A Sharp Lieb–Thirring Inequality for Functional Difference Operators

A Sharp Lieb–Thirring Inequality
for Functional Difference Operators††This paper is a contribution to the
Special Issue on Mathematics of Integrable Systems: Classical and Quantum in
honor of Leon Takhtajan. The full collection is available at
https://www.emis.de/journals/SIGMA/Takhtajan.html

Ari LAPTEV ab and Lukas SCHIMMER c

A. Laptev and L. Schimmer

a) Department of Mathematics, Imperial College London, London SW7 2AZ, UK
<EMAIL_ADDRESS>b) Saint Petersburg State University, Saint
Petersburg, Russia

c) Institut Mittag–Leffler, The Royal Swedish Academy of Sciences, 182 60
Djursholm, Sweden<EMAIL_ADDRESS>
Received September 12, 2021, in final form November 25, 2021; Published online
December 06, 2021

We prove sharp Lieb–Thirring type inequalities for the eigenvalues of a class
of one-dimensional functional difference operators associated to mirror
curves. We furthermore prove that the bottom of the essential spectrum of
these operators is a resonance state.

Lieb–Thirring inequality; functional difference operator; semigroup property

47A75; 81Q10

To our friend and coauthor Leon Takhtajan
on the occasion of his 70th birthday

##  1 Introduction

Let $P$ be the self-adjoint quantum mechanical momentum operator on
$L^{2}(\mathbb{R})$, i.e., $P=-\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}x}$ and
for $b>0$ denote by $U(b)$ the Weyl operator $U(b)=\exp(-bP)$. By using the
Fourier transform

$\displaystyle\widehat{\psi}(k)=(\mathcal{F}\psi)(k)=\int_{\mathbb{R}}\mathrm{e}^{-2\pi\mathrm{i}kx}\psi(x)\,{\rm
d}x$

we can write the domain of $U(b)$ as

$\displaystyle\operatorname{dom}(U(b))=\big{\\{}\psi\in
L^{2}(\mathbb{R})\colon\mathrm{e}^{-2\pi bk}\widehat{\psi}(k)\in
L^{2}(\mathbb{R})\big{\\}}.$

Equivalently, $\operatorname{dom}(U(b))$ consists of those functions $\psi(x)$
which admit an analytic continuation to the strip
$\\{z=x+\mathrm{i}y\in\mathbb{C}\colon 0<y<b\\}$ such that
$\psi(x+\mathrm{i}y)\in L^{2}(\mathbb{R})$ for all $0\leq y<b$ and there is a
limit $\psi(x+\mathrm{i}b-\mathrm{i}0)=\lim_{\varepsilon\to
0^{+}}\psi(x+\mathrm{i}b-\mathrm{i}\varepsilon)$ in the sense of convergence
in $L^{2}(\mathbb{R})$, which we will denote simply by $\psi(x+\mathrm{i}b)$.
The domain of the inverse operator $U(b)^{-1}$ can be characterised similarly.

For $b>0$ we define the operator $W_{0}(b)=U(b)+U(b)^{-1}=2\cosh(bP)$ on the
domain

$\displaystyle\operatorname{dom}(W_{0}(b))=\big{\\{}\psi\in
L^{2}(\mathbb{R})\colon 2\cosh(2\pi bk)\widehat{\psi}(k)\in
L^{2}(\mathbb{R})\big{\\}}.$

The operator $W_{0}(b)$ is self-adjoint and unitarily equivalent to the
multiplication operator $2\cosh(2\pi bk)$ in Fourier space. Its spectrum is
thus absolutely continuous covering the interval $[2,\infty)$ doubly.

Let $V\geq 0$, $V\in L^{1}(\mathbb{R})$ now be a real-valued potential
function. The scalar inequality $2\cosh(2\pi bk)-2\geq(2\pi bk)^{2}$ implies
the operator inequality

$\displaystyle W_{0}(b)-2\geq-b^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}$
(1.1)

on $\operatorname{dom}(W_{0}(b))$. By Sobolev’s inequality, we can conclude
that the operator

$\displaystyle W_{V}(b)=W_{0}(b)-V$

is symmetric and bounded from below on the common domain of $W_{0}(b)$ and
$V$. We can thus consider its Friedrichs extension, which we continue to
denote by $W_{V}(b)$. This operator acts as

$\displaystyle(W_{V}(b)\psi)(x)=\psi(x+\mathrm{i}b)+\psi(x-\mathrm{i}b)-V(x)\psi(x).$

Furthermore, by an application of Weyl’s theorem (in a version for quadratic
forms) and Rellich’s lemma together with the fact that the form domain of
$W_{0}(b)$ is continuously embedded in $H^{1}(\mathbb{R})$ (as discussed at
the beginning of Section 4) the spectrum of $W_{V}(b)$ consists of essential
spectrum $[2,\infty)$ and discrete finite-multiplicity eigenvalues below.
Details of this argument in the similar case of a Schrödinger operator can be
found in the upcoming book [2, Proposition 4.14].

We will show that the discrete spectrum satisfies a version of Lieb–Thirring
inequalities for $1/2$-Riesz means. When formulating the main result of the
paper it is convenient to parametrise the eigenvalues (repeated with
multiplicities) as $\lambda_{j}=-2\cos(\omega_{j})$, where
$\omega_{j}\in[0,\pi]$ for $\lambda_{j}\in[-2,2]$ and
$\omega_{j}\in\mathrm{i}[0,\infty)$ for $\lambda_{j}\leq-2$. Note that in the
latter case $\lambda_{j}=-2\cosh(|\omega_{j}|)$.

######  Theorem 1.1.

Let $V\geq 0$ and let $V\in L^{1}(\mathbb{R})$. If $W_{V}(b)\geq-2$, then the
discrete eigenvalues $\lambda_{j}=-2\cos(\omega_{j})\in[-2,2)$ $($repeated
with multiplicities$)$ satisfy

$\displaystyle\sum_{j\geq 1}\frac{\sin\omega_{j}}{\omega_{j}}\leq\frac{1}{2\pi
b}\int_{\mathbb{R}}V(x)\,{\rm d}x.$ (1.2)

The constant in the inequality (1.2) is sharp in the sense that there is a
potential $V$ such that (1.2) becomes equality.

######  Remark 1.2.

Note that Theorem 1.1 does not allow to estimate eigenvalues below $-2$. In
fact, from the proof of this theorem, the case of one eigenvalue below $-2$
could be included in the inequality (1.2). We expect that the inequality holds
true for all eigenvalues below $-2$. However, the method we use in the proof
prevents us from including all eigenvalues due to oscillating properties of
the resolvent $(W_{0}(b)-\lambda)^{-1}$ for $\lambda<-2$.

Lieb–Thirring inequalities were first established for Schrödinger operators in
[15]. For a one-dimensional Schrödinger operator
${-}\frac{\mathrm{d}}{\mathrm{d}x^{2}}-V$ on $L^{2}(\mathbb{R})$ with negative
eigenvalues $\mu_{1}\leq\mu_{2}\allowbreak\leq\\!\cdots\\!<0,\\!$ these bounds
state that for any $\gamma\geq 1/2$ there is a constant $L_{\gamma}>0$ such
that

$\displaystyle\sum_{j\geq 1}|\mu_{j}|^{\gamma}\leq
L_{\gamma}\int_{\mathbb{R}}V(x)^{\gamma+1/2}\,{\rm d}x$ (1.3)

for all $V\geq 0$, $V\in L^{\gamma+1/2}(\mathbb{R})$. The condition
$\gamma\geq 1/2$ is optimal. Inequality (1.1) implies that

$\displaystyle\sum_{j\geq
1}|\lambda_{j}-2|^{\gamma}\leq\frac{L_{\gamma}}{b}\int_{\mathbb{R}}V(x)^{\gamma+1/2}\,{\rm
d}x$ (1.4)

for all eigenvalues $\lambda_{j}\leq 2$ of $W_{V}(b)$. Under the additional
assumption $W_{V}(b)\geq-2$, our bound (1.2) presents an improvement of (1.4)
for $\gamma=1/2$. This can be seen from the fact that for $\gamma=1/2$ the
sharp constant in (1.3) is given by $L_{1/2}=1/2$ [7] and from the strict
inequality

$\displaystyle|\lambda_{j}-2|^{\frac{1}{2}}=|2\cos\omega_{j}+2|^{\frac{1}{2}}<\frac{\pi\sin\omega_{j}}{\omega_{j}}$

for $\omega_{j}\in[0,\pi)$. The difference of the terms above vanishes as
$\omega_{j}\to\pi$, implying that (1.4) is asymptotically optimal for small
coupling. While the necessity of $\gamma\geq 1/2$ in the Lieb–Thirring
inequality for Schrödinger operators does not allow us to conclude that (1.4)
fails for $0\leq\gamma<1/2$, we will prove the following.

######  Theorem 1.3.

Let $b>0$. If $V\in L^{1}(\mathbb{R})$ with $\int_{\mathbb{R}}V\,{\rm d}x>0$,
then $W_{V}(b)$ has at least one eigenvalue below $2$. Furthermore, if
$0\leq\gamma<1/2$, then there is no constant $L_{\gamma}$ such that (1.4)
holds for all compactly supported $V$. This conclusion holds even under the
assumption that $W_{V}(b)\geq-2$.

The study of different properties of functional difference operators
$W_{V}(b)$ was considered before. In the case when $-V=V_{0}=\mathrm{e}^{2\pi
bx}$ is an exponential function, the operator $W_{V_{0}}(b)$ first appeared in
the study of the quantum Liouville model on the lattice [1] and plays an
important role in the representation theory of the non-compact quantum group
$\mathrm{SL}_{q}(2,\mathbb{R})$. The spectral analysis of this operator was
first studied in [9], see also [17]. In the case when $-V=2\cosh(2\pi bx)$ the
spectrum of $W_{V}(b)$ is discrete and converges to $+\infty$. Its Weyl
asymptotics were obtained in [13]. This result was extended to a class of
growing potentials in [14]. More information on spectral properties of
functional difference operators can be found in papers [4, 5, 10, 11, 16].

The proof method of Theorem 1.1 is similar to the proof of the sharp
Lieb–Thirring inequality (1.3) for a one-dimensional Schrödinger operator in
the case $\gamma=1/2$ as presented in [6]. It relies on a property of
convolutions of the resolvent kernels of the operator under consideration.
Such a semigroup property was also recently established for Jacobi operators
where it was again used to prove sharp Lieb–Thirring type inequalities [12].
With a different proof (not using the convolution property) the sharp
inequalities for the Schrödinger operator and the Jacobi operator were first
obtained in [7] and in [8], respectively. Despite formal similarity to the
case of Jacobi operators, it is still surprising that the proof method works
for functional difference operators $W_{V}(b)$. These operators could be
considered as differential operators of infinite order since the symbol
$\cosh(2\pi bk)$ can be written as an infinite Taylor series of symbols of
even degree w.r.t. the variable $k$.

##  2 Free resolvent

Since $W_{0}(b)\geq 2$ we conclude that $W_{0}(b)-\lambda$ is an invertible
operator for $\lambda<2$. Let $\lambda=-2\cos(\omega)$ with $\omega\in[0,\pi]$
if $\lambda\in[-2,2]$ and $\omega\in\mathrm{i}[0,\infty)$ if $\lambda<-2$.
Then in Fourier space the inverse of $W_{0}(b)-\lambda$ is given by the
multiplication operator $(2\cosh(2\pi bk)+2\cos(\omega))^{-1}$.

Applying the inverse Fourier transform $\mathcal{F}^{-1}$ to $(2\cosh(2\pi
bk)+2\cos(\omega))^{-1}$ we find the kernel of the free resolvent
$G_{\lambda}=(W_{0}(b)-\lambda)^{-1}$ that is

$\displaystyle
G_{\lambda}(x,y)=G_{\lambda}(x-y)=\frac{1}{2b\sin\omega}\frac{\sinh\big{(}\frac{\omega}{b}(x-y)\big{)}}{\sinh\big{(}\frac{\pi}{b}(x-y)\big{)}}.$
(2.1)

######  Remark 2.1.

Note that $G_{\lambda}(x-y)$ is an even and positive kernel for
$\omega\in[0,\pi]$ and it becomes oscillating if
$\omega\in\mathrm{i}(0,\infty)$. This fact is one of the reasons why we are
able to study Lieb–Thirring inequalities only for the eigenvalues
$\lambda_{j}\in[-2,2]$. This interval contains all of the discrete spectrum if
the potential $V$ is “small” enough. However, if $V$ generates eigenvalues
lying in $(-\infty,-2)$, then the oscillating property of the Green’s function
prevents us from obtaining the desired inequality for all eigenvalues.

Note that the value of $G_{\lambda}$ on the diagonal $x=y$ takes the form

$\displaystyle G_{\lambda}(0)=\frac{1}{2\pi b}\frac{\omega}{\sin\omega}$ (2.2)

and we can see the relation between the right-hand side of (2.2) and the
expression in the left-hand side of (1.2). Due to our parameterisation of the
spectral parameter, the convergence $\lambda\to 2^{-}$ implies
$\omega\to\pi^{-}$ and thus

$\displaystyle
G_{\lambda}(0)\sim\frac{1}{2b}\frac{1}{\sqrt{1-\cos^{2}\omega}}\sim\frac{1}{2b}\frac{1}{\sqrt{2-\lambda}}\qquad{\rm
as}\quad\lambda\to 2^{-}.$

If $\lambda\to-\infty$, then $\omega\to\mathrm{i}\infty$ and

$\displaystyle G_{\lambda}(0)\sim\frac{1}{\pi b}|\lambda|^{-1}\log|\lambda|.$

In [17] L. Faddeev and L.A. Takhtajan studied the resolvent in a slightly
different form

$\displaystyle
G_{\lambda}(x,y)=\frac{\sigma}{\sinh\big{(}\frac{\pi\mathrm{i}\varkappa}{\sigma}\big{)}}\bigg{(}\frac{\mathrm{e}^{-2\pi\mathrm{i}\varkappa(x-y)}}{1-\mathrm{e}^{-4\pi\mathrm{i}\sigma(x-y)}}+\frac{\mathrm{e}^{2\pi\mathrm{i}\varkappa(x-y)}}{1-\mathrm{e}^{4\pi\mathrm{i}\sigma(x-y)}}\bigg{)},$

which coincides with (2.1) with $\sigma=\mathrm{i}/2b$,
$\lambda=2\cosh(2b\pi\varkappa)$ and
$\varkappa=\frac{\omega-\pi}{2\pi\mathrm{i}b}$. It was pointed out that the
free resolvent can be written using the analogues of the Jost solutions

$\displaystyle f_{-}(x,\varkappa)=\mathrm{e}^{-2\pi\mathrm{i}\varkappa
x}\qquad{\rm and}\qquad f_{+}(x,\varkappa)=\mathrm{e}^{2\pi\mathrm{i}\varkappa
x}$

that appear in the theory of one-dimensional Schrödinger operators. Namely

$\displaystyle
G_{\lambda}(x-y)=\frac{2\sigma}{C(f_{-},f_{+})(\varkappa)}\bigg{(}\frac{f_{-}(x,\varkappa)f_{+}(y,\varkappa)}{1-\mathrm{e}^{\frac{\pi\mathrm{i}}{\sigma^{\prime}}(x-y)}}+\frac{f_{-}(y,\varkappa)f_{+}(x,\varkappa)}{1-\mathrm{e}^{-\frac{\pi\mathrm{i}}{\sigma^{\prime}}(x-y)}}\bigg{)},$

where $\sigma^{\prime}\sigma=-1/4$ and where $C(f,g)$ is the so-called
Casorati determinant (a difference analogue of the Wronskian) of the solutions
of the functional-difference equation

$\displaystyle
C(f,g)(x,\varkappa)=f(x+2\sigma^{\prime},\varkappa)g(x,\varkappa)-f(x,\varkappa)g(x+2\sigma^{\prime},\varkappa).$

For the Jost solutions
$C(f_{-},f_{+})(x,\varkappa)=2\sinh\big{(}\frac{\pi\mathrm{i}\kappa}{\sigma}\big{)}$.

The equality $(W_{0}(b)-\lambda)G(x-y)=\delta(x-y)$ could be interpreted as an
equation of distributions. Since the functions $f_{\pm}(x,k)$ are Jost
solutions, the distribution defined by $(W_{0}(b)-\lambda)\times G(x-y)$ is
supported only at $x=y$, and its singular part coincides with the singular
part of the distribution

$\displaystyle-\frac{2\sigma\sigma^{\prime}}{\pi\mathrm{i}C(f_{-},f_{+})(\varkappa)}\bigg{(}\frac{f_{-}(x+2\sigma^{\prime},\varkappa)f_{+}(y,\varkappa)-f_{-}(y,\varkappa)f_{+}(x+2\sigma^{\prime},\varkappa)}{x-y-\mathrm{i}0}$
$\displaystyle\hphantom{-\frac{2\sigma\sigma^{\prime}}{\pi\mathrm{i}C(f_{-},f_{+})(\varkappa)}\bigg{(}}{}+\frac{f_{-}(x-2\sigma^{\prime},\varkappa)f_{+}(y,\varkappa)-f_{-}(y,\varkappa)f_{+}(x-2\sigma^{\prime},\varkappa)}{x-y+\mathrm{i}0}\bigg{)}$

in the neighbourhood of $x=y$. This singular part is equal to

$\displaystyle-\frac{2\sigma\sigma^{\prime}}{\pi\mathrm{i}}\bigg{(}\frac{1}{x-y-\mathrm{i}0}-\frac{1}{x-y+\mathrm{i}0}\bigg{)}=\delta(x-y),$

where the authors used the Sokhotski–Plemelj formula. This formula is similar
to the respective formula for a Schrödinger operator when the Dirac
$\delta$-function appears by differentiating a step function.

##  3 Proof of inequality (1.2)

###  3.1 Some auxiliary results

We first collect some results from [6] verbatim. Let $A$ be a compact operator
on a Hilbert space $\mathcal{G}$ and let us denote

$\displaystyle\|A\|_{n}=\sum_{j=1}^{n}\sqrt{\lambda_{j}(A^{*}A)},$

where $\lambda_{j}(A^{*}A)$ are the eigenvalues of $A^{*}A$ in decreasing
order. Then by Ky Fan’s inequality (see for example [3, Lemma 4.2]) the
functionals $\|\cdot\|_{n}$, $n=1,2,\dots$, are norms and thus for any unitary
operator $Y$ in $\mathcal{G}$ we have

$\displaystyle\|Y^{*}AY\|_{n}=\|A\|_{n}.$

######  Definition 3.1.

Let $A$, $B$ be two compact operators on $\mathcal{G}$. We say that $A$
majorises $B$ or $B\prec A$, iff

$\displaystyle\|B\|_{n}\leq\|A\|_{n},\qquad\text{for all}\quad
n\in\mathbb{N}.$

######  Lemma 3.2.

Let $A$ be a nonnegative compact operator acting in $\mathcal{G}$,
$\\{Y(k)\\}_{k\in\mathbb{R}}$ be a family of unitary operators on
$\mathcal{G}$, and let $g(k)\,{\rm d}k$ be a probability measure on
$\mathbb{R}$. Then the operator

$\displaystyle B=\int_{\mathbb{R}}Y(k)^{*}AY(k)g(k)\,{\rm d}k$

is majorised by $A$.

###### Proof.

This is a simple consequence of the triangle inequality

$\displaystyle\|B\|_{n}\leq\int_{\mathbb{R}}\|Y^{*}(k)AY(k)\|_{n}g(k)\,{\rm
d}k=\|A\|_{n}\int_{\mathbb{R}}g(k)\,{\rm d}k=\|A\|_{n}.$ ∎

Let $\lambda_{j}=-2\cos\omega_{j}\leq 2$ be the eigenvalues of $W_{0}(b)-V$
with $V\geq 0$. In order to slightly simplify the notations it is convenient
to write

$\displaystyle\lambda_{j}=-2\cos\big{(}\sqrt{\theta_{j}}\big{)}$

with $\theta_{j}\in\big{(}{-}\infty,\pi^{2}\big{]}$ and
$\omega_{j}^{2}=\theta_{j}$.

Let us denote by $K_{\lambda}$ the Birman–Schwinger operator

$\displaystyle K_{\lambda}=V^{1/2}G_{\lambda}V^{1/2}.$ (3.1)

Let $\mu_{j}(K_{\lambda})$ be the eigenvalues (in decreasing order) of the
Birman–Schwinger operator $K_{\lambda}$ defined in (3.1). Then due to the
Birman–Schwinger principle we have

$\displaystyle 1=\mu_{j}(K_{\lambda_{j}}).$ (3.2)

Let us define the operator

$\displaystyle
L_{\theta}:=\frac{1}{G_{-2\cos\sqrt{\theta}}(0)}K_{-2\cos\sqrt{\theta}},$

where $G_{-2\cos\sqrt{\theta}}(0)=\frac{1}{2\pi
b}\frac{\sqrt{\theta}}{\sin\sqrt{\theta}}$ is given in (2.2). Then from (3.2)
we obtain

$\displaystyle\sum_{j\geq 1}\frac{1}{G_{\lambda_{j}}(0)}=\sum_{j\geq
1}\frac{1}{G_{\lambda_{j}}(0)}\mu_{j}(K_{\lambda_{j}})=\sum_{j\geq
1}\mu_{j}(L_{\theta_{j}}).$

The integral kernel of the operator $L_{\theta}$ is given by
$\sqrt{V(x)}g_{\pi^{2},\theta}(x-y)\sqrt{V(y)}$, where

$\displaystyle
g_{\pi^{2},\theta}(x):=\frac{\pi}{\sqrt{\theta}}\frac{\sinh\big{(}\frac{\sqrt{\theta}}{b}x\big{)}}{\sinh\big{(}\frac{\pi}{b}x\big{)}}.$

Consider a more general function

$\displaystyle
g_{\varphi,\theta}(x):=\frac{\sqrt{\varphi}}{\sqrt{\theta}}\frac{\sinh\big{(}\frac{\sqrt{\theta}}{b}x\big{)}}{\sinh\big{(}\frac{\sqrt{\varphi}}{b}x\big{)}}.$

Since $g_{\varphi,\theta}(0)=1$ its Fourier transform
$\widehat{g}_{\varphi,\theta}=\mathcal{F}(g_{\varphi,\theta})$ satisfies the
equation

$\displaystyle\int_{\mathbb{R}}\widehat{g}_{\varphi,\theta}(k)\,{\rm d}k=1.$

Moreover, for any $-\infty<\theta<\varphi$ with $0<\varphi<\pi^{2}$ we have

$\displaystyle\widehat{g}_{\varphi,\theta}(k)=\mathcal{F}\Bigg{(}\frac{\sqrt{\varphi}}{\sqrt{\theta}}\frac{\sinh\big{(}\frac{\sqrt{\theta}}{b}x\big{)}}{\sinh\big{(}\frac{\sqrt{\varphi}}{b}x\big{)}}\Bigg{)}(k)=\frac{2\pi\sin\big{(}\pi\frac{\sqrt{\theta}}{\sqrt{\varphi}}\big{)}}{\sqrt{\theta}}\frac{b}{2\cosh\big{(}\frac{2\pi^{2}bk}{\sqrt{\varphi}}\big{)}+2\cos\big{(}\frac{\pi\sqrt{\theta}}{\sqrt{\varphi}}\big{)}},$

and the right-hand side is positive. Thus $\widehat{g}_{\varphi,\theta}\,{\rm
d}k$ is a probability measure for such values.

Note also that importantly

$\displaystyle\frac{g_{\pi^{2},\theta}(x)}{g_{\pi^{2},\theta^{\prime}}(x)}=\frac{\sqrt{\theta^{\prime}}}{\sqrt{\theta}}\frac{\sinh\big{(}\frac{\sqrt{\theta}}{b}x\big{)}}{\sinh\big{(}\frac{\sqrt{\theta^{\prime}}}{b}x\big{)}}=g_{\theta^{\prime},\theta}(x)$

and therefore

$\displaystyle\big{(}\widehat{g}_{\pi^{2},\theta^{\prime}}*\widehat{g}_{\theta^{\prime},\theta}\big{)}(k)=\widehat{g}_{\pi^{2},\theta}(k).$

This is the interesting convolution/semigroup property mentioned in the
introduction. In the special case $-\infty<\theta<0=\theta^{\prime}$ analogous
computations lead to the same result with
$\widehat{g}_{0,\theta}(k)=\chi_{[-1,1]}\big{(}2\pi
bk/\sqrt{|\theta|}\big{)}\pi b/\sqrt{|\theta|}$.

######  Lemma 3.3 (monotonicity).

For $(\theta,\theta^{\prime})$ such that $-\infty<\theta\leq\theta^{\prime}$
and $0\leq\theta^{\prime}<\pi^{2}$ we have $L_{\theta}\prec
L_{\theta^{\prime}}$.

###### Proof.

Let $Y(k)\colon L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the unitary
multiplication operator

$\displaystyle(Y(k)\psi)(x)=\mathrm{e}^{-2\pi\mathrm{i}kx}\psi(x)$

and let $T$ be the projection onto $V^{1/2}$, i.e.,

$\displaystyle(T\psi)(x)=V^{1/2}(x)\int_{\mathbb{R}}V^{1/2}(y)\psi(y)\,{\rm
d}y.$

Using $Y(k^{\prime}+k^{\prime\prime})=Y(k^{\prime})Y(k^{\prime\prime})$ and
Lemma 3.2 we obtain

$\displaystyle L_{\theta}$
$\displaystyle=\int_{\mathbb{R}}Y(k)^{*}TY(k)\widehat{g}_{\pi^{2},\theta}(k)\,{\rm
d}k$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}}Y(k)^{*}TY(k)\widehat{g}_{\pi^{2},\theta^{\prime}}(k^{\prime})\widehat{g}_{\theta^{\prime},\theta}(k-k^{\prime})\,{\rm
d}k^{\prime}{\rm d}k$
$\displaystyle=\int_{\mathbb{R}}Y(k^{\prime\prime})^{*}\bigg{(}\int_{\mathbb{R}}Y(k^{\prime})^{*}TY(k^{\prime})\widehat{g}_{\pi^{2},\theta^{\prime}}(k^{\prime})\,{\rm
d}k^{\prime}\bigg{)}Y(k^{\prime\prime})\widehat{g}_{\theta^{\prime},\theta}(k^{\prime\prime})\,{\rm
d}k^{\prime\prime}\prec L_{\theta^{\prime}},$

where we have used that $\widehat{g}_{\theta^{\prime},\theta}\,{\rm d}k$ is a
probability measure. ∎

######  Remark 3.4.

With a slight abuse of notations, Lemma 3.3 says that $L_{\lambda}\prec
L_{\lambda^{\prime}}$ for any $\lambda<2$ as long as
$\lambda\leq\lambda^{\prime}$ and $-2\leq\lambda^{\prime}<2$.

###  3.2 Proof of inequality (1.2)

We now enumerate the eigenvalues of the operator $W_{V}(b)$ belonging to the
interval $[-2,2)$ such that
$-2\leq\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\cdots$ repeated with
multiplicity. By using the monotonicity established in Lemma 3.3 we have a
sequence of inequalities

$\displaystyle\frac{1}{G_{\lambda_{1}}(0)}=2\pi
b\frac{\sin\omega_{1}}{\omega_{1}}=\mu_{1}(L_{\theta_{1}})\leq\mu_{1}(L_{\theta_{2}}),$
$\displaystyle\sum_{j=1}^{2}\frac{1}{G_{\lambda_{j}}(0)}=2\pi
b\sum_{j=1}^{2}\frac{\sin\omega_{j}}{\omega_{j}}\leq\sum_{j=1}^{2}\mu_{j}(L_{\theta_{2}})\leq\sum_{j=1}^{2}\mu_{j}(L_{\theta_{3}}),$
$\displaystyle\sum_{j=1}^{3}\frac{1}{G_{\lambda_{j}}(0)}=2\pi
b\sum_{j=1}^{3}\frac{\sin\omega_{j}}{\omega_{j}}\leq\sum_{j=1}^{3}\mu_{j}(L_{\theta_{3}})\leq\sum_{j=1}^{3}\mu_{j}(L_{\theta_{4}}),\qquad{\rm
etc}.$

Note that we do not use any assumptions on the multiplicities of the
eigenvalues, other than their finiteness. Furthermore, by Lemma 3.3 the same
results also hold true if a single eigenvalue is below $-2$. Continuing the
above process and noting that the trace of $L_{\theta}$ is
$\int_{\mathbb{R}}V\,{\rm d}x$ for all $\theta$, we finally obtain

$\displaystyle\sum_{j\geq 1}\frac{\sin\omega_{j}}{\omega_{j}}\leq\frac{1}{2\pi
b}\int_{\mathbb{R}}V(x)\,{\rm d}x.$

The proof is complete.

######  Remark 3.5.

Note that $\frac{2\cosh(2\pi bk)-2}{b^{2}}\to(2\pi k)^{2}$ tends to the symbol
of the second derivative as $b\to 0$ and that $W_{b^{2}V}(b)\geq-2$ for
sufficiently small $b$. We thus expect that it should be possible to recover
the Lieb–Thirring inequality (1.3) for a Schrödinger operator with the sharp
constant $L_{1/2}=1/2$ from Theorem 1.1.

##  4 Sharpness of inequality (1.2)

Similarly to the case of Schrödinger operators, we aim to prove that the
Lieb–Thirring inequality becomes an equality for Dirac-delta potentials. To
this end let $c>0$ and consider the potential $V_{c}(x)=c\delta(x)$. To
properly define $W_{V_{c}}(b)$, we first note that the quadratic form
${\left\langle\psi,(W_{0}(b)-2)\psi\right\rangle}$ can be written as

$\displaystyle{\left\langle\psi,(W_{0}(b)-2)\psi\right\rangle}=\int_{\mathbb{R}}\big{|}2\sinh(\pi
bk)\widehat{\psi}(k)\big{|}^{2}\,{\rm
d}k=\int_{\mathbb{R}}|\psi(x+\mathrm{i}b/2)-\psi(x-\mathrm{i}b/2)|^{2}\,{\rm
d}x.$ (4.1)

This can be seen by introducing the self-adjoint operator
$D(b)=U(b/2)-U(b/2)^{-1}=2\sinh\big{(}\frac{bP}{2}\big{)}$ and checking that
$D(b)^{2}=W_{0}(b)-2$ either directly or by means of the identity $\cosh(2\pi
bk)-1\allowbreak=2\sinh(\pi bk)^{2}$. The form domain of $W_{0}(b)$ is thus
$\operatorname{dom}(D(b))=\operatorname{dom}(W_{0}(b/2))\subset
H^{1}(\mathbb{R})$ and on this domain Sobolev’s inequality yields that

$\displaystyle|\psi(0)|^{2}$
$\displaystyle\leq\varepsilon\int_{\mathbb{R}}|\psi^{\prime}(x)|^{2}\,{\rm
d}x+\frac{1}{\varepsilon}\int_{\mathbb{R}}|\psi(x)|^{2}\,{\rm d}x$
$\displaystyle\leq\frac{\varepsilon}{b^{2}}\int_{\mathbb{R}}\big{|}2\sinh(\pi
bk)\widehat{\psi}(k)\big{|}^{2}\,{\rm
d}k+\frac{1}{\varepsilon}\int_{\mathbb{R}}|\psi(x)|^{2}\,{\rm d}x$

for any choice of $\varepsilon>0$. The KLMN theorem thus allows us to define
$W_{0}(b)-V_{c}$. As a rank one perturbation of the operator $W_{0}(b)$ the
potential $V_{c}$ generates no more than one eigenvalue below the continuous
spectrum $[2,\infty)$.

In Fourier space the eigenequation
$(W_{0}(b)-c\delta)\psi_{c}=\lambda\psi_{c}$ becomes

$\displaystyle 2\cosh(2\pi
bk)\widehat{\psi_{c}}(k)-c\psi_{c}(0)=\lambda\widehat{\psi_{c}}(k)$

by means of the formal identity $\mathcal{F}(\delta\psi_{c})=\psi_{c}(0)$.
Writing again $\lambda=-2\cos\omega$ we obtain

$\displaystyle\widehat{\psi_{c}}(k)=\frac{c\psi_{c}(0)}{2\cosh(2\pi
bk)+2\cos\omega}$ (4.2)

and therefore

$\displaystyle\psi_{c}(x)=c\psi_{c}(0)G_{-2\cos\omega}(x)=\frac{c\psi_{c}(0)}{2b\sin\omega}\frac{\sinh\big{(}\frac{\omega}{b}x\big{)}}{\sinh\big{(}\frac{\pi}{b}x\big{)}}.$
(4.3)

Of course we could have seen this immediately by using the equation for the
Green’s function

$\displaystyle(W_{0}(b)+2\cos\omega)G_{-2\cos\omega}(x)=\delta(x).$

Letting $x\to 0$ in (4.3) we find

$\displaystyle 1=\frac{c}{2b\sin\omega}\frac{\omega}{\pi}$

or equivalently

$\displaystyle\frac{\sin\omega}{\omega}=\frac{c}{2\pi b}.$ (4.4)

Since $\frac{\sin\sqrt{\theta}}{\sqrt{\theta}}$ is a monotone decreasing
function of $\theta=\omega^{2}\in\big{(}{-}\infty,\pi^{2}\big{]}$ that takes
all values in $[0,\infty)$, for any $c>0$ there is a unique solution
$\omega_{c}$ to (4.4) and vice versa. If $c/(2\pi b)<1$ then
$\omega_{c}\in(0,\pi)$ and otherwise $\omega_{c}\in\mathrm{i}[0,\infty)$.
Since $\int V_{c}\,{\rm d}x=c$, the identity (4.4) can be rewritten as

$\displaystyle\frac{\sin\omega}{\omega}=\frac{1}{2\pi
b}\int_{\mathbb{R}}V_{c}(x)\,{\rm d}x$

showing that the Lieb–Thirring inequality is satisfied for potentials
$-c\delta$ with a single eigenvalue that can be placed anywhere in
$(-\infty,2)$ by choosing $c>0$ suitably.

######  Remark 4.1.

If we choose the normalising constant $\psi(0)>0$ then the eigenfunction
defined in (4.3)

$\displaystyle\psi_{c}(x)=\frac{c\psi(0)}{2b\sin\omega_{c}}\frac{\sinh\big{(}\frac{\omega_{c}}{b}x\big{)}}{\sinh\big{(}\frac{\pi}{b}x\big{)}}$

is positive assuming that the coupling constant $c$ is small enough satisfying
the inequality $c/(2\pi b)\leq 1$ and thus $\omega_{c}\in[0,\pi)$. Note that
if $c/(2\pi b)=1$ then $\omega_{c}=0$ and

$\displaystyle\psi_{c}(x)=\frac{\pi\psi(0)x}{b\sinh\big{(}\frac{\pi}{b}x\big{)}}>0.$

However, if the coupling constant $c>2\pi b$ then
$\omega_{c}\in\mathrm{i}(0,\infty)$ and hence

$\displaystyle\psi_{c}(x)=\frac{c\psi(0)}{2b\sinh|\omega_{c}|}\frac{\sin\big{(}\frac{|\omega_{c}|}{b}x\big{)}}{\sinh\big{(}\frac{\pi}{b}x\big{)}}$

is an oscillating function and in particular has an infinite number of zeros.
This contradicts a possible conjecture that the eigenfunction for the lowest
eigenvalue is strictly positive.

Open problem. Assume that the discrete spectrum $\sigma_{\rm d}(W_{V}(b))$ of
the operator $W_{V}(b)$ satisfies the property $\sigma_{\rm
d}(W_{V}(b))\subset[-2,2)$. Is it true that the eigenfunction corresponding to
the lowest eigenvalue could be chosen strictly positive?

##  5 Necessity of $\boldsymbol{\gamma\geq 1/2}$

The following argument is similar to that presented in the upcoming book [2,
Propositions 4.41 and 4.42] for the case of a Schrödinger operator. For
$\varepsilon>0$ let $\psi_{\varepsilon}(x)=1/\cosh(2\varepsilon x/b)$. If
$\varepsilon$ is sufficiently small, say $\varepsilon\leq\varepsilon_{0}$,
then $\psi_{\varepsilon}\in\operatorname{dom}(W_{0}(b))$. Using (4.1) we
compute that

$\displaystyle{\left\langle\psi_{\varepsilon},(W_{0}(b)-2)\psi_{\varepsilon}\right\rangle}=\frac{b\sin^{2}\varepsilon}{2\varepsilon}\int_{\mathbb{R}}\bigg{|}\frac{2\sinh
x}{\cos^{2}\varepsilon\cosh^{2}x+\sin^{2}\varepsilon\sinh^{2}x}\bigg{|}^{2}\,{\rm
d}x\leq Cb\varepsilon$ (5.1)

for a constant $C>0$ independent of $\varepsilon\leq\varepsilon_{0}$. For any
potential $V\in L^{1}(\mathbb{R})$ it holds that
${\left\langle\psi_{\varepsilon},V\psi_{\varepsilon}\right\rangle}\to\int_{\mathbb{R}}V\,{\rm
d}x$ as $\varepsilon\to 0$ by dominated convergence and thus for sufficiently
small $\varepsilon$

$\displaystyle{\left\langle\psi_{\varepsilon},(W_{V}(b)-2)\psi_{\varepsilon}\right\rangle}<0.$

By the min-max principle this proves the first part of Theorem 1.3.

For the second assertion of the theorem we choose more specifically the
compactly supported potential $V(x)=c\chi_{[-1/2,1/2]}(x/b)$. By Sobolev’s
inequality $W_{V}(b)\geq-2$ for sufficiently small $c\leq c_{0}$ such that all
the discrete eigenvalues of $W_{V}(b)$ are contained in $[-2,2)$. Furthermore
${\left\|\psi_{\varepsilon}\right\|}^{2}=b/\varepsilon$ and, since $\tanh
x\geq x/2$ for $0\leq x\leq 1$,

$\displaystyle{\left\langle\psi_{\varepsilon},V\psi_{\varepsilon}\right\rangle}=cb\int_{-1/2}^{1/2}|\cosh(2\varepsilon
x)|^{-2}\,{\rm d}x=\frac{cb\tanh\varepsilon}{\varepsilon}\geq\frac{1}{2}cb$
(5.2)

for $\varepsilon\leq 1$. We now choose $\varepsilon=c\delta$. If
$\delta\leq\min(\varepsilon_{0}/c_{0},1/c_{0})$ such that
$\varepsilon\leq\min(\varepsilon_{0},1)$, then (5.1) and (5.2) both hold and

$\displaystyle\frac{{\left\langle\psi_{\varepsilon},(W_{V}(b)-2)\psi_{\varepsilon}\right\rangle}}{{\left\|\psi_{\varepsilon}\right\|}^{2}}\leq
C\varepsilon^{2}-\frac{1}{2}c\varepsilon=c^{2}\delta\bigg{(}C\delta-\frac{1}{2}\bigg{)}.$

Choosing $\delta<\min(\varepsilon_{0}/c_{0},1/c_{0},1/2C)$ we can conclude by
the min-max principle that $W_{V}(b)-2$ has a negative eigenvalue
$\lambda_{1}\leq-c^{2}\delta\big{(}\frac{1}{2}-C\delta\big{)}$. If a
Lieb–Thirring inequality (1.4) were to hold for $\gamma<1/2$ then for some
finite $L_{\gamma}$

$\displaystyle
c^{2\gamma}\delta^{\gamma}\bigg{(}\frac{1}{2}-C\delta\bigg{)}^{\gamma}\leq\frac{L_{\gamma}}{b}\int_{\mathbb{R}}V(x)^{\gamma+\frac{1}{2}}\,{\rm
d}x=L_{\gamma}c^{\gamma+\frac{1}{2}},$

which is clearly a contradiction if $c\to 0$.

### Acknowledgements

A. Laptev was partially supported by RSF grant 18-11-0032. L. Schimmer was
supported by VR grant 2017-04736 at the Royal Swedish Academy of Sciences. The
authors would like to thank the anonymous referees for their useful comments
to improve the article.

## References

  * [1] Faddeev L.D., Takhtajan L.A., Liouville model on the lattice, in Field Theory, Quantum Gravity and Strings (Meudon/Paris, 1984/1985), Lecture Notes in Phys., Vol. 246, Springer, Berlin, 1986, 166–179.
  * [2] Frank R.L., Laptev A., Weidl T., Schrödinger operators: eigenvalues and Lieb–Thirring inequalities, in preparation.
  * [3] Gohberg I.C., Kreĭn M.G., Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, Amer. Math. Soc., Providence, R.I., 1969.
  * [4] Grassi A., Hatsuda Y., Mariño M., Topological strings from quantum mechanics, Ann. Henri Poincaré 17 (2016), 3177–3235, arXiv:1410.3382.
  * [5] Gu J., Klemm A., Mariño M., Reuter J., Exact solutions to quantum spectral curves by topological string theory, J. High Energy Phys. 2015 (2015), no. 10, 025, 69 pages, arXiv:1506.09176.
  * [6] Hundertmark D., Laptev A., Weidl T., New bounds on the Lieb–Thirring constants, Invent. Math. 140 (2000), 693–704, arXiv:math-ph/9906013.
  * [7] Hundertmark D., Lieb E.H., Thomas L.E., A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator, Adv. Theor. Math. Phys. 2 (1998), 719–731, arXiv:math-ph/9806012.
  * [8] Hundertmark D., Simon B., Lieb–Thirring inequalities for Jacobi matrices, J. Approx. Theory 118 (2002), 106–130, arXiv:math-ph/0112027.
  * [9] Kashaev R., The quantum dilogarithm and Dehn twists in quantum Teichmüller theory, in Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory (Kiev, 2000), NATO Sci. Ser. II Math. Phys. Chem., Vol. 35, Kluwer Acad. Publ., Dordrecht, 2001, 211–221.
  * [10] Kashaev R., Mariño M., Operators from mirror curves and the quantum dilogarithm, Comm. Math. Phys. 346 (2016), 967–994, arXiv:1501.01014.
  * [11] Kashaev R., Mariño M., Zakany S., Matrix models from operators and topological strings, 2, Ann. Henri Poincaré 17 (2016), 2741–2781, arXiv:1505.02243.
  * [12] Laptev A., Loss M., Schimmer L., On a conjecture by Hundertmark and Simon, arXiv:2012.13793.
  * [13] Laptev A., Schimmer L., Takhtajan L.A., Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves, Geom. Funct. Anal. 26 (2016), 288–305, arXiv:1510.00045.
  * [14] Laptev A., Schimmer L., Takhtajan L.A., Weyl asymptotics for perturbed functional difference operators, J. Math. Phys. 60 (2019), 103505, 10 pages.
  * [15] Lieb E.H., Thirring W.E., Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, in Studies in Mathematical Physics, Princeton University Press, Princeton, 1976, 269–303.
  * [16] Takhtajan L.A., Trace formulas for the modified Mathieu equation, in Partial Differential Equations, Spectral Theory, and Mathematical Physics, Editors P. Exner, R. Frank, F. Gesztesy, H. Holden, T. Weidl, European Mathematical Society, Berlin, 2021, 427–443, arXiv:2103.00038.
  * [17] Takhtajan L.A., Faddeev L.D., The spectral theory of a functional-difference operator in conformal field theory, Izv. Math. 79 (2015), 388–410, arXiv:1408.0307.


# Local conductor bounds for modular abelian varieties

Kimball Martin<EMAIL_ADDRESS>Department of Mathematics, University of
Oklahoma, Norman, OK 73019 USA

###### Abstract.

Brumer and Kramer gave bounds on local conductor exponents for an abelian
variety $A/\mathbb{Q}$ in terms of the dimension of $A$ and the localization
prime $p$. Here we give improved bounds in the case that $A$ has maximal real
multiplication, i.e., $A$ is isogenous to a factor of the Jacobian of a
modular curve $X_{0}(N)$. In many cases, these bounds are sharp. The proof
relies on showing that the rationality field of a newform for $\Gamma_{0}(N)$,
and thus the endomorphism algebra of $A$, contains
$\mathbb{Q}(\zeta_{p^{r}})^{+}$ when $p$ divides $N$ to a sufficiently high
power. We also deduce that certain divisibility conditions on $N$ determine
the endomorphism algebra when $A$ is simple.

The author was partially supported by the Japan Society for the Promotion of
Science (Invitational Fellowship L22540), and the Osaka Central Advanced
Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and
Theoretical Physics JPMXP0619217849).

##  1\. Introduction

Let $A/\mathbb{Q}$ be a $d$-dimensional abelian variety of conductor $N_{A}$.
Brumer and Kramer [brumer-kramer] gave upper bounds $v_{p}(N_{A})\leq B(p,d)$
on local conductor exponents in terms of $p$ and $d$. Moreover, their bounds
are sharp in the sense for any $p,d$, there exists an abelian variety $A$ with
$v_{p}(N_{A})=B(p,d)$. The abelian variety $A$ is said to have real
multiplication (RM) if the endomorphism algebra
$\mathrm{End}^{0}(A):=\mathrm{End}(A)\otimes\mathbb{Q}$ contains a totally
real number field $K\supsetneq\mathbb{Q}$. Brumer and Kramer suggested that
stronger bounds on local conductor exponents may exist when restricting to
abelian varieties with RM.

We say $A$ has maximal RM if $\mathrm{End}^{0}(A)$ contains a totally real
field $K$ of degree $d=\dim A$. (Necessarily, $[K:\mathbb{Q}]\mid d$ for any
subfield $K$ of $\mathrm{End}^{0}(A)$.) Abelian varieties with maximal RM are
of GL(2)-type, and they are simple over $\mathbb{Q}$ if and only if
$\mathrm{End}^{0}(A)\simeq K$ is a totally real field of degree $d$
[ribet:korea, Theorem 2.1]. Up to isogeny, the simple abelian varieties with
maximal RM are precisely the simple factors of Jacobians
$J_{0}(N)=\mathrm{Jac}(X_{0}(N))$ of the modular curves $X_{0}(N)$. If $N$ is
minimal such that $A$ is isogenous to a simple factor of $J_{0}(N)$, then
$N_{A}=N^{d}$.

We improve the Brumer–Kramer bounds for abelian varieties with maximal RM.
Define

(1.1) $B_{0}(p,d)=\begin{cases}8+2v_{2}(d)&\text{if }p=2,\\\
5+2v_{3}(d)&\text{if }p=3,\\\ 4+2v_{p}(d)&\text{if }p\geq 5\text{ and
}(p-1)\mid 2d,\\\ 2&\text{else}.\end{cases}$

######  Theorem 1.1.

Let $A/\mathbb{Q}$ be a $d$-dimensional simple abelian variety with maximal RM
and conductor $N^{d}$.

  1. (1)

We have $v_{p}(N)\leq B_{0}(p,d)$, i.e., $v_{p}(N_{A})\leq dB_{0}(p,d)$.

  2. (2)

The bounds in (1) are stronger than what the Brumer–Kramer bounds imply.
Namely, $B_{0}(p,d)\leq\lfloor\frac{B(p,d)}{d}\rfloor$ for all $p,d$, and this
is a strict inequality if either (a) $5\leq p<2d+1$ and $(p-1)\nmid 2d$, or
(b) $p\leq 3$, $d>3$ and $p\nmid d$. It is an equality when $p\geq 2d+1$. (In
other cases, this is sometimes an equality and sometimes not.)

  3. (3)

The bound in (1) sharp, i.e., $v_{p}(N)=B_{0}(p,d)$ occurs for some such $A$,
for all $(p,d)$ such that $d\leq 10$, with the possible exclusion of the
following 5 cases: $B_{0}(5,10)=6$, $B_{0}(11,10)=4$, and $B_{0}(2d+1,d)=4$
for $d=6,7,8$.

######  Remark 1.2.

If $A$ is a $d$-dimensional simple abelian variety of
$\operatorname{GL}(2)$-type, i.e., it is isogenous to a factor of some
$J_{1}(N)=\mathrm{Jac}(X_{1}(N))$, then one similarly has the bounds
$v_{p}(N_{A})\leq dB_{0}(p,d)$ for $p$ odd, and $v_{2}(N_{A})\leq
d(B_{0}(2,d)+1)$. This improves the Brumer–Kramer bounds for odd $p$, and for
certain values of $d$ when $p=2$. See Section 3.4.

######  Remark 1.3.

If $A$ is not simple, then applying the above bounds to its simple factors
yield even stronger bounds in terms of $d$. E.g., suppose $A\simeq A_{1}\times
A_{2}$ where $A_{1}$ and $A_{2}$ are isogenous simple abelian varieties with
endomorphism algebra $K_{0}$, a totally real field of degree $d/2$. Then
$\mathrm{End}^{0}(A)\simeq M_{2}(K_{0})$, which contains totally real fields
of degree $d$, and thus $A$ has maximal RM. Since $N_{A}=N_{A_{1}}^{2}$, one
sees that $v_{p}(N_{A})\leq 2v_{p}{N_{A_{1}}}\leq dB_{0}(p,d/2)$. This is
better than the bound $v_{p}(N_{A})\leq dB_{0}(p,d)$ for simple $A$.

The precise formula for the Brumer–Kramer bounds is slightly complicated—see
Section 3.1 for details—but we list the bounds
$B^{\prime}(p,d):=\lfloor\frac{B(p,d)}{d}\rfloor$ (i.e., the Brumer–Kramer
bounds applied to $v_{p}(N)$ with $N$ and $A$ as in 1.1) for $d\leq 10$ in
Table 1. For $p>2d+1$, $B^{\prime}(p,d)=B_{0}(p,d)=2$, and we omit these
entries. We write the bound $B_{0}(p,d)$ from 1.1 in parentheses when it is
smaller.

In this table, we bolded all of the cases where we have checked the upper
bound is sharp, by finding associated modular forms. We also starred the cases
$B_{0}(19,9)$ and $B_{0}(11,10)$ to indicate the upper bounds are at least
“almost sharp”, in the sense that $v_{p}(N)=B_{0}(p,d)-1$ is attained. Note
that by quadratic twisting at $p$, an upper bound of $B_{0}(p,d)=2$ is always
sharp, provided there exists at least one simple $d$-dimensional abelian
variety $A/\mathbb{Q}$ with maximal RM. It seems plausible that the bound
$B_{0}(p,d)$ is always sharp, but we do not know of constructions of simple
abelian varieties with maximal RM (or more generally of GL(2) type) in higher
dimensions which would shed light on this.

| $p=2$ | 3 | 5 | 7 | 11 | 13 | 17 | 19
---|---|---|---|---|---|---|---|---
$d=1$ | 8 | 5 |  |  |  |  |  |
2 | 10 | 5 | 4 |  |  |  |  |
3 | 9 (8) | 7 | 3 (2) | 4 |  |  |  |
4 | 12 | 6 (5) | 4 | 3 (2) |  |  |  |
5 | 11 (8) | 6 (5) | 4 (2) | 3 (2) | 4 |  |  |
6 | 11 (10) | 7 | 4 | 4 | 3 (2) | 4 |  |
7 | 10 (8) | 6 (5) | 4 (2) | 4 (2) | 3 (2) | 3 (2) |  |
8 | 14 | 6 (5) | 4 | 3 (2) | 3 (2) | 3 (2) | 4 |
9 | 13 (8) | 9 | 4 (2) | 4 | 3 (2) | 3 (2) | 3 (2) | 4*
10 | 13 (10) | 8 (5) | 6 | 4 (2) | 4* | 3 (2) | 3 (2) | 3 (2)
Table 1. Brumer–Kramer bounds $B^{\prime}(p,d)$ for simple maximal RM, with
the bounds $B_{0}(p,d)$ in parentheses when different; bounds known to be
sharp (resp. almost sharp) are bolded (resp. starred)

The proof of part (1) relies on a result about rationality fields of modular
forms. Denote by $\zeta_{m}$ a primitive $m$-th root of unity in
$\bar{\mathbb{Q}}$. Let
$\mathbb{Q}(\zeta_{m})^{+}=\mathbb{Q}(\zeta_{m}+\zeta_{m}^{-1})$ be the
maximal totally real subfield of $\mathbb{Q}(\zeta_{m})$. Write
$K_{f}=\mathbb{Q}((a_{n})_{n})$ for the rationality field of a cuspidal
newform $f=\sum a_{n}q^{n}$.

Let $f\in S_{2k}(N)$ be a newform. (Here $S_{*}(N)$ indicates level
$\Gamma_{0}(N)$ with trivial nebentypus). Then 2.1 states that when $v_{p}(N)$
is sufficiently large, $K_{f}\supset\mathbb{Q}(\zeta_{p^{r}})^{+}$, where $r$
is a certain function of $v_{p}(N)$ that grows like $v_{p}(N)/2$.
Consequently, if we fix the (rationality) degree $d:=[K_{f}:\mathbb{Q}]$ of
$f$, then this places bounds on $v_{p}(N)$, precisely $v_{p}(N)\leq
B_{0}(p,N)$. Applying these bounds for a weight 2 newform $f$ associated to
$A$ leads to the first part of the theorem.

2.1 is not entirely novel. Brumer [brumer, Theorem 5.5] proved a version of it
in terms of abelian varieties of GL(2)-type, which suffices to deduce 1.2.
However, to our knowledge, 1.2 has not appeared in the literature. Moreover,
2.1 is sharper for $p=2$, and our proof also allows us to say something more
about the local representations at $p=3$ (see 2.3).

Apart from sharpening Brumer’s result when $p=2$, what is new here is an
explicit formulation of the bounds $B_{0}(p,d)$, a direct comparison with the
Brumer–Kramer bounds, and a computational investigation of whether they are
sharp. In fact, 2.1 tells us more than just the bounds in 1.1. We explicitly
spell out the stronger conclusions one can make for $2\leq d\leq 6$.

######  Proposition 1.4.

Let $A$ be a $d$-dimensional simple abelian variety with maximal RM of
conductor $N^{d}$. Set $K=\mathrm{End}^{0}(A)$, which is a totally real number
field of degree $d$.

  1. (i)

Suppose $d=2$. If $2^{9}\mid N$, then $K=\mathbb{Q}(\sqrt{2})$. If $5^{3}\mid
N$, then $K=\mathbb{Q}(\sqrt{5})$. In particular, $2^{9}\cdot 5^{3}\mid N$ is
impossible.

  2. (ii)

Suppose $d=3$. If $3^{6}\mid N$, then $K=\mathbb{Q}(\zeta_{9})^{+}$. If
$7^{3}\mid N$, then $K=\mathbb{Q}(\zeta_{7})^{+}$. Hence $3^{6}\cdot 7^{3}\mid
N$ is impossible.

  3. (iii)

Suppose $d=4$. If $2^{11}\mid N$, then $K=\mathbb{Q}(\zeta_{16})^{+}$. If
$2^{9}\mid N$ (resp $5^{3}\mid N$), then $K$ contains $\mathbb{Q}(\sqrt{2})$
(resp. $\mathbb{Q}(\sqrt{5})$). Hence if $2^{9}\cdot 5^{3}\mid N$, then
$K=\mathbb{Q}(\sqrt{2},\sqrt{5})$. Further, $N$ cannot be divisible by
$2^{11}\cdot 5^{3}$.

  4. (iv)

Suppose $d=5$. If $11^{3}\mid N$, then $K=\mathbb{Q}(\zeta_{11})^{+}$.

  5. (v)

Suppose $d=6$. If $2^{9}\cdot 3^{6}$, $2^{9}\cdot 7^{3}$, $3^{6}\cdot 5^{3}$,
$5^{3}\cdot 7^{3}$, or $13^{3}$ divides $N$, then
$K=\mathbb{Q}(\sqrt{2},\zeta_{9}+\zeta_{9}^{-1})$,
$\mathbb{Q}(\sqrt{2},\zeta_{7}+\zeta_{7}^{-1})$,
$\mathbb{Q}(\sqrt{5},\zeta_{9}+\zeta_{9}^{-1})$,
$\mathbb{Q}(\sqrt{5},\zeta_{7}+\zeta_{7}^{-1})$, or
$\mathbb{Q}(\zeta_{13})^{+}$, respectively. Further, $N$ cannot be divisible
by $2^{9}\cdot 5^{3}$, $2^{9}\cdot 13^{3}$, $3^{6}\cdot 7^{3}$, $3^{6}\cdot
13^{3}$, or $7^{3}\cdot 13^{3}$.

######  Corollary 1.5.

Let $C/\mathbb{Q}$ be a genus $2$ curve of conductor $N$. Let $A$ be the
Jacobian of $C$. Suppose $C$ has RM (i.e., $A$ has RM). If $N$ is divisible by
$5^{6}$ (resp. $2^{18}$), then $A$ is simple and
$\mathrm{End}^{0}(A)\simeq\mathbb{Q}(\sqrt{5})$ (resp.
$\mathrm{End}^{0}(A)\simeq\mathbb{Q}(\sqrt{2})$).

Parts (i)–(v) of 1.4 also hold verbatim when $f\in S_{2k}(N)$ is a degree $d$
newform with $K=K_{f}$. These statements are proved in Section 3.3.

These consequences were initially quite striking to us for the following
reason. One can specify a finite number of local discrete series components
(and thus local conductors) of automorphic representations in the trace
formula, and asymptotically count such representations (e.g., see
[weinstein]). Each of these fixed local components merely contribute
independent local densities (specified by the Plancherel measure) to this
asymptotic. By analogy, one might guess that, when counting weight 2 newforms
of a fixed degree $d$ with a finite number of local conductors specified, the
asymptotic may be a product of local densities for each of these local
conductors. Indeed, the work [SSW] on counting elliptic curves suggests this
is the case for $d=1$.

In contrast, there exist degree 2 newforms (or, if one prefers, genus 2 curves
with RM) whose level is divisible by 125, and also degree 2 newforms with
level divisible by 512. A naive guess is that each of these sets make up some
positive proportions, say $\delta_{125}$ and $\delta_{512}$, of all degree 2
newforms, and that the proportion of all degree 2 newforms with level
divisible by $125\cdot 512$ should be $\delta_{125}\cdot\delta_{512}>0$. But
the corollary tells us there are no such forms! (In fact, [CM] suggests 100%
of all degree 2 forms may have rationality field $\mathbb{Q}(\sqrt{5})$, so it
may be that $\delta_{512}=0$, but a local-global counting principle would
still suggest there should be infinitely many such forms.) Hence 1.4 and 1.5
say that, for $d\geq 2$, local conductor behavior is not independent, at least
in the case of “sufficiently wild” ramification.

##  2\. Rationality subfields for modular forms

######  Proposition 2.1.

Let $f\in S_{2k}(N)$ be a newform and fix a prime $p$. Suppose $p^{3}\mid N$
if $p$ is odd and $p^{9}\mid N$ if $p=2$. Set $r=\lceil
v_{p}(N)/2-v_{p}(3)/2\rceil-1-v_{p}(2)$. Then
$\mathbb{Q}(\zeta_{p^{r}})^{+}\subset K_{f}$. In particular,
$\frac{1}{2}p^{r-1}(p-1)\mid[K_{f}:\mathbb{Q}]$.

The lower bound of 3 or 9 on $v_{p}(N)$ in the hypothesis is the minimum
needed to get a nontrivial conclusion, except when $p=3$ where one needs
$3^{6}\mid N$.

This proposition refines earlier results of Saito and Brumer. Namely, [saito,
Corollary 3.4] uses certain operators on $S_{2k}(N)$ to obtain a weaker
version of this proposition with $r=\lfloor v_{p}(N)/3\rfloor-v_{p}(2)$. In
the case $2k=2$, [brumer, Theorem 5.5] obtains a version of this result in the
context of abelian varieties of $\operatorname{GL}(2)$-type with $r=\lceil
v_{p}(N)/2-1-\frac{1}{p-1}\rceil-v_{p}(2)$.111There is a typo in the $p=2$
case of [brumer, Theorem 5.5(i)]—the field in the $p=2$ case should read as in
part (ii) of _loc. cit._ Brumer remarks that his approach also applies to
higher weights. This is almost the same as our value of $r$, but is smaller by
1 when $p=2$ and $v_{2}(N)$ is odd. E.g., when $p=2$, Brumer’s result only
implies $\mathbb{Q}(\sqrt{2})\subset K_{f}$ if $v_{2}(N)\geq 10$, but this
proposition says $v_{2}(N)\geq 9$ suffices. So this proposition should be
viewed as a slight sharpening of Brumer’s result when $p=2$.

We will prove this proposition by examining local Galois types of modular
forms, following [DPT]. In fact, the proof amounts to making some arguments in
[DPT] slightly more precise, and correcting an error therein. The basic idea
is the same as Brumer’s: show $\zeta_{p^{r}}+\zeta_{p^{r}}^{-1}$ appears as a
trace of an appropriate representation. However, our proof uses an explicit
description of possible representations, and we obtain slightly more
information when $p=2,3$.

In fact neither Saito’s nor Brumer’s result, nor the argument we give,
requires trivial nebentypus. We assume it for simplicity so that we can
directly apply [DPT].

###### Proof.

Let $\pi_{p}$ be the smooth irreducible representation of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$ associated to $f$. Since we assumed
$v_{p}(N)\geq 3$, either $\pi_{p}$ is an irreducible principal series
representation or a supercuspidal representation (i.e., twisted Steinberg is
not possible). For a local representation $\sigma$, we write $c(\sigma)$ for
its conductor.

First assume $\pi_{p}=\pi(\mu,\mu^{-1})$ is an irreducible principal series.
Then $c(\pi_{p})=2n$, where $n=c(\mu)$. Now $\mu$ factors through
$\mathbb{Z}_{p}^{\times}/(1+p^{n}\mathbb{Z}_{p}^{\times})\simeq(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$,
which is cyclic if $p$ is odd and abstractly isomorphic to $C_{2}\times
C_{2^{n-2}}$ if $p=2$. Since $\mu|_{\mathbb{Z}_{p}^{\times}}$ does not factor
though $1+p^{n-1}\mathbb{Z}_{p}^{\times}$, necessarily the order of
$\mu|_{\mathbb{Z}_{p}^{\times}}$ is a multiple of $p^{n-1-v_{2}(p)}$. Now
[DPT, Lemma 3.1] implies that $\zeta_{p^{r}}+\zeta_{p^{r}}^{-1}\in K_{f}$,
where $r=n-1-v_{2}(p)=v_{p}(N)/2-1-v_{2}(p)$. (This lemma is proved by looking
at the images of the associated $\ell$-adic Galois representations
$\rho_{f,\lambda}$.) This proves the proposition in the principal series case.

Now assume $\pi_{p}$ is supercuspidal. All the details we will need about
supercuspidal representations are recalled or proved in [DPT, Section 2], with
a correction noted below. Necessarily, $\pi_{p}$ is dihedral if $p$ is odd. If
$p=2$, there are non-dihedral supercuspidal representations $\pi$, but they
have conductor exponent $c(\pi)\leq 8$ (in fact $\leq 7$), in which case the
proposition is vacuous. Thus for any $p$ we may assume that $\pi_{p}$ is
dihedral. This means that $\pi=\pi_{p}$ corresponds to the induction
$\operatorname{Ind}_{W_{\mathbb{Q}_{p}}}^{W_{E}}\theta$ of a regular character
$\theta$ of the Weil group of some quadratic extension $E/\mathbb{Q}_{p}$.
Write $n=c(\theta)$.

First suppose $E/\mathbb{Q}_{p}$ is unramified. Then $c(\pi)=2n$. As explained
in the proof of [DPT, Lemma 2.11], the order of $\theta$ is a multiple of
$p^{n-1-v_{2}(p)}$.

Next suppose $E/\mathbb{Q}_{p}$ is ramified with $p$ odd. Then $c(\pi)=n+1$
and $n\geq 2$. _Loc. cit._ states that $n$ must be even and the order of
$\theta$ is a multiple of $p^{n/2}$. This is correct except in the special
case that $E\simeq\mathbb{Q}_{3}(\sqrt{-3})$, where we can only say that the
order of $\theta$ is a multiple of $3^{n/2-1}$. The difference in the exponent
in this case arises from difference in the structure of
$\mathfrak{o}_{E}/\mathfrak{p}_{E}^{j}$ when
$E\simeq\mathbb{Q}_{3}(\sqrt{-3})$ due to the presence of extra roots of
unity. The structure of $\mathfrak{o}_{E}/\mathfrak{p}_{E}^{j}$ is described
in [DPT, Theorem 2.10] and implies the above fact about the order of $\theta$.
(We remark that there is a typo in this theorem: $a$ and $b$ should be swapped
in third line of data in [DPT, Table 2], which corresponds to
$E\neq\mathbb{Q}_{3}(\sqrt{-3})$ is ramified with $p\neq 2$, and gives the
exponent $\frac{n}{2}$ in this case. Neither this typo nor the above
correction affect the other proofs or results in [DPT].)

Finally suppose $p=2$ and $E/\mathbb{Q}_{2}$ is ramified. Then
$c(\pi)=n+\delta$, where $\delta\in\\{2,3\\}$ is the 2-adic valuation of the
discriminant of $E$. As remarked above, when $p=2$, we may assume
$c(\pi)=n+\delta\geq 9$, in which case $n$ is necessarily even. When
$\delta=2$, i.e., $E\simeq\mathbb{Q}_{2}(\sqrt{-1})$ or
$\mathbb{Q}_{2}(\sqrt{3})$, then $\theta$ has order a multiple of $2^{n/2-1}$.
If $\delta=3$, i.e., $E\simeq\mathbb{Q}_{2}(\sqrt{\pm 2})$ or
$\mathbb{Q}_{2}(\sqrt{\pm 6})$, then $\theta$ has order a multiple of
$2^{n/2}$.

Thus in all dihedral supercuspidal cases of relevance, the order of $\theta$
is a multiple of $p^{r}$, where $r$ is as in the statement of the proposition.
Now [DPT, Lemma 3.2] implies $\zeta_{p^{r}}+\zeta_{p^{-r}}\in K_{f}$, as
desired. ∎

######  Corollary 2.2.

Suppose $f\in S_{2k}(N)$ is a newform of degree $d$. Then $v_{p}(N)\leq
B_{0}(p,d)$, where $B_{0}(p,d)$ is given in (1.1).

###### Proof.

Suppose $p\geq 5$. Since $B_{0}(p,d)\geq 2$, we may assume $v_{p}(N)\geq 3$.
Then 2.1 implies $\frac{1}{2}p^{r-1}(p-1)\mid d$, where
$r=\lceil\frac{v_{p}(N)}{2}\rceil-1\geq 1$. Hence $\frac{p-1}{2}\mid d$ and
$\lceil\frac{v_{p}(N)}{2}\rceil-2\leq v_{p}(d)$. Thus $v_{p}(N)\leq
B_{0}(p,d)$.

The same reasoning gives the result for $p=2$ and $p=3$. ∎

We point out one more consequence of the proof of 2.1. Given a newform $f\in
S_{2k}(N)$, denote by $\pi_{p}(f)$ the local admissible representation of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$ associated to $f$. An algorithm for
determining $\pi_{p}(f)$ is given in [LW]. The main difficulty is
distinguishing supercuspidal representations, which [LW] carries out using the
cohomology of the modular curve. Simply knowing $v_{p}(N)$ places many
restrictions on the possibilities for $\pi_{p}(f)$—in particular if
$v_{p}(N)\geq 3$ is odd, $\pi_{p}(f)$ must be supercuspidal. The following
gives an elementary way to partially distinguish supercuspidal components
using $K_{f}$ when $p=3$.

######  Corollary 2.3.

Let $f\in S_{2k}(N)$ be a newform and suppose $3^{2m+1}\parallel N$ with
$m\geq 1$. If $\mathbb{Q}(\zeta_{3^{m}})^{+}\not\subset K_{f}$, then
$\pi_{3}(f)$ is a supercuspidal representation dihedrally induced from
$E=\mathbb{Q}_{3}(\sqrt{-3})$.

###### Proof.

Since $c(\pi_{3}(f))=2m+1\geq 3$ is odd, $\pi_{3}(f)$ must be a dihedral
supercuspidal induced from a character $\theta$ along a ramified quadratic
extension $E/\mathbb{Q}_{3}$. Now the corollary follows from the difference in
exponents for our lower bounds for the order of $\theta$ in the cases
$E\not\simeq\mathbb{Q}_{3}(\sqrt{-3})$ and $E\simeq\mathbb{Q}_{3}(\sqrt{-3})$.
∎

For instance, up to Galois conjugacy, $S_{2}(243)$ has 2 rational newforms, 1
newform with rationality field $\mathbb{Q}(\sqrt{3})$, 1 newform with
rationality field $\mathbb{Q}(\sqrt{6})$, and 2 newforms with rationality
field $\mathbb{Q}(\zeta_{9})^{+}$. The 4 Galois orbits of newforms with
rationality fields of degree $\leq 2$ must all have local components $\pi_{3}$
being dihedral supercuspidal representations induced from
$\mathbb{Q}_{3}(\sqrt{-3})$.

##  3\. Local conductor bounds for modular abelian varieties

In this section we will prove the results stated in the introduction.

Given a weight 2 newform $f$ for $\Gamma_{0}(N)$ or $\Gamma_{1}(N)$, there is
an associated simple abelian variety $A_{f}$ which is a subquotient of
$J_{0}(N)$ or $J_{1}(N)$. Moreover, $\dim A_{f}=[K_{f}:\mathbb{Q}]$,
$\mathrm{End}^{0}(A_{f})\simeq K_{f}$ and $L(s,A)=\prod L(s,f^{\sigma})$,
where $f^{\sigma}$ runs over the Galois conjugates of $f$. We say an abelian
variety $A/\mathbb{Q}$ corresponds to a weight 2 newform $f$ if it is
isogenous to $A_{f}$ (over $\mathbb{Q}$), in which case we say $A$ is modular.
Recall that isogenies preserve endomorphism algebras.

######  Lemma 3.1.

Suppose $A/\mathbb{Q}$ is a simple $d$-dimensional abelian variety with
maximal RM. Then $A$ corresponds to a newform $f\in S_{2}(N)$, where $N^{d}$
is the conductor of $A$.

This lemma together with 2.2 implies part (1) of 1.1.

As we do not know a reference that explicitly concludes the newform $f$ must
have trivial nebentypus, we provide a proof.

###### Proof.

Since Serre’s conjecture is known, [ribet:korea, Theorem 4.4] implies that $A$
is a simple factor of $J_{1}(N)$. Thus $A$ is isogenous to an abelian variety
$A_{f}$ associated to a newform $f\in S_{2}(M,\varepsilon)$ for some $M\mid N$
and nebentypus $\varepsilon$. Comparing conductors shows $M=N$. Now the
rationality field of $f$ equals $K=\mathrm{End}_{0}(A)$, and thus contains the
values of $\varepsilon$. Hence $\varepsilon$ is quadratic. If $\varepsilon=1$,
then $f$ is a newform in $S_{2}(N)$, as desired.

We will use several facts about CM forms, all of which can be found in
[ribet:survey]. Suppose $\varepsilon$ is nontrivial. Then $f$ has CM by
$\varepsilon$, i.e., $f\otimes\varepsilon=f$. We claim this is impossible
because $f$ has weight 2.

More generally, take a newform $f\in S_{k}(N,\varepsilon)$ of any weight
$k\geq 2$, and assume $f$ has CM by $\varepsilon$. Then $f$ is induced from a
Grossencharacter $\psi$ of an imaginary quadratic field $F$, and $f$ has CM by
the quadratic Dirichlet character $\chi_{F}$ attached to $f$. Since the
character of CM is unique, $\varepsilon=\chi_{F}$. This forces $\varepsilon$
to be odd, as $F$ is imaginary quadratic. But because $\varepsilon$ is also
the nebentypus character, we must have $\varepsilon(-1)=(-1)^{k}$. Hence $k$
is odd, proving the claim. ∎

###  3.1. Comparison with Brumer–Kramer bounds

Suppose $A/\mathbb{Q}$ is a $d$-dimensional simple factor of the new part of
$J_{0}(N)$. Thus $A$ has conductor $N^{d}$. Put $e=v_{p}(N)$. Brumer and
Kramer’s conductor exponent bounds for abelian varieties [brumer-kramer,
Theorem 6.2] states that

$dv_{p}(N)\leq B(p,d):=2d+pt+(p-1)\lambda_{p}(t),$

where $t=\lfloor 2d/(p-1)\rfloor$ and $\lambda_{p}$ is defined by

$\lambda_{p}(m)=\sum_{i=0}^{s}ic_{i}p^{i},\quad\text{for
}m=\sum_{i=0}^{s}c_{i}p^{i},\quad 0\leq c_{i}<p.$

Let us rewrite this as

$v_{p}(N)\leq
B^{\prime}(p,d):=2+\lfloor\frac{pt+(p-1)\lambda_{p}(t)}{d}\rfloor.$

Thus $B^{\prime}(p,d)$ can be thought of as the Brumer–Kramer bound for
abelian varieties of GL(2)-type. We want to compare $B^{\prime}(p,d)$ with
$B_{0}(p,d)$.

######  Lemma 3.2.

First suppose $p\geq 5$. When $p\geq d$, $B^{\prime}(p,d)$ is given by

$B^{\prime}(p,d)=\begin{cases}2&p>2d+1\\\ 4&p=2d+1\text{ or }p=d\\\
3&d+1<p<2d+1\end{cases}$

If $5\leq p\leq d$, then $B^{\prime}(p,d)\geq 3$.

When $p=3$, we have $B^{\prime}(3,1)=B^{\prime}(3,2)=5$ and
$B^{\prime}(3,d)\geq 6$ for $d\geq 3$.

When $p=2$, we have $B^{\prime}(2,1)=8$, $B^{\prime}(2,2)=10$,
$B^{\prime}(2,3)=9$ and $B^{\prime}(2,d)\geq 9$ for $d\geq 4$.

For any $p$, if $(p-1)\mid 2d$, then $B^{\prime}(p,d)\geq
4+2v_{p}(d)+4v_{2}(p)+v_{2}(3)$, with equality when
$\frac{2d}{p^{v_{p}(2d)}(p-1)}<p$.

###### Proof.

Note that $\lambda_{p}(t)=0$ if and only if $t<p$, i.e., $2d<p(p-1)$.

First suppose $p\geq 5$. Since $t=0\iff p>2d+1$, we see
$B^{\prime}(p,d)=2=B_{0}(p,d)$ when $p>2d+1$. We also note that if $p=2d+1$,
so $t=1$, then $B^{\prime}(p,d)=4=B_{0}(p,d)$. If $d+1<p<2d+1$, so $t=1$, then
$B^{\prime}(p,d)=3>B_{0}(p,d)=2$. When $p=d$, so $t=2$,
$B^{\prime}(p,d)=4>B_{0}(p,d)=2$.

Next suppose $5\leq p<d$. Then $t\geq 2$. Since $t+1>\frac{2d}{p-1}$, we have
$(p-1)t+(p-1)>2d$, and thus $pt>2d+1+t-p>d+1$. Hence $\frac{pt}{d}>1$ and
$B^{\prime}(p,d)\geq 3$.

For $p=3$, note that $t=d$ so
$B^{\prime}(p,d)=5+\lfloor\frac{2\lambda_{3}(d)}{d}\rfloor$. Since we always
have $\lambda_{p}(m)\geq m-p+1$, one gets $2\lambda_{3}(d)\geq 3$ for all
$d\geq 3$. This gives the $p=3$ statement.

When $p=2$, we have $t=2d$ and
$B^{\prime}(p,d)=6+\lfloor\frac{\lambda_{2}(2d)}{d}\rfloor$. One can compute
$B^{\prime}(2,d)$ case-by-case for $d\leq 3$. If
$s=\lfloor\log_{2}(d)\rfloor$, then $2^{s+1}>d$ so
$\lambda_{2}(2d)\geq(s+1)2^{s+1}>(s+1)d\geq 3d$. This implies the $p=2$ lower
bound.

Lastly, consider any $p\geq 2$. Suppose $2d=ap^{m}(p-1)$ for some $m\geq 0$
and integer $a$ coprime to $p$. Then $t=ap^{m}$ so
$\frac{pt}{d}=2+\frac{2}{p-1}$. If $a<p$, then $\lambda_{p}(t)=map^{m}$, and
thus $\frac{(p-1)\lambda_{p}(t)}{d}=2m=2(v_{p}(d)+v_{2}(p))$ If $a>p$, then
$\lambda_{p}(t)>map^{m}$. This proves the assertion when $(p-1)\mid 2d$. ∎

The lower bounds are not intended to be optimal, but merely sufficient to
conclude the following, which implies part (2) of 1.1.

######  Corollary 3.3.

We have $B_{0}(p,d)\geq B^{\prime}(p,d)$.

This is an equality if (i) $d\leq 2$; (ii) $p\geq 2d+1$; or (iii) if
$2d=ap^{m}(p-1)$ for some $m\geq 0$ and $a<p$.

We have a strict inequality $B_{0}(p,d)<B^{\prime}(p,d)$ if (i)
$\max\\{5,d\\}\leq p<2d+1$; (ii) $5\leq p<d$ and $(p-1)\nmid d$; or (iii)
$p\in\\{2,3\\}$, $d>p$, and $p\nmid d$.

###### Proof.

This follows from the previous lemma. The cases $d\leq 2$ are more easily seen
from examining Table 1. ∎

###  3.2. Sharpness of bounds

Here we consider when the bounds $v_{p}(N)\leq B_{0}(p,d)$ of 1.1 are sharp.
Note that once there exists, say, a weight 2 newform $f$ of degree $d$, one
can replace $f$ with a quadratic twist to ensure $v_{p}(N)\geq 2$. Thus we
will look for examples where $v_{p}(N)=B_{0}(p,d)>2$ is attained for $d\leq
10$. This will justify the bolded entries in Table 1.

First we searched the LMFDB [LMFDB] for weight 2 newforms with trivial
nebentypus which attain such bounds. The LMFDB contains data on all weight 2
newforms of level up to 10000. This data shows the bounds are sharp when (i)
$d\leq 4$; (ii) $d=5$ and $p\neq 11$; (iii) $d=6$ and $p\neq 13$; (iv) $d=7$
and $p\neq 2$; (v) $d=8$ and $p\neq 2,17$; (vi) $d=9$ and $p\neq 2,3,19$; and
(vii) $d=10$ and $p\neq 2,5,11$.

Note that $11^{4}=14641$ is already outside of the LMFDB range, so the LMFDB
data is necessarily insufficient to search for $v_{p}(N)=B_{0}(p,d)$ when
$p=2d+1\geq 11$. Similarly, $2^{14}$, $3^{9}$ and $5^{6}$ are also outside of
the LMFDB range.

For the other cases in Table 1, we used Magma [magma] to compute newform
decompositions (i.e., degrees of all newforms) for $S_{2}(N)$ for various
$N>10000$. Andrew Sutherland has also done some such calculations, and at
least some of our data is contained in his. In particular, we found the
following, which justifies the remaining bolded entries in Table 1. We
indicate approximate runtimes and RAM usage for each newform decomposition
calculation in parentheses.

  * •

$N=12032=2^{8}\cdot 47$ has a degree 7 newform (32s, 245MB)

  * •

$N=14592=2^{8}\cdot 57$ has a degree 9 newform (4min, 1.3GB)

  * •

$N=11264=2^{10}\cdot 11$ has a degree 10 newform (52s, 325MB)

  * •

$N=16384=2^{14}$ has a degree 8 newform (65s, 444MB)

  * •

$N=19683=3^{9}$ has a degree 9 newform (4min, 763MB)

  * •

$N=14641=11^{4}$ has a degree 5 newform (9min, 1.5GB)

The remaining cases to check are when $d=10$ and $p=5,11$, or when $d=6,8,9$
and $p=2d+1$. In these cases, we searched for newforms where
$v_{p}(N)=B_{0}(p,d)$ or $v_{p}(N)=B_{0}(p,d)-1$ is attained, to the extent we
could with moderate computational resources. We did not find any levels
outside of the LMFDB range where either of these bounds is attained, but we
summarize our attempts. We remark that these computations tend to be more
accessible for levels of the form $2^{m}p^{r}$ than for $q\cdot p^{r}$ where
$q$ is an odd prime of comparable size to $2^{m}$.

For $p=5$, we computed newform decompositions for levels $N=5^{6}\cdot M$
where $M=2,4,8$, as well as $N=5^{5}\cdot M$ for $M\leq 12$ and $M=14,16,18$.
The decomposition for $N=125000=8\cdot 5^{6}$ took over 7 hours and used 46 GB
of RAM. None of these levels have degree $d=10$ forms, though there are degree
20 forms in levels $18750=6\cdot 5^{5}$, $28125=9\cdot 5^{5}$, $50000=16\cdot
5^{5}$ and $56250=18\cdot 5^{5}$. There are degree 10 newforms in the LMFDB
with level $N=8750=14\cdot 5^{4}$, so at least $v_{5}(N)=B_{0}(5,10)-2$ is
attained.

For $p=11$, we computed newform decompositions for levels $N=11^{4}\cdot M$
for $M=1,2,4,8$. None of these levels have degree $d=10$ forms. There are
however many degree 10 forms in the LMFDB with $v_{11}(N)=3$, e.g., there is
one in level $1331=11^{3}$, so at least $v_{11}(N)=B_{0}(11,10)-1$ is
attained.

For $p=13$, we computed newform decompositions for levels $N=13^{4}\cdot M$
with $M=1,4$, as we well as levels $N=13^{3}\cdot M$ for $M\leq 16$ and
$M=18,20,24,32$. The decomposition for $N=114244=4\cdot 13^{4}$ took 11 hours
and used 61 GB of RAM. None of these levels have degree 6 newforms, though
there are degree 12 newforms in levels $35152=16\cdot 13^{3}$, $39546=18\cdot
13^{3}$ and $70304=32\cdot 13^{3}$.

For $p=17$, we computed newform decompositions for levels $N=17^{3}\cdot M$
for $M\leq 9$ and $M=12,16$. (For level $7\cdot 17^{3}$, we had to use an
alternate method of factoring Hecke polynomials.) None of these levels have
degree 8 newforms, though there are degree 16 newforms in level $44217=9\cdot
17^{3}$.

For $p=19$, the LMFDB lists degree 9 newforms in level $6859=19^{3}$.

###  3.3. Restrictions on RM fields

Let $A$ be simple abelian variety with maximal RM of conductor $N^{d}$, where
$d=\dim A$. A further application of 2.1 (combined with 3.1) is that large
powers of $p$ dividing $N$ force restrictions on the endomorphism algebra
$K=\mathrm{End}^{0}(A)$, sometimes determining it completely. Moreover,
different primes can interact in this phenomenon, creating “global”
restrictions on local conductor exponents.

We can think of this as follows. Say $v_{p}(N)$ is large, and let $r$ be as in
2.1. This forces $K=\mathrm{End}^{0}(A)$ to contain
$\mathbb{Q}(\zeta_{p^{r}})^{+}$, and leaves only
$d^{\prime}=d/(\frac{1}{2}p^{r-1}(p-1))$ “degrees of freedom” inside $K$ for
other cyclotomic subfields. So if $q$ is another prime dividing $N$, we have
the stronger conductor bound $v_{q}(N)\leq B_{0}(q,d^{\prime})$, since the
cyclotomic fields $\mathbb{Q}(\zeta_{p^{r}})$ and $\mathbb{Q}(\zeta_{q^{s}})$
are disjoint for $q\neq p$.

Applying this reasoning case-by-case for $2\leq d\leq 6$ proves 1.4.

The $d=2$ case of this proposition immediately implies 1.5 once one knowns
$A=\mathrm{Jac}(C)$ is simple. Indeed, if it were not, it would be a product
of elliptic curves $E_{1}$ and $E_{2}$, and then $N$ is the product of the
conductors of $E_{1}$ and $E_{2}$. But by local conductor bounds for elliptic
curves, this is impossible if $v_{2}(N)>16$ or $v_{5}(N)>4$.

###  3.4. Abelian varieties of GL(2)-type

Now suppose $A$ is a $d$-dimensional simple abelian variety of GL(2)-type, not
necessarily with maximal RM. Then $A$ is isogenous to a newform $f\in
S_{2}(N,\varepsilon)$. Suppose $p,r$ are as in 2.1. When $p$ is odd or $p=2$
and $v_{2}(N)$ is even, [brumer, Theorem 5.5] tells us that
$\mathbb{Q}(\zeta_{p}^{r})^{+}\mathbb{Q}(\varepsilon)\subset K_{f}$. If $p=2$
and $v_{2}(N)$ is odd, Brumer’s theorem says
$\mathbb{Q}(\zeta_{p}^{r-1})^{+}\mathbb{Q}(\varepsilon)\subset K_{f}$. Thus
using [brumer, Theorem 5.5] in place of 2.1 in the proof of 1.1(1) yields the
statement in 1.2.

### Acknowledgements

We thank Armand Brumer and Alex Cowan for discussions which led to this note.
We also thank Ariel Pacetti for clarifications on [DPT]. Andrew Sutherland
kindly explained how to compute newform decompositions in Magma, and shared
with us some of his data. We are grateful to the referee for numerous
suggestions and comments which substantially improved the manuscript. Some of
the computing for this project was performed at the OU Supercomputing Center
for Education & Research (OSCER) at the University of Oklahoma (OU).

## References