Vanishing of quartic twists of
L-Functions of elliptic curves:

experiments and conjectures

JMM Special Session on Rethinking Number Theory

January 2023

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L-functions: What and why?

Let $E$ be an elliptic curve over $\mathbb{Q}$.

Example: The curve 11.a1 has equation $y^2=x^3-10135152x-12419196912$

Often want to study arithmetic of $E$ and, in particular, gain info about $\small E(\mathbb{Q})$

The L-function associated to $E$ is the Euler product$\small ^\ast$
\[ \small L(E,s) = \prod_p \frac{1}{1 - a_p p^{-s} + p^{1-2s}} = \sum_{n \ge 1} \frac{a_n}{n^s}\]

• $a_p$ encode info about $\#E(\mathbb{F}_p)$, the number of points of $E$ mod $p$
• has analytic continuation to $\mathbb{C}$
• satisfies functional equation relating $\small L(E,s)$ and $\small L(E,2-s)$

$^\ast$ ignoring primes $p$ of bad reduction for simplicity

So the $L$-function $L(E,s)$ of an elliptic curve $E/\mathbb{Q}$ packages local information about $E$ into a global object.

The Birch and Swinnerton Dyer Conjecture (BSD) predicts that vanishing of $L(E,s)$ at $s=1$ implies $E$ has infinitely many $\mathbb{Q}$-rational points.

Generalizations: Points over number fields

Given $E/\mathbb{Q}$ and a Dirichlet character $\chi \colon \mathbb{Z} \to \mathbb{C}$, we can form the twisted $L$-function

\[ \small L(E,s,\chi) = \sum_{n \ge 1} \frac{\chi(n) a_n}{ n^{s}} \]

$L(E,s,\chi)$ also has an analytic continuation to $\mathbb{C}$ and satisfies a functional equation relating $L(E,s,\chi)$ and $L(E,2-s, \overline{\chi})$.

BSD predicts vanishing of $\small L(E,s,\chi)$ at $s=1$ is related to the existence of rational points $E(K)$ over the abelian extension $\small K/\mathbb{Q}$ associated to $\small \chi$.

Theme: Interesting arithmetic data is encoded in the location of the zeroes (and poles) of L-functions. Since L-functions are analytic objects, one can use analytic tools to study arithmetic.

Dirichlet Characters

A Dirichlet character is a function $\chi \colon \mathbb{Z} \to \mathbb{C}$ together with a positive integer $d$ called the modulus, such that for all integers $a,b$

These imply $\chi(a)$ is a root of unity if $\gcd(a,d) = 1$.

LMFDB Dirichlet characters database

Example

The character $\chi$ with LMFDB label 65.k.8 has modulus $d = 65$.

The first few values on integers $a$ with $\gcd(a,65) = 1$ are

The order of $\chi$ is $4$ and we say $\chi$ is a quartic character.

Moreover, $\chi$ is primitive since there is no character of smaller modulus whose values agree with $\small \chi$ on integers coprime to $\small d$.

Vanishing of twisted L-functions

GOAL: Given an elliptic curve $\small E/\mathbb{Q}$ and a family of twisted $L$-functions by Dirichlet characters $\chi$, describe how likely it is that $\small L(E,1,\chi) = 0$ to gain arithmetic info about $\small E$.

The Functional Equation

The functional equation for the twisted L-function by a character $\small \chi$ is \[ \small L(E,s,\chi) = \varepsilon \, L(E,2-s,\overline{\chi}),\] where the sign $\varepsilon$ is a complex number with $\small |\varepsilon|=1$ depending $\small E$ and $\small \chi$

Random Matrix Theory

One fruitful approach via Katz and Sarnak is to model such L-functions by a suitable classical (compact) matrix/symmetry group (unitary, orthogonal, symplectic,...)

In this context, the symmetries of the family of L-functions determine the appropriate group of matrices; the sign $\varepsilon$ of the functional equation is analogous to the determinant of the matrix
(unitary: $\small |\det| = 1$, orthogonal: $\small \det = \pm 1$, symplectic: $\small \det = 1$)

Many properties of twisted L-functions are analogous to those of characteristic polynomials $\small p_A(\lambda) = \det(A - \lambda I)$ of unitary$\small ^\ast$ matrices $\small A$.

Keating and Snaith conjectured that the distribution of the values $\small |L(E,1,\chi)|$ is related to the distribution of the values of the characteristic polynomials $\small |p_A(1)|$

Thus to obtain conjectural asymptotics for the vanishing of $\small L(E,1,\chi)$ in families, one can relate the probability distribution of $\small |p_A(1)|$ to that of the values of the twisted L-functions

$^\ast$A matrix $U$ is unitary if $U^{-1}$ is equal to the conjugate transpose of $U$.

Comparisons

Conjectures and evidence

Let $\small E/\mathbb{Q}$ be an elliptic curve.

Conjecture (David, Fearnley, Kisilevsky; 2007)

There are infinitely many primitive Dirichlet characters $\small \chi$ of order $\small 3$ and $\small 5$ (ordered by conductor) such that $\small L(E,1,\chi) = 0$. However, for any fixed prime $\small k \ge 7$, there are only finitely many $\small \chi$ of order $k$ such that $\small L(E,1,\chi) = 0$.

Conjecture (Berg, Ryan, Young; 2022)

There are infinitely many primitive Dirichlet characters $\chi$ of order $\small 4$ and $\small 6$ such that $\small L(E,1,\chi) = 0$.

To provide numerical evidence for the conjecture, we ran a combination of Sage and PARI/GP code on the Open Science Grid.

The combined wall time of all computations was more than 70 years.

Quartic case

Ratio of predicted vanishings to emperical vanishings of $\small |L(E,1,\chi)|$ by primitive quartic characters $\small \chi$ of modulus $\small \le 700000$ for the curve $\small E$ with label 11.a1

Sextic case

Ratio of predicted vanishings to emperical vanishings of $\small |L(E,1,\chi)|$ by primitive sextic characters $\small \chi$ of modulus $\small \le 200000$ for the curve $\small E$ with label 11.a1

Insights and distinguishing features

To apply the random matrix theory model to our family of twisted L-functions, we need to know

• the symmetry type of the family of twisted L-functions, in particular, the distribution of (normalized) Gauss sums
  \[ \footnotesize \tau(\chi) = \sum \limits_{1 \le a \le d} \chi(a) e^{2 \pi i a/d}\]
  on the unit circle
• $\small \#$ of primitive characters $\chi$ of order $k$ with modulus $\small \le X$
• a discretization of the values $\small |L(E,1,\chi)|$

Distribution of Gauss sums

From left to right: Histograms of the argument (angle) of $\footnotesize \tau(\chi)^2/\text{cond}(\chi)$ of cubic, quartic, and quintic characters of modulus $\footnotesize \le 200000$

Quartic characters differ from those of prime order since they may decompose into a product of characters of strictly smaller order.

We say that $\small \chi$ is a totally quartic character if it decomposes into a product of characters of exact order 4.

Non-example: The primitive quartic character with label 65.h.12 decomposes as the product of a quartic character 65.i.27 and a quadratic character 65.c.51.

PROPOSITION (BERG, RYAN, YOUNG) The normalized Gauss sums of the family of totally quartic and of all quartic Dirichlet characters $\small \chi$ equidistribute on the unit circle. The latter does so slowly; for such characters $\footnotesize \chi$ of modulus $\footnotesize \le X$ the rate of equidistribution is $\footnotesize O(\log(X)^{-1})$.

Totally quartic case

The family of totally quartic characters has unitary symmetry. Applying the random matrix model, we also conjecture that there are infinitely many totally quartic characters with $\footnotesize |L(E,1,\chi)| = 0$