field $K$, elements of the Brauer group can obstruct the existence of
$K$-points in [Man71].

Let $F$ be a field of characteristic coprime to $\ell$ containing a primitive
$\ell$-th root of unity $\zeta_{\ell}$. A symbol algebra is a central simple
algebra of the form

$(a,b)_{\ell,F}=F\left\langle
x,y:x^{\ell}=a,y^{\ell}=b,xy=\zeta_{{}_{\ell}}yx\right\rangle.$

A theorem by Merkurjev and Suslin [MS82] implies that every element in the
$\ell$-torsion of the Brauer group of $F$, written as
$\mathbin{{}_{\ell}\operatorname{Br}(F)}$, can be described as a tensor
product of symbol algebras over $F$. In particular, there is an isomorphism
$K_{2}(F)/\ell K_{2}(F)\xrightarrow{\sim}\mathbin{{}_{\ell}\text{Br}(F)},$
explicitly given by $(a,b)\mapsto(a,b)_{\ell,F}$.

For the $\ell$-torsion of the Brauer group of a field, the symbol length is
the minimal number $n$ such that some representative of every Brauer class can
be written as a tensor product of at most $n$ symbol algebras. Bounds on the
symbol length may be used to give upper bounds for the degree of abelian (or
meta-abelian) Galois splitting fields. For a survey on symbol length, see
[ABGV11, Section 3].

The symbol length of Brauer classes of prime index is unknown in general, but
is conjectured to be $1$. If $F$ is a local or global field containing a
primitive $\ell$-th root of unity, then Albert, Brauer, Hasse, and Noether
showed that every algebra of exponent $\ell$ is cyclic and has symbol length 1
[AH32, BNH32]. For $F$ a $C_{2}$ field containing a certain primitive root of
unity, Artin showed that every Brauer class of degree $2$ or $3$ has symbol
length 1 [Art82]. Similar results have been proven for Brauer classes defined
over a function field of an $\ell$-adic curve $F$. Saltman showed that every
Brauer class defined over $F$ of prime index coprime to $\ell$ is cyclic
[Sal07]. Over the same such $F$, work by Suresh for prime order [Sur10], and
Brussel, McKinnie, and Tengan [BMT16] for order coprime to $\ell$, showed that
every Brauer class has symbol length at most 2. Matzri gave bounds on the
symbol lengths of elements in the Brauer group of a field that contains a
$C_{m}$ field [Mat16].

The Brauer group $\operatorname{Br}(E)$ of an elliptic curve $E$ defined over
a field $K$ is a torsion abelian group and a subgroup of the Brauer group of
its function field $\operatorname{Br}(K(E))$. It is also isomorphic to the
unramified Brauer group of $K(E)$ by purity. Applying the Merkurjev-Suslin
theorem in the setting $F=K(E)$, one can express
$\mathbin{{}_{\ell}\operatorname{Br}(E)}$ as a tensor product of symbol
algebras. A description of $\mathbin{{}_{\ell}\operatorname{Br}(E)}$ in terms
of generators and relations is given by Skorobogatov in [Sko01, Chapter 4]
provided that the $\ell$-torsion of $E$ is $K$-rational. When $E(K)$ does not
contain $E[\ell]$ and $K$ is not of characteristic $2$ or $3$, the explicit
generators and relations are given for $\ell=2$ by Chernousov and Guletskii in
[CG01] and for odd primes $\ell$ coprime to the characteristic of $K$ by the
last author in [Ure19]. We will rely on these algorithms to obtain bounds on
the symbol length.

Let $\ell$ be an odd prime, and let $K$ be a field of characteristic not 2 or
3 and coprime to $\ell$ containing an $\ell$-th primitive root of unity
$\zeta_{\ell}$. We consider an elliptic curve $E$ defined over $K$. Under
these assumptions, we have

$\mathbin{{}_{\ell}\operatorname{Br}(E)}=\mathbin{{}_{\ell}\operatorname{Br}(K)}\oplus
I,$

where $I$ is the tensor product of certain explicit symbol algebras [Sko01,
CG01, Ure19]. More details are given in Section 2.4. Denote by

$\rho_{E,\ell}:\operatorname{Gal}(\overline{K}/K)\to\text{Aut}(E[\ell])\left(\xrightarrow{\sim}\operatorname{GL}_{2}(\operatorname{\mathbb{F}}_{\ell})\right)$

the standard Galois representation obtained by letting elements of the Galois
group act on the $\ell$-torsion points of $E$. Recall that since
$E[\ell]\simeq\mathbb{Z}/\ell\mathbb{Z}\times\mathbb{Z}/\ell\mathbb{Z}$,
picking a $\mathbb{Z}/\ell\mathbb{Z}$-basis $P,Q$ of $E[\ell]$ gives us a
natural map from
$\operatorname{Gal}(\overline{K}/K)\to\operatorname{GL}_{2}(\operatorname{\mathbb{F}}_{\ell})$.
Denote by $L$ the fixed field of the kernel of $\rho_{E,\ell}$. Equivalently,
$L$ is the smallest Galois extension of $K$ such that the $\ell$-torsion
points of $E$ are $L$-rational.

The algorithm presented by the last author in [Ure19] implies that the symbol
length in
$\mathbin{{}_{\ell}\operatorname{Br}(E)}/\mathbin{{}_{\ell}\operatorname{Br}(K)}$
is bounded above by $2[L:K]$ if $\ell\nmid[L:K]$ and $\frac{2}{\ell}[L:K]$ if
$\ell\mid[L:K]$. In particular, since $G$ is a subgroup of
$\mathrm{SL}_{2}\left(\mathbb{F}_{\ell}\right)$, the symbol length is always
bounded above by $2(\ell-1)(\ell+1)$ [Ure19, Corollary 1.2]. If
$\ell\nmid[L:K]$, we improve this bound significantly in a series of
statements. Write $G:=\operatorname{Gal}(L/K)$ and assume throughout that
$\ell\nmid[L:K]$.

######  Theorem 1.1.

Suppose $\ell\nmid[L:K]$ and $[L:K]>2$. Then the symbol length of
$\mathbin{{}_{\ell}\operatorname{Br}(E)}/\mathbin{{}_{\ell}\operatorname{Br}(K)}$
is bounded above by $[L:K]-1$.

When there exists some intermediate field $K\subset K^{\prime}\subset L$, we
can further improve our bound on the symbol length of
$\mathbin{{}_{\ell}\operatorname{Br}(E)}/\mathbin{{}_{\ell}\operatorname{Br}(K)}$.

######  Theorem 1.2.

Suppose $\ell\nmid[L:K]$, $[L:K]>2$, and $G$ contains a subgroup of order
$d>1$. Then the symbol length of
$\mathbin{{}_{\ell}\operatorname{Br}(E)}/\mathbin{{}_{\ell}\operatorname{Br}(K)}$
is bounded above by $\left(1-\frac{1}{d}\right)[L:K]$.

######  Remark 1.3.

When $K$ is a local or global field, we obtain the following bounds on the
symbol length of $\mathbin{{}_{\ell}\operatorname{Br}(E)}$. By [AH32], the
symbol length of $\operatorname{Br}(K)$ is $1$, so Theorem 1.1 implies that
when $\ell\nmid[L:K]$ and $[L:K]>2$, the symbol length of
$\mathbin{{}_{\ell}\operatorname{Br}(E)}$ is bounded above by $[L:K]$. If we
additionally assume that $G$ contains an element of order $d>1$, then Theorem
1.2 implies that the symbol length is bounded above by
$\left(1-\frac{1}{d}\right)[L:K]+1$.

Theorem 1.2 is best applicable when the Galois group $G$ has an element of
order $2.$ In this case, the symbol length becomes bounded by
$\frac{1}{2}[L:K].$

######  Corollary 1.4.

If $K$ is a number field and the image of $\rho_{E,\ell}$ lies inside the
normalizer of a Cartan subgroup, then
$\mathbin{{}_{\ell}\text{Br}(E)}/\mathbin{{}_{\ell}\text{Br}(K)}$ has symbol
length bounded above by $\ell-1$ or $\ell+1$, corresponding to a split and
nonsplit Cartan respectively.

We note that the assumptions of 1.4 are satisfied by CM elliptic curves over a
number field $K$, and we refer the reader to Remark 2.2 for a complete
discussion.

This paper is organized as follows. In Section 2, we introduce preliminary
definitions and describe the algorithm from [Ure19] to explicitly compute the
generators of
$\mathbin{{}_{\ell}\operatorname{Br}(E)}/\mathbin{{}_{\ell}\operatorname{Br}(K)}$.
In Section 3, we prove our first bounds on the symbol length. In Section 4, we
improve these bounds on the symbol length and prove our main theorems. We
conclude with some illustrative examples of our main theorems in Section 5 and
compute bounds on the symbol length of
$\mathbin{{}_{5}\operatorname{Br}(E)}/\mathbin{{}_{5}\operatorname{Br}(K)}$
for a CM elliptic curve $E$.

Throughout this paper, let $\ell$ be an odd prime, $K$ a number field of
characteristic different from $2$, $3$, and $\ell$ containing a primitive
$\ell$-th root of unity $\zeta_{\ell}$. Denote a separable closure of $K$ by
$\overline{K}$ and for any separable extension $K^{\prime}$ of $K$ let
$G_{K^{\prime}}$ be the absolute Galois group
$\operatorname{Gal}(\overline{K}/K^{\prime})$. Fix an elliptic curve $E$ over
$K$ and let $L$ be the smallest field extension of $K$ so that the
$\ell$-torsion $E[\ell]$ is $L$-rational.

##  2\. Preliminaries

We begin by discussing the theory needed to bound the symbol length in the
Brauer group of an elliptic curve.

###  2.1. Elliptic Curves and Galois Representations

Fix a basis $P,Q$ of the torsion subgroup $E[\ell]$ so that the Weil-pairing
$\text{e}(-,-)$ satisfies $\text{e}(P,Q)=\zeta_{\ell}$. Recall that by letting
elements of the Galois group act on $E[\ell]$, we obtain the standard Galois
representation by $\ell$:

$\rho_{E,\ell}:\operatorname{Gal}(\overline{K}/K)\to\operatorname{GL}_{2}(\operatorname{\mathbb{F}}_{\ell}).$

######  Remark 2.1.

Since we assume that $K$ contains a primitive $\ell$-th root of unity
$\zeta_{{}_{\ell}}$, we have $\mathrm{Im}\rho_{E,\ell}\leq
SL_{2}(\mathbb{F}_{\ell})$. To see this, let $\sigma\in G_{K}$ with
$\rho_{E,\ell}(\sigma)=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$. Noting that
the Weil pairing commutes with the Galois action through its determinant
[Sil09, Chapter 3, Section 8], we have

$\zeta_{{}_{\ell}}=\sigma(\zeta_{{}_{\ell}})=\sigma\text{e}(P,Q)=\text{e}(\sigma(P),\sigma(Q))=\text{e}(aP+cQ,bP+dQ)=\text{e}(P,Q)^{ad-
bc}=\zeta_{{}_{\ell}}^{ad-bc},$

so $ad-bc=1\mod\ell$ and $\rho_{E,\ell}(\sigma)\in
SL_{2}(\operatorname{\mathbb{F}}_{\ell})$.

By examining admissible subgroups of
$\operatorname{GL}_{2}(\operatorname{\mathbb{F}}_{\ell})$, we can determine
the structure of $G$, which will play a large role in improving bounds on the
symbol length. A Borel subgroup of $\operatorname{GL}_{2}(\mathbb{F}_{\ell})$
is any group conjugate to the group of matrices of the form
$\begin{pmatrix}a&b\\\ 0&d\end{pmatrix}.$ A Cartan subgroup of
$\operatorname{GL}_{2}(\mathbb{F}_{\ell})$ is a maximal abelian subgroup that
is semisimple. A Cartan subgroup is of index 2 in its normalizer. There are
two types of Cartan subgroups. A split Cartan subgroup is a Cartan subgroup
that is conjugate to the subgroup of diagonal matrices of the form
$\begin{pmatrix}a&0\\\ 0&d\end{pmatrix},$ and is isomorphic to
$\operatorname{\mathbb{F}}_{\ell}^{\times}\times\operatorname{\mathbb{F}}_{\ell}^{\times}$.
Let $\varepsilon\in\operatorname{\mathbb{F}}_{\ell}^{\times}$ be a non-square.
A nonsplit Cartan subgroup is a Cartan subgroup that is conjugate to the group
of matrices of the form $\begin{pmatrix}a&\varepsilon b\\\ b&a\end{pmatrix},$
and is isomorphic to $\operatorname{\mathbb{F}}_{\ell^{2}}^{\times}$ [Lan95,
LMF21]. If $G\neq\operatorname{SL}_{2}(\mathbb{F}_{\ell})$, then it satisfies
one of the following cases [Ser79, p. 261]:

  1. (1)

$G\subset$ Borel,

  2. (2)

$G\subset$ Cartan,

  3. (3)

$G\subset$ normalizer of a Cartan but not the Cartan,

  4. (4)

$PG\cong S_{4}$, where $PG$ denotes the image of $G$ under the
projectivization
$\operatorname{GL}_{2}(\operatorname{\mathbb{F}}_{\ell})\mapsto\mathrm{PGL}_{2}(\operatorname{\mathbb{F}}_{\ell})$.

######  Remark 2.2.

By [Lan95] we know that each of the three cases above corresponds to $G$
having a different image $PG$ in the projectivization
$\mathrm{PGL}_{2}(\operatorname{\mathbb{F}}_{\ell})$. If we assume that $E$
has CM, by comparing the number of order two elements in each of these images
with the expected number of order two elements in $PG$ from the Chebotarev
density theorem, we see that $G$ is always contained in the normalizer of a
Cartan but not the Cartan. Hence if $E$ is a CM elliptic curve, we always have
$\ell\nmid[L:K]$.

####  2.1.1. Divisors

Recall that for a point $P\in\overline{E}$, we denote by $t_{P}$ any function
in $\overline{K}(E)$ with divisor equal to $\ell[P]-\ell[\mathcal{O}]$. Recall
also that $\sigma(t_{P})=t_{\sigma(P)}$ for any $\sigma\in G_{K}$. Denote by
$L_{P,Q}$ the line in $\mathbb{P}^{2}$ which passes through $P$ and $Q$. The
divisor of $L_{P,Q}$ is equal to $[P]+[Q]+[-P-Q]-3[\mathcal{O}]$.

######  Proposition 2.3.

Up to $\ell$th powers in $L(E)$, we may assume that

$t_{P}^{-1}=t_{-P}\qquad\text{and}\qquad t_{P}\cdot t_{Q}=t_{P+Q}.$

###### Proof.

These identities follow from

$\operatorname{Div}\left(\tfrac{L_{P,-P}^{\ell}}{t_{P}}\right)=\ell[-P]-\ell[\mathcal{O}]\qquad\text{and}\qquad\operatorname{Div}\left(\tfrac{t_{P}\cdot
t_{Q}\cdot L_{-P,-Q}^{\ell}}{L_{P,-P}^{\ell}\cdot
L_{Q,-Q}^{\ell}}\right)=\ell[P+Q]-\ell[\mathcal{O}].\qed$

###  2.2. Brauer Groups and Group Cohomology

We review some facts about cohomology. For more details, see [NSW08] and
[Lic69].

####  2.2.1. Brauer Groups

Recall that for a field $F$ with separable closure $\overline{F}$, the Brauer
group is $H^{2}(G_{F},\overline{F})$. The Brauer group $\operatorname{Br}(E)$
of an elliptic curve $E$ can also be defined in terms of étale cohomology.
However, we forgo this definition in favor of identifying
$\operatorname{Br}(E)$ as a subgroup of the Brauer group of its function field
$K(E)$. We give this identification explicitly as follows.

Denoting by $\text{Div}(\overline{E})$ the divisors of $\overline{E}$, we have
a map $\overline{K}(E)^{\times}\to\text{Div}(\overline{E})$ by sending each
function to its divisor. This induces a map in cohomology as

$H^{2}(G_{K},\overline{K}(E)^{\times})\to
H^{2}(G_{K},\text{Div}(\overline{E})).$

The kernel of this map is $\operatorname{Br}(E)$. One can verify that this is
equivalent to the étale cohomology definition in [Lic69, Section 2].

####  2.2.2. Restriction and Corestriction

If $\mathcal{G}$ is any group, $\mathcal{H}$ is a normal finite index subgroup
of $\mathcal{G}$, and $A$ is a $\mathcal{G}$-module, then we have the
corestriction map $\operatorname{Cor}:H^{n}(\mathcal{H},A)\to
H^{n}(\mathcal{G},A)$ and restriction map
$\operatorname{Res}:H^{n}(\mathcal{G},A)\to H^{n}(\mathcal{H},A)$.

######  Proposition 2.4.

[NSW08] The map

$\operatorname{Cor}\circ\operatorname{Res}:H^{n}(\mathcal{G},A)\to
H^{n}(\mathcal{G},A)$

is equal to the map of multiplication by $[\mathcal{G}:\mathcal{H}]$. When
$n=0$ so that $H^{0}(\mathcal{G},A)=A^{\mathcal{G}}$, the elements of $A$
fixed by $\mathcal{G}$, the map

$\operatorname{Res}\circ\operatorname{Cor}:H^{0}(\mathcal{H},A)\to
H^{0}(\mathcal{H},A)$

sends an element $a$ to $\sum_{s\in\mathcal{G}/\mathcal{H}}s\cdot a$.

Thus we see that corestriction acts like a norm map with respect to the group
$\mathcal{G}/\mathcal{H}$. When $\mathcal{G}=G_{K}$ and $\mathcal{H}=G_{L}$
are absolute Galois groups of a field extension $L/K$, we write these maps as
$\operatorname{Cor}_{L/K}$ and $\operatorname{Res}_{K/L}$. However, when the
fields $L$ and $K$ are implied, we will ignore the subscripts. Using explicit
descriptions of these maps to bound the symbol length will be the chief goal
of this paper.

####  2.2.3. Galois Action on Symbols

For any element $\sigma\in G_{K}$ we have an action $\sigma^{*}$ on the Brauer
groups $\operatorname{Br}(L(E))$ that fixes $\operatorname{Br}(K(E))$.
Furthermore, the elements of $G_{L}$ act trivially on $\text{Br}(L(E))$ and
$\text{Br}(K(E))$, so in fact we have an action of
$G_{K}/G_{L}=\operatorname{Gal}(L/K)=G$, commuting with corestriction [NSW08,
Chapter 1]. This action restricts to give us an action of $G$ on
$\text{Br}(E_{L})$ and $\text{Br}(E)$. We can explicitly determine the action
on symbols by the following lemma.

######  Lemma 2.5.

[Dra83, p. 85] For any $\sigma\in\text{Gal}(\overline{K(E)}/K(E))$, the action
of $\sigma^{*}$ on cohomology groups is given explicitly by

$\sigma^{*}(\alpha,\beta)_{\ell,L(E)}=(\sigma(\alpha),\sigma(\beta))_{\ell,L(E)}.$

In our setting, since we only consider the action of elements of $G$, which
act trivially on $\operatorname{Br}(K(E))$, we observe the following.

######  Remark 2.6.

For any $(\alpha,\beta)\in\operatorname{Br}(E_{L})$, we have

$\operatorname{Cor}(\sigma(\alpha),\sigma(\beta))_{\ell,L(E)}=\operatorname{Cor}\left(\sigma^{*}(\alpha,\beta)_{\ell,L(E)}\right)=\sigma^{*}\operatorname{Cor}(\alpha,\beta)_{\ell,L(E)}=\operatorname{Cor}(\alpha,\beta)_{\ell,L(E)}.$

######  Remark 2.7.

[Ser79, p. 209] If $\alpha\in K(E)^{\times}$, $\beta\in L(E)^{\times}$, we
have

$\operatorname{Cor}_{L/K}(\alpha,\beta)_{\ell,L(E)}=\left(\alpha,\operatorname{Norm}_{L(E)/K(E)}(\beta)\right)_{\ell,K(E)}.$

###  2.3. Explicit Descriptions of Restriction and Corestriction

We now explicitly apply the Galois action on cohomology. By our assumption
$\ell\nmid[L:K]$ and Proposition 2.4, we see that for any $G_{K}$-module $A$,
the corestriction map
$\mathbin{{}_{\ell}H^{n}(G_{L},A)}\to\mathbin{{}_{\ell}H^{n}(G_{K},A)}$ is
surjective.

####  2.3.1. Describing Restriction

We will explicitly describe the image of the restriction
$H^{1}(G_{K},E[\ell])\to H^{1}(G_{L},E[\ell])$. Since corestriction is
surjective, it is equal to the image of
$\operatorname{Res}\circ\operatorname{Cor}$.

######  Definition 2.8.

Denote by $\mathcal{N}$ the map

$\operatorname{Res}\circ\operatorname{Cor}:H^{1}(G_{L},E[\ell])\to
H^{1}(G_{L},E[\ell]).$

For a different choice of subfield $K^{\prime}\subset L$ with
$H=\operatorname{Gal}(L/K^{\prime})$, we write

$\operatorname{Res}_{L/K^{\prime}}\circ\operatorname{Cor}_{L/K^{\prime}}=\mathcal{N}_{H}.$

If we pick an $\mathbb{F}_{\ell}$-basis $P$, $Q$ of $E[\ell]$, then we have an
isomorphism

$H^{1}(G_{L},E[\ell])\cong_{P,Q}H^{1}(G_{L},\mathbb{Z}/\ell\mathbb{Z}\times\mathbb{Z}/\ell\mathbb{Z})\cong
H^{1}(G_{L},\mathbb{Z}/\ell\mathbb{Z})\times
H^{1}(G_{L},\mathbb{Z}/\ell\mathbb{Z}).$ (2.1)

Kummer theory gives us an isomorphism of
$H^{1}(G_{L},\mathbb{Z}/\ell\mathbb{Z})$ with
$L^{\times}/(L^{\times})^{\ell}$. Since we have an action of $G$ on
$H^{1}(G_{L},E[\ell])$, we may therefore transport it via these isomorphisms
to an action of $G$ on $L^{\times}/(L^{\times})^{\ell}\times
L^{\times}/(L^{\times})^{\ell}$. Explicitly, this action is given by

$g\cdot(a,b)=\left(\left(g^{-1}(a)\right)^{c_{1}^{g}}\left(g^{-1}(b)\right)^{c_{3}^{g}},\left(g^{-1}(a)\right)^{c_{2}^{g}}\left(g^{-1}(b)\right)^{c_{4}^{g}}\right),$

where $g^{-1}=\begin{pmatrix}c_{1}^{g}&c_{3}^{g}\\\
c_{2}^{g}&c_{4}^{g}\end{pmatrix}$ in terms of the basis $P$, $Q$ of $E[\ell]$
[Ure19, Remark 5.4].

######  Definition 2.9.

We denote by $\mathcal{N}_{P,Q}$ or $\mathcal{N}_{G,P,Q}$ the composition

$\tfrac{L^{\times}}{(L^{\times})^{\ell}}\times\tfrac{L^{\times}}{(L^{\times})^{\ell}}\xrightarrow{\sim_{P,Q}}H^{1}(G_{L},E[\ell])\xrightarrow{\mathcal{N}_{G}}H^{1}(G_{L},E[\ell])\xrightarrow{\sim_{P,Q}}\tfrac{L^{\times}}{(L^{\times})^{\ell}}\times\tfrac{L^{\times}}{(L^{\times})^{\ell}},$

where $\sim_{P,Q}$ comes from the map above in eq. 2.1 and Kummer theory. We
denote the image of the map $\mathcal{N}_{P,Q}$ by

$M_{G,P,Q}=\left\\{\mathcal{N}_{P,Q}(a,b)=\prod_{g\in G}g\cdot(a,b):a,b\in
L^{\times}/(L^{\times})^{\ell}\right\\}.$

Note that this map is a norm with respect to the action of $G$ on
$\tfrac{L^{\times}}{(L^{\times})^{\ell}}\times\tfrac{L^{\times}}{(L^{\times})^{\ell}}$.

####  2.3.2. Describing Corestriction

Our next goal is to describe the image of the corestriction map in the case of
symbol algebras. We can explicitly describe the corestriction of a pair of
symbols $(\alpha,\beta)\in K_{2}(F)$ using the proposition from [RT83, Section
3]. By Corollary 2 of the same section, this restricts to give us the
corestriction on the $\ell$-torsion of Brauer groups. We reproduce the
algorithm here for clarity of exposition.

######  Definition 2.10.

For a polynomial $p(t)=a_{n}t^{n}+a_{n-1}t^{n-1}+\cdots+a_{m}t^{m}$ of degree
$n$, define $p^{*}(t)=\frac{p(t)}{a_{m}t^{m}}$ and $c(p)=(-1)^{n}a_{n}$.

######  Algorithm 2.11 (Rosset-Tate Algorithm [RT83, p. 44] ).

Let $F^{\prime}/F$ be a finite extension of fields over $K$, and let
$x,y\in(F^{\prime})^{*}$. Let $g(t)\in F[t]$ be the minimal polynomial of $x$
over $F$, and let $f(t)\in F[t]$ be the polynomial of smallest degree such
that $\text{Norm}_{F^{\prime}/F(x)}(y)=f(x)$. Define a sequence of nonzero
polynomials $g_{0},\dots,g_{m}$ as follows. Set $g_{0}=g$, $g_{1}=f$, and for
$i\geq 1$, let $g_{i+1}$ be the remainder of the division of $g_{i-1}^{*}$ by
$g_{i}$ as long as $g_{i}\neq 0$. Then the corestriction of the symbol
$(x,y)_{\ell,F^{\prime}}$ is given explicitly by

$\operatorname{Cor}_{F^{\prime}/F}(x,y)_{F^{\prime}}=-\sum_{i=1}^{m}(c(g_{i-1}^{*}),c(g_{i}))_{F}.$

Note that the degree of $g_{i}$ strictly decreases on each step, so $m\leq\deg
g_{0}\leq[F^{\prime}:F]$.

###  2.4. Brauer Group of an Elliptic Curve

We seek to understand the prime torsion of $\operatorname{Br}(E)$ through the
exact sequence

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This exact sequence comes from the Hochschild-Serre spectral sequence, with
the map $i$ taking the class of an algebra $A$ to the class of $A\otimes
K(E)$. For details, see [Sko01, p. 63] or [Lic69, Section 2].

We will now describe a split to $r$ that gives us the desired decomposition of
${\mathbin{{}_{\ell}}\text{Br}(E)}$. For this, we will use a map
$\varepsilon:H^{1}(G_{K},E[\ell])\to\mathbin{{}_{\ell}}\operatorname{Br}(E)$.
For reasons given by [CG01] and [Ure19], this will induce a split to $r$,
allowing us to write
$\mathbin{{}_{\ell}\text{Br}(E)}=\mathbin{{}_{\ell}\text{Br}(K)}\oplus\text{Im}(\varepsilon)$.
There is an interpretation of this map as a cup product given in [Sko99]. By
[Len76], when $K$ is a number field all elements of $\text{Br}(K)$ are
symbols, and in general, $\operatorname{Br}(K)$ will not be too complicated,
so we instead examine the image of $\varepsilon$.

In the special case where $K$ contains all $\ell$-torsion points (i.e.,
$K=L$), we can define our map $\varepsilon$ as follows. First, fix a basis
$P,Q$ for $E[\ell]$, which gives us an isomorphism
$E[\ell]\xrightarrow{\sim}\mathbb{Z}/\ell\mathbb{Z}\times\mathbb{Z}/\ell\mathbb{Z}$
as $G_{K}$-modules. Then there is an isomorphism from $H^{1}(K,E[\ell])$ to
$\left((K^{\times}/(K^{\times})^{\ell}\right)^{2}$ by Kummer theory. The map
$\varepsilon$ is given by

$(a,b)\mapsto(a,t_{P})_{\ell,K(E)}\otimes(b,t_{Q})_{\ell,K(E)}.$

In the more general case where $K\neq L$, but we still have that
$\ell\nmid[L:K]$, so that restriction is injective and corestriction is
surjective, we define $\bar{\varepsilon}$ as before and use the following
diagram to define $\varepsilon$ as the composition
$[L:K]\circ\varepsilon=\operatorname{Cor}\circ\bar{\varepsilon}\circ\operatorname{Res}$
[Ure19]:

${0}$${{E_{K}/\ell
E_{K}}}$${{H^{1}\left(G_{K},E[\ell]\right)}}$${0}$${0}$${{\mathbin{{}_{\ell}\text{Br}}(K)}}$${{\mathbin{{}_{\ell}\text{Br}}(E)}}$${{\mathbin{{}_{\ell}H^{1}\left(G_{K},\overline{E}\right)}}}$${0}$${{E_{L}/\ell
E_{L}}}$${0}$${{H^{1}\left(G_{L},E[\ell]\right)\cong_{P,Q}\left((L^{\times}/(L^{\times})^{\ell}\right)^{2}}}$${0}$${{\mathbin{{}_{\ell}\text{Br}}(L)}}$${{\mathbin{{}_{\ell}\text{Br}}(E_{L})}}$${{\mathbin{{}_{\ell}H^{1}\left(G_{L},\overline{E}\right)}}}$${0}$${0}$$\scriptstyle{\definecolor{.}{rgb}{0.83921568627451,0.36078431372549,0.36078431372549}\color[rgb]{0.83921568627451,0.36078431372549,0.36078431372549}\definecolor[named]{pgfstrokecolor}{rgb}{0.83921568627451,0.36078431372549,0.36078431372549}\operatorname{Res}}$$\scriptstyle{\definecolor{.}{rgb}{0.83921568627451,0.36078431372549,0.36078431372549}\color[rgb]{0.83921568627451,0.36078431372549,0.36078431372549}\definecolor[named]{pgfstrokecolor}{rgb}{0.83921568627451,0.36078431372549,0.36078431372549}\operatorname{Cor}}$$\scriptstyle{\bar{\varepsilon}}$$\scriptstyle{\varepsilon}$

Remark that the first vertical exact sequence of the diagram comes from the
long exact sequence in cohomology arising from the short exact sequence of
$G_{K}$-modules:

$0\to E[\ell]\to\overline{E}\xrightarrow{\ell}\overline{E}\to 0$

and similarly for the second vertical exact sequence. Following the diagram
above, we take
$(\alpha,\beta)\in\left(L^{\times}/(L^{\times})^{\ell}\right)^{2}$ in the
image of the restriction map and apply
$\operatorname{Cor}\circ\overline{\varepsilon}$ to it. Specifically, for any
element $(\alpha,\beta)\in M_{G,P,Q}$, we use 2.11 to compute the
corestriction of
$\overline{\varepsilon}(\alpha,\beta)=(\alpha,t_{P})_{L(E)}\otimes(\beta,t_{Q})_{L(E)}$.
We use this to give bounds on the symbol length in Theorem 1.1 and Theorem
1.2.

######  Remark 2.12.

When we have $\ell\mid[L:K],$ one can use the restriction-inflation sequence
to define $\bar{\varepsilon}$ over $K^{\prime}$ with $[L:K^{\prime}]=\ell$,
and then use the previous construction to pass from the map
$\bar{\varepsilon}$ over $K^{\prime}$ to the $\varepsilon$ over $K$ [Ure19,
Theorems 5.2-5.15]. However, our setup no longer applies since the extension
$K^{\prime}$ over $K$ is not necessarily Galois. Hence we restrict ourselves
to the case $\ell\nmid[L:K]$.

##  3\. Image of the Restriction Map to $H^{1}(L,E[\ell])$

Our first objective is to understand the image of the restriction
$H^{1}(G_{K},E[\ell])\to H^{1}(G_{L},E[\ell])$. As mentioned earlier, we will
use the fact that if we pick a basis $P,Q$ of $E[\ell]$, we may apply
isomorphisms from Kummer theory to understand the image as pairs of elements
$(\alpha,\beta)\in L^{\times}/(L^{\times})^{\ell}\times
L^{\times}/(L^{\times})^{\ell}$. To prove our first bounds on the symbol
length, we require the following lemma.

######  Lemma 3.1.

Let
$A\leqslant\operatorname{SL}_{2}(\overline{\operatorname{\mathbb{F}}}_{\ell})$,
with $\ell\neq 2$ and $1\neq\left|A\right|<\infty$. Then

$\sum_{a\in A}a=0,$

where the sum is an element of the matrix ring
$M_{2}(\overline{\operatorname{\mathbb{F}}}_{\ell})$.

###### Proof.

We first prove this in the case where $A$ is a cyclic group of prime order
$p$.

Let $\sigma$ be a generator of $A$. Assume first that the generator $\sigma$
of $A$ is in Jordan normal form. We have two cases:

$\sigma_{1}=\begin{pmatrix}\lambda_{1}&0\\\
0&\lambda_{2}\end{pmatrix}\qquad\text{ or
}\qquad\sigma_{2}=\begin{pmatrix}\lambda&1\\\ 0&\lambda\end{pmatrix},$

where $\lambda_{1}^{p}=\lambda_{2}^{p}=\lambda^{p}=1$. Furthermore, the latter
case occurs only when $p=\ell$, since

$\begin{pmatrix}1&0\\\
0&1\end{pmatrix}=\sigma_{2}^{p}=\begin{pmatrix}\lambda^{p}&p\lambda^{p-1}\\\
0&\lambda^{p}\end{pmatrix}$

In the first case, we must have $\lambda_{1},\lambda_{2}\neq 1$ since
$\det\sigma=1$ as
$\sigma\in\operatorname{SL}_{2}(\overline{\operatorname{\mathbb{F}}}_{\ell})$.
Hence,

$\sum_{0\leq
n<p}\sigma_{1}^{n}=\begin{pmatrix}1+\lambda_{1}+\dots+\lambda_{1}^{p-1}&0\\\
0&1+\lambda_{2}+\dots+\lambda_{2}^{p-1}\end{pmatrix}=\begin{pmatrix}\frac{\lambda_{1}^{p}-1}{\lambda_{1}-1}&0\\\
0&\frac{\lambda_{2}^{p}-1}{\lambda_{2}-1}\end{pmatrix}=0.$

In the second case, we observe the polynomial identity
$x^{\ell}-1=(x-1)^{\ell}$ in $\overline{\operatorname{\mathbb{F}}}_{\ell},$
implying that the unique (multiple) root of this polynomial is $\lambda=1.$
Then a direct computation shows that

$\sum_{0\leq
n<\ell}\sigma_{2}^{n}=\begin{pmatrix}\ell&\nicefrac{{\ell(\ell+1)}}{{2}}\\\
0&\ell\end{pmatrix}=0.$

If $\sigma$ is not in Jordan form, we know that $g\sigma g^{-1}$ is in Jordan
form for some $g\in\operatorname{GL}_{2}(\overline{\mathbb{F}}_{\ell})$. The
theorem holds in this case by the following calculation:

$\sum_{0\leq n<p}\sigma^{n}=g^{-1}\left(\sum_{0\leq
n<p}g\sigma^{n}g^{-1}\right)g=g^{-1}\left(\sum_{0\leq n<p}\left(g\sigma
g^{-1}\right)^{n}\right)g=0.$

Finally, for $A$ any nontrivial finite subgroup of
$\operatorname{SL}_{2}(\overline{\operatorname{\mathbb{F}}}_{\ell})$, we note
that $\left|A\right|$ is divisible by some prime $p$, so it contains a cyclic
subgroup $\langle\sigma\rangle$ of order $p$. Denoting by $S$ a set of
representatives of left cosets of $\langle\sigma\rangle$, we have

$\sum_{a\in A}a=\sum_{s\in S}\sum_{0\leq n<p}s\sigma^{n}=\sum_{s\in
S}s\left(\sum_{0\leq n<p}\sigma^{n}\right)=0.\qed$

Using Lemma 3.1 we can determine the image of the restriction map $M_{G,P,Q}$
defined in 2.9. Recall that the algorithm in [Ure19] implies a bound on the
symbol length of $2[L:K]$ if $\ell\nmid[L:K]$ and $\frac{2}{\ell}[L:K]$ if
$\ell\mid[L:K]$. Combining the following result carefully with the algorithm,
we may reduce these bounds significantly.

######  Proposition 3.2.

Let $L$ over $K$ be a finite Galois extension with $G=\operatorname{Gal}(L/K)$
and suppose $\ell\neq 2$, $\left|G\right|\neq 1$, and
$G\leq\operatorname{SL}_{2}(\operatorname{\mathbb{F}}_{\ell})$. Then for any
$K^{\prime}\subsetneq L$ we have that $(\alpha,\beta)\in M_{G,P,Q}$ satisfies

$\operatorname{Norm}_{L/K^{\prime}}(\alpha)=\operatorname{Norm}_{L/K^{\prime}}(\beta)=1,$

as elements of $(K^{\prime})^{\times}/((K^{\prime})^{\times})^{\ell}$.

###### Proof.

Choose $(a,b)\in L^{\times}/\left(L^{\times}\right)^{\ell}\times
L^{\times}/\left(L^{\times}\right)^{\ell}$ such that
$\mathcal{N}_{G,P,Q}(a,b)=(\alpha,\beta)$. We write $H$ for
$\text{Gal}(L/K^{\prime})$ and $S$ for a set of representatives of right
cosets of $H$ in $G$. Observe that

$\mathcal{N}_{G,P,Q}(a,b)=\prod_{g\in G}g\cdot(a,b)=\prod_{h\in
H}h\cdot\left(\prod_{s\in S}s\cdot(a,b)\right)=\prod_{h\in
H}h\cdot(a^{\prime},b^{\prime})=\mathcal{N}_{H,P,Q}(a^{\prime},b^{\prime}),$

where $(a^{\prime},b^{\prime})=\prod_{s\in S}s\cdot(a,b)\in
L^{\times}/(L^{\times})^{\ell}\times L^{\times}/(L^{\times})^{\ell}$. Fix a
basis $P,Q$ of $E[\ell]$, and let $h\in H$ with
$h^{-1}=\begin{pmatrix}c_{1}^{h}&c_{3}^{h}\\\
c_{2}^{h}&c_{4}^{h}\end{pmatrix}$. We have

$\displaystyle\mathcal{N}_{H,P,Q}(a,b)$ $\displaystyle=\left(\prod_{h\in
H}(h^{-1}(a))^{c_{1}^{h}}(h^{-1}(b))^{c_{3}^{h}},\prod_{h\in
H}(h^{-1}(a))^{c_{2}^{h}}(h^{-1}(b))^{c_{4}^{h}}\right).$

Since the field norm $\operatorname{Norm}_{L/K^{\prime}}$ commutes with
products and is invariant under $H$-action, we see that

$\displaystyle\operatorname{Norm}_{L/K^{\prime}}(\alpha)$
$\displaystyle=\left(\operatorname{Norm}_{L/K^{\prime}}(a^{\prime})\right)^{\sum_{h\in
H}c_{1}^{h}}\cdot\left(\operatorname{Norm}_{L/K^{\prime}}(b^{\prime})\right)^{\sum_{h\in
H}c_{3}^{h}}$ $\displaystyle\operatorname{Norm}_{L/K^{\prime}}(\beta)$
$\displaystyle=\left(\operatorname{Norm}_{L/K^{\prime}}(a^{\prime})\right)^{\sum_{h\in
H}c_{2}^{h}}\cdot\left(\operatorname{Norm}_{L/K^{\prime}}(b^{\prime})\right)^{\sum_{h\in
H}c_{4}^{h}}.$

Since $H\leq\operatorname{SL}_{2}(\operatorname{\mathbb{F}}_{\ell})$,
$\ell\neq 2$, and $1\neq\left|H\right|<\infty$, by Lemma 3.1 we have
$\sum_{h\in H}c_{i}^{h}=0$. The statement of the proposition follows
immediately. ∎

######  Corollary 3.3.

If $\left|G\right|=\left|\langle\sigma\rangle\right|=2$, then $M_{G,P,Q}$ is
exactly equal to the elements $(\alpha,\beta)$ of
$L^{\times}/(L^{\times})^{\ell}\times L^{\times}/(L^{\times})^{\ell}$ such
that $\operatorname{Norm}_{L/K}(\alpha)$ and
$\operatorname{Norm}_{L/K}(\beta)$ are 1 modulo $\ell^{th}$ powers. This is
because

$\mathcal{N}_{G,P,Q}(a,b)=\left(\frac{a}{\sigma(a)},\frac{b}{\sigma(b)}\right),$

and by Hilbert’s Theorem 90 [Hil97], all norm 1 elements are of this form.

The fact that the image of the restriction map on any subfield will always
give us elements of field norm 1 implies the following observation on how the
symbol length changes upon applying the corestriction map.

######  Lemma 3.4.

Suppose $\ell\nmid[L:K^{\prime}]$ and $(s,t)\in\operatorname{Br}(L(E))$ with
$\operatorname{Norm}_{L(E)/K^{\prime}(E)}{s}=1$ modulo $\ell^{\text{th}}$
powers. Then the length of the symbol

$\operatorname{Cor}_{L(E)/K^{\prime}(E)}(s,t)_{\ell,L(E)}$

is at most $[L:K^{\prime}]-1$.

###### Proof.

We apply 2.11. Let
$p_{s}:=x^{d}+e_{d-1}x^{d-1}+\dots+e_{1}x+(-1)^{d}e_{0}^{\ell}$ denote the
minimum polynomial of $s$ over $K^{\prime}(E)$. Let $g_{0},g_{1},\dots,g_{m}$
be the sequence of polynomials as in 2.11 with $g_{0}=p_{s}$. Then

$\displaystyle\operatorname{Cor}_{L(E)/K^{\prime}(E)}(s,t)_{\ell,L(E)}$
$\displaystyle=-\sum_{i=1}^{m}\left(c(g_{i-1}^{*}),c(g_{i})\right)_{\ell,K^{\prime}(E)}$
$\displaystyle=-((-1)^{2d}\cdot
e_{0}^{-\ell},c(g_{1}))_{\ell,K^{\prime}(E)}\otimes-\sum_{i=2}^{m}\left(c(g_{i-1}^{*}),c(g_{i})\right)_{\ell,K^{\prime}(E)}$
$\displaystyle=-\sum_{i=2}^{m}\left(c(g_{i-1}^{*}),c(g_{i})\right)_{\ell,K^{\prime}(E)}.$

2.11 implies that $m\leq\deg(g_{0})\leq[L:K^{\prime}]$, so the length of the
symbol

$\operatorname{Cor}_{L(E)/K^{\prime}(E)}(\alpha,\beta)_{L(E)}$

is at most $m-1\leq[L:K^{\prime}]-1$. ∎

We can apply this lemma to obtain our first upper bound on the symbol length
of $\mathbin{{}_{\ell}\text{Br}(E)}/\mathbin{{}_{\ell}\text{Br}(K)}$.

######  Corollary 3.5.

Suppose that $K\neq L$, that $\ell\nmid[L:K]$, and $(\alpha,\beta)\in
M_{G,P,Q}.$ Then the symbol length of

$\operatorname{Cor}_{L(E)/K(E)}(\alpha,t_{P})_{L(E)}\otimes\operatorname{Cor}_{L(E)/K(E)}(\beta,t_{Q})_{L(E)}$

and hence of $\mathbin{{}_{\ell}\text{Br}(E)}/\mathbin{{}_{\ell}\text{Br}(K)}$
is at most $2[L:K]-2$.

###### Proof.

Suppose $\ell\nmid[L:K]$ and $(\alpha,\beta)\in M_{G,P,Q}$. Since $K\neq L$,
we may apply Proposition 3.2 to see that
$\text{Norm}_{L/K}(\alpha)=\text{Norm}_{L/K}(\beta)=1$ modulo $\ell$-th
powers. By Lemma 3.4, the symbol length of

$\operatorname{Cor}_{L(E)/K(E)}(a,t_{P})_{L(E)}$

is at most $[L:K]-1$ and similarly for
$\operatorname{Cor}_{L(E)/K(E)}(b,t_{Q})_{L(E)}$. Hence the symbol length of
their tensor product is at most $2[L:K]-2$. ∎

##  4\. Bounds on the Symbol Length

In the previous section, we used 2.11 to obtain a bound on the symbol length
of corestriction. Our next step is to choose judiciously a basis $P,Q$ of
$E[\ell]$ in order to significantly improve the bound on the symbol length of
$\mathbin{{}_{\ell}\operatorname{Br}(E)}/\mathbin{{}_{\ell}\operatorname{Br}(K)}$.

######  Proposition 4.1.

If $G\leq\operatorname{SL}_{2}(\operatorname{\mathbb{F}}_{\ell})$ and there
exists $\sigma\in G$ with order $d>2$, then there exists a choice of basis
$P,Q$ such that $Q=\sigma(P)$ and any element $(\alpha,\beta)\in M_{G,P,Q}$
satisfies $\beta=\frac{1}{\sigma^{-1}(\alpha)}.$

###### Proof.

We first reduce to the case that $G$ is cyclic. Assume that the proposition
holds for some subgroup $H\leq G$, then the same also holds for $G$. If we
write $S$ for a set of representatives of right cosets of $H$, then for any
$(a,b)\in L^{\times}/(L^{\times})^{\ell}\times
L^{\times}/(L^{\times})^{\ell}$, we have

$\mathcal{N}(a,b)=\prod_{g\in G}g\cdot(a,b)=\prod_{h\in H}h\cdot\prod_{s\in
S}s\cdot(a,b)=\prod_{h\in H}h\cdot(a^{\prime},b^{\prime})=(\alpha,\beta).$

However, this is just $\mathcal{N}_{H}(a^{\prime},b^{\prime})$, so there
exists some $\sigma\in H$ such that $\beta=\frac{1}{\sigma^{-1}(\alpha)}$.

Suppose from now on that $G$ is cyclic of order $d$. Since $\left|G\right|>2$
and there is only one element in
$\operatorname{SL}_{2}(\operatorname{\mathbb{F}}_{\ell})$ of order equal to
$2$, we may assume that $d>2$. Each of the eigenvalues of $\sigma$ is a $d$-th
root of unity $\zeta_{d}$. Since $\sigma$ has determinant $1,$ its other
eigenvalue is $\zeta_{d}^{-1}.$ Note that we cannot have $\zeta_{d}=\pm 1$
since $d\neq 2$. Since $\ell\nmid\left|G\right|,$ we must have $\sigma=\pm I.$
In particular, we have $\zeta_{d}\neq\zeta_{d}^{-1}$ so the invariant factor
is $x^{2}-(\zeta_{d}+\zeta_{d}^{-1})x+1.$ We thus obtain a rational canonical
form

$\sigma=\begin{pmatrix}0&-1\\\ 1&\zeta_{d}+\zeta_{d}^{-1}\end{pmatrix}.$

A direct computation shows that for $k\geq 2,$

$\displaystyle\sigma^{k}$
$\displaystyle=\begin{pmatrix}-\sum\limits_{l=0}^{k-2}\zeta_{d}^{k-2-2l}&-\sum\limits_{l=0}^{k-1}\zeta_{d}^{k-1-2l}\\\
\sum\limits_{l=0}^{k-1}\zeta_{d}^{k-1-2l}&\sum\limits_{l=0}^{k}\zeta_{d}^{k-2l}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-(\zeta_{d}^{k-2}+\zeta_{d}^{k-4}+\cdots+\zeta_{d}^{4-k}+\zeta_{d}^{2-k})&-(\zeta_{d}^{k-1}+\zeta_{d}^{k-3}+\cdots+\zeta_{d}^{3-k}+\zeta_{d}^{1-k})\\\
\zeta_{d}^{k-1}+\zeta_{d}^{k-3}+\cdots+\zeta_{d}^{3-k}+\zeta_{d}^{1-k}&\zeta_{d}^{k}+\zeta_{d}^{k-2}+\cdots+\zeta_{d}^{2-k}+\zeta_{d}^{-k}\end{pmatrix}.$

By abuse of notation, we will denote by $\zeta_{d}+\zeta_{d}^{-1}$ both an
element of $\operatorname{\mathbb{F}}_{\ell}$ as well as any lift of this to
$\mathbb{Z}$. Thus we may raise any element of
$L^{\times}/(L^{\times})^{\ell}$ to the power of $\zeta_{d}+\zeta_{d}^{-1}$ in
a well-defined manner. In particular, the image of $(a,b)$ is $(\alpha,\beta)$
with

$\displaystyle\alpha$
$\displaystyle=\left(a^{-1}\sigma(b)\sigma^{2}(a)\sigma^{2}\left(b^{\zeta_{d}+\zeta_{d}^{-1}}\right)\cdots\sigma^{d-1}\left(a^{\zeta_{d}^{d-3}+\cdots+\zeta_{d}^{3-d}}\right)\sigma^{d-1}\left(b^{\zeta_{d}^{d-2}+\cdots+\zeta_{d}^{2-d}}\right)\right)^{-1},$
$\displaystyle\beta$
$\displaystyle=b\sigma(a)\sigma\left(b^{\zeta_{d}+\zeta_{d}^{-1}}\right)\cdots\sigma^{d-1}\left(a^{\zeta_{d}^{d-2}+\cdots+\zeta_{d}^{2-d}}\right)\sigma^{d-1}\left(b^{\zeta_{d}^{d-1}+\cdots+\zeta_{d}^{1-d}}\right).$

Noting the identities $\zeta_{d}^{d-1}+\cdots+\zeta_{d}^{1-d}=0$ (for even
$d,$ this is because for every element $a$ in this sum there is $-a,$ and for
odd $d$, all exponents occur exactly once) and
$\zeta_{d}^{d-2}+\cdots+\zeta_{d}^{2-d}=-1,$ we see that
$\beta=\frac{1}{\sigma^{-1}(\alpha)}$ as required. ∎

######  Remark 4.2.

If $G$ contains an element $\sigma$ of order 4, then picking a basis $P,Q$ as
in Proposition 4.1, we have $\beta=\sigma(\alpha)$. In this case, the rational
canonical form of $\sigma$ is

$\sigma=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}.$

Applying $\mathcal{N}$, we get that an element $(a,b)$ is mapped to

$(\alpha,\beta)=\left(\frac{a\sigma^{3}(b)}{\sigma^{2}(a)\sigma(b)},\frac{\sigma(a)b}{\sigma^{3}(a)\sigma^{2}(b)}\right).$

From here, it is clear that $\beta=\sigma(\alpha)$.

Using Proposition 4.1, we can now prove our main result.

See 1.1

###### Proof.

The image of $\varepsilon$ consists of corestrictions of elements of the form

$(\alpha,t_{P})_{\ell,L(E)}\otimes(\beta,t_{Q})_{\ell,L(E)}$

for $(\alpha,\beta)\in M_{G,P,Q}$. Fix $\sigma,P,Q$ as in Proposition 4.1.
Then by the remarks at the end of Section 2 and Proposition 4.1, the image
consists of corestrictions of elements of the form

$(\alpha,t_{P})_{\ell,L(E)}\otimes((\sigma^{-1}(\alpha))^{-1},t_{\sigma(P)})_{\ell,L(E)}.$

By Remark 2.6, the corestriction is invariant under $G$-action, so that the
corestriction of

$((\sigma^{-1}(\alpha))^{-1},t_{\sigma(P)})_{\ell,L(E)}$

is equal to the corestriction of

$\sigma^{*}((\sigma^{-1}(\alpha))^{-1},t_{\sigma(P)})=\left(\alpha^{-1},t_{\sigma^{2}(P)}\right)_{\ell,L(E)}=\left(\alpha,t_{\sigma^{2}(P)}^{-1}\right)_{\ell,L(E)}.$

Hence

$\displaystyle\operatorname{Cor}\left((\alpha,t_{P})_{\ell,L(E)}\otimes\left(\tfrac{1}{\sigma^{-1}(\alpha)},t_{\sigma(P)}\right)_{\ell,L(E)}\right)$
$\displaystyle=\operatorname{Cor}(\alpha,t_{P})_{\ell,L(E)}\otimes\operatorname{Cor}\left(\alpha,t_{\sigma^{2}(P)}^{-1}\right)_{\ell,L(E)}$
$\displaystyle=\operatorname{Cor}\left(\alpha,\tfrac{t_{P}}{\sigma^{2}(t_{P})}\right)_{\ell,L(E)}.$

By Proposition 3.2, $\operatorname{Norm}_{L/K}(\alpha)=1$ modulo
$\ell^{\text{th}}$ powers so by Lemma 3.4, we see that the corestriction of
this symbol is the sum of at most $[L:K]-1$ symbols. ∎

When $L/K$ contains an intermediate field extensions $K^{\prime}$, we can
factor the corestriction map through $K^{\prime}$ to further improve the bound
and prove our second main theorem.

See 1.2

###### Proof.

Let $H$ be an order $d$-subgroup of $G$. We write $K^{\prime}=L^{H}$ the fixed
field of $H$, so that $[L:K^{\prime}]=d$. We will factor the corestriction map
as

$\operatorname{Cor}_{\operatorname{Br}(E_{L})/\operatorname{Br}(E)}=\operatorname{Cor}_{\operatorname{Br}(E_{K^{\prime}})/\operatorname{Br}(E)}\circ\operatorname{Cor}_{\operatorname{Br}(E_{L})/\operatorname{Br}(E_{K^{\prime}})}.$

Let $\sigma$ be an element of $G$ of order $>2$. By Proposition 4.1, it
suffices to bound the symbol length of the corestriction of symbols of the
form

$\left(\alpha,\frac{t_{P}}{\sigma^{2}(t_{P})}\right)_{\ell,L(E)}=\left(\alpha,t_{P-\sigma^{2}(P)}\right)_{\ell,L(E)}$

Again by Proposition 3.2, $\operatorname{Norm}_{L/K^{\prime}}(\alpha)=1$
modulo $\ell^{\text{th}}$ powers, so using Lemma 3.4, we can bound the symbol
length of the corestriction of $(\ast)$ to $\operatorname{Br}(E_{K^{\prime}})$
by $[L:K^{\prime}]-1=d-1$. By applying 2.11, the corestriction of a symbol in
$\operatorname{Br}(E_{K^{\prime}})$ to $\operatorname{Br}(E)$ is the sum of at
most $[K^{\prime}:K]$ symbols, so that the corestriction of $(\ast)$ to
$\operatorname{Br}(E)$ is the sum of at most
$(d-1)[K^{\prime}:K]=(1-\frac{1}{d})[L:K]$ symbols. ∎

By Remark 2.2, we can apply Theorem 1.1 and Theorem 1.2 for any CM elliptic
curve over number fields.

See 1.4

###### Proof.

Since we assume that $\ell$ is an odd prime, we may ignore the case $[L:K]=2$.
In this case, 3.5 gives us an upper bound of 2 on the symbol length. Recall
that the order of the normalizer of a Cartan subgroup is either
$2(\ell^{2}-1)$ or $2(\ell-1)^{2}$, depending on whether we have a nonsplit or
split Cartan. Since $K/\mathbb{Q}$ is of degree at least $\ell-1$, we
therefore see that $G$ is a subgroup of a group of order $2(\ell+1)$ or
$2(\ell-1)$. If it is a strict subgroup of either, then the statement holds by
Theorem 1.1. Otherwise $G$ must contain an element of order 2, so the bound
holds by Theorem 1.2. ∎

##  5\. Examples

We provide an example to illustrate 1.4. Let $K$ be a number field, and
consider the infinite family of CM elliptic curves

$\mathcal{F}:y^{2}=x^{3}+c,\qquad c\in K.$

Let $E$ be a CM elliptic curve in the infinite family $\mathcal{F}$. We
compute the symbols in
$\mathbin{{}_{5}\operatorname{Br}(E)}/\mathbin{{}_{5}\operatorname{Br}(K)}$.
To see explicit generators of
$\mathbin{{}_{5}\operatorname{Br}(E)}/\mathbin{{}_{5}\operatorname{Br}(K)}$,
as well as the code written to perform these computations, please visit
https://github.com/team-brauer/brauer-groups 111All computations were done
using SageMath [Sag21].. Our example requires the following two propositions.
The first is a restatement of Theorem 1.4 (ii) in [LR19], and the second is an
application of Theorem 1.1 and Theorem 1.2 to the case $\ell=5$.

######  Proposition 5.1.

[LR19, Theorem 1.4(ii)] Let $E/\mathbb{Q}$ be an elliptic curve with $j(E)=0$.
If $\ell\equiv 2,5\mod 9$, then the image of the associated Galois
representation $\rho_{E,\ell}$ is contained in the normalizer of a nonsplit
Cartan.

######  Proposition 5.2.

If the image of the Galois representation

$\rho_{E,5}:\operatorname{Gal}(\overline{K}/K)\rightarrow\operatorname{SL}_{2}(\operatorname{\mathbb{F}}_{5})$

is contained in the normalizer of a nonsplit Cartan, then $[L:K]\mid 12$. In
particular, the symbol length of
$\mathbin{{}_{\ell}\operatorname{Br}(E)}/\mathbin{{}_{\ell}\operatorname{Br}(K)}$
is bounded above by 6.

###### Proof.

The order of a nonsplit Cartan subgroup is equal to $\ell^{2}-1=24$ since it
is isomorphic to $\operatorname{\mathbb{F}}_{\ell^{2}}^{\times}$. By [Lan95],
we know that a Cartan subgroup has index at most 2 in its normalizer, so the
image of $\rho_{E,5}$ is contained inside a group of order at most 48.
Therefore $[L:\mathbb{Q}]\mid 48$. However, since $K$ contains $\zeta_{5}$ we
see that $[K:\mathbb{Q}]$ is a multiple of 4, hence $[L:K]\mid 12$. The bound
on the symbol length follows from 1.4. ∎

Since an elliptic curve $E$ in the infinite family of CM elliptic curves
$\mathcal{F}$ satisfies $j(E)=0$, 1.4, Proposition 5.1, and Proposition 5.2
immediately imply that the symbol length of
$\mathbin{{}_{5}\operatorname{Br}(E)}/\mathbin{{}_{5}\operatorname{Br}(K)}$ is
bounded above by 6. In the following example, we use our code to compute the
symbols in
$\mathbin{{}_{5}\operatorname{Br}(E)}/\mathbin{{}_{5}\operatorname{Br}(K)}$
for a specific CM elliptic curve $E$ in the infinite family $\mathcal{F}$ and
illustrate Theorem 1.2.

######  Example 5.3.

Let $\zeta_{5}$ be a primitive 5th root of unity, and consider the number
field $K=\mathbb{Q}(\zeta_{5})$. Consider the CM elliptic curve over $K$ given
by the affine equation

$E:y^{2}=x^{3}+10.$

All computations for this example were done in SageMath [Sag21] and can be
found at https://github.com/team-brauer/brauer-groups. The division field
$L=K(E[5])$ is an extension of degree $4$ defined by adjoining a root $\mu$ of
the polynomial
$z^{4}+\left(6\zeta_{5}^{3}-6\zeta_{5}^{2}-6\zeta_{5}+6\right)z^{2}+24\zeta_{5}^{3}-12\zeta_{5}-12$.
We have an element $\sigma$ of order $4$ which sends

$\mu\to\left(\tfrac{1}{2}\zeta_{5}^{2}+\tfrac{1}{2}\zeta_{5}+\tfrac{1}{2}\right)\mu^{3}+\left(-\zeta_{5}^{3}-\zeta_{5}^{2}+4\right)\mu.$

Using SageMath, we choose a basis $P,Q$ of $E[5]$ as in Proposition 4.1 given
by

$\displaystyle P$
$\displaystyle=\left(-\zeta_{5}^{3}\mu^{2}-6\zeta_{5}^{3}-6\zeta_{5}+2,\left(-\zeta_{5}^{3}+\tfrac{1}{2}\zeta_{5}^{2}-\tfrac{1}{2}\zeta_{5}+1\right)\mu^{3}+(3\zeta_{5}^{3}+6\zeta_{5}^{2}-9\zeta_{5}+15)\mu\right),$
$\displaystyle Q$
$\displaystyle=\left(\zeta_{5}^{3}\mu^{2}+6\zeta_{5}^{3}+6\zeta_{5}^{2}+6\zeta_{5}+2,\left(\tfrac{1}{2}\zeta_{5}^{2}-\tfrac{1}{2}\zeta_{5}\right)\mu^{3}+\left(3\zeta_{5}^{3}+6\zeta_{5}^{2}+3\zeta_{5}+3\right)\mu\right).$

We observe that $\sigma(P)=Q$. Choose $(\alpha,\beta)\in M_{G,P,Q}$ to be the
image of $(\zeta_{5}+\mu,\mu)$ after applying $\mathcal{N}_{P,Q}$.
Specifically,

$\displaystyle\alpha$
$\displaystyle=(-16188/319831\zeta_{5}^{3}+15910/319831\zeta_{5}^{2}-4980/319831\zeta_{5}+14076/319831)\mu^{3}$
$\displaystyle+(32098/319831\zeta_{5}^{3}+11208/319831\zeta_{5}^{2}+30264/319831\zeta_{5}+16188/319831)\mu^{2}$
$\displaystyle+(-70930/319831\zeta_{5}^{3}+67262/319831\zeta_{5}^{2}-199162/319831\zeta_{5}+273158/319831)\mu$
$\displaystyle+138192/319831\zeta_{5}^{3}-128232/319831\zeta_{5}^{2}+344088/319831\zeta_{5}+390761/319831.$

Observe that $\sigma(\alpha)^{-1}=\beta$. Now by Theorem 1.2, it suffices to
compute the corestriction of

$(\alpha,t_{P}/t_{\sigma^{2}(P)})_{5,L(E)}.$

By Proposition 2.3, this is in fact equal to the corestriction of

$(\alpha,t_{2P})_{5,L(E)},$

since $\sigma^{2}(S)=2S$ for any $S\in E[\ell]$. We calculate this
corestriction by first passing through the intermediate field
$K^{\prime}=K(\zeta_{3})=\mathbb{Q}(\zeta_{15})$. The corestriction of the
symbol $(\alpha,t_{2P})_{5,L(E)}$ to $\mathbb{Q}(\zeta_{15})$ as calculated by
2.11 gives us two symbols but the first is trivial as $\alpha$ has norm $1$ by
Proposition 3.2. The second symbol is given by

$\displaystyle((($ $\displaystyle
551984402727902346319004580861749039896316402068750345026\cdot\zeta_{15}^{7}+$
$\displaystyle
134424070611518537543552069107213590707472267793980007392\cdot\zeta_{15}^{6}-\cdots)\cdot
x^{2}y^{10}+\cdots)$ $\displaystyle/((-$ $\displaystyle
134396143139335966898002857906327476642111742609439665612\zeta_{15}^{7}-$
$\displaystyle
358389715038229245061340954416873271045631313625172441632\zeta_{15}^{6}+\cdots)\cdot
x^{2}y^{10}+\cdots),$ $\displaystyle(($ $\displaystyle
86565057808308862398284296950\zeta_{15}^{7}-15301839790878217253755018864\zeta_{15}^{6}-\cdots)\cdot
x^{2}y^{4}+\cdots)$ $\displaystyle/(($ $\displaystyle
36812512953998458768246189661\zeta_{15}^{7}-23426144607089928307065757057\zeta_{15}^{5}+\cdots)\cdot
x^{2}y^{4}+\cdots)).$

The symbol in its entirety is approximately 53000 characters long and can be
viewed at https://github.com/team-brauer/brauer-groups.

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