## AROUND FROBENIUS DISTRIBUTIONS AND RELATED TOPICS III Online October 5-6-7, 2022

 This is the third edition of a conference on the theme of Frobenius distributions. The first edition was organized by Victoria Cantoral Farfán and Seoyoung Kim in 2020; see here. The second edition was organized by Alina Carmen Cojocaru and Francesc Fité in 2021; see here.

## Organizers

Alina Carmen Cojocaru (University of Illinois at Chicago, USA)
Florent Jouve (Université de Bordeaux, France)
Elisa Lorenzo García (Université de Neuchâtel, Switzerland & Université de Rennes, France)

## Confirmed speakers

 Jonas Bergström Philippe Michel Dante Bonolis Peter Sarnak Chantal David Kaneenika Sinha David Kohel Yunqing Tang Peter Koymans Jesse Thorner Wanlin Li David Zywina

## Registration

In order to participate, fill in the registration form .
Note that registration is free, but required in order to be admitted in the conference.

## Schedule of talks

 Chicago time Wednesday, October 5 Thursday, October 6 Friday, October 7 Paris time 8:30am-9:20am Sinha David Kohel 3:30pm-4:20pm 9:25am-10:15am Zywina Li Michel 4:25pm-5:15pm 10:20am-11:10am Bergström Thorner Bonolis 5:20pm-6:10pm 11:10am-1:00pm Break 6:10pm-8:00pm 1:00pm-1:50pm Tang 8:00pm-8:50pm 1:55pm-2:45pm Sarnak 8:55pm-9:45pm 2:50pm-3:40pm Koymans 9:50pm-10:40pm

## Titles and abstracts

• Jonas Bergström (Stockholms universitet, Sweden)

Lower bounds on the maximal number of points on curves over finite fields
Abstract: In this talk I will present three approaches to finding lower bounds on the maximal number of points on curves over finite fields. We will focus on the one involving the cohomology of moduli spaces of curves. Using a variant of this approach we will also get information on Serre's obstruction problem (which concerns the asymmetry in the distribution of traces of Frobenius for curves of genus at least three).
This is joint work with E. Howe, E. Lorenzo García and C. Ritzenthaler.

• Dante Bonolis (University of Basel, Switzerland)

On the density of rational points on some quadric bundle threefolds
Abstract: In this talk, we present a proof of the Manin-Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree $(1,2)$. This is a joint work with Tim Browning and Zhizhong Huang.

• Chantal David (Concordia University, Canada)

On the vanishing of twisted $L$-functions of elliptic curves over function fields
Joint work with A. Comeau-Lapointe (Concordia University), M. Lalin (Université de Montréal) and W. Li (Washington University).
Abstract: Let $E$ be an elliptic curve over $\mathbb Q$, and let $\chi$ be a Dirichlet character of order $\ell$ for some prime $\ell\geq 3$. Heuristics based on the distribution of modular symbols and random matrix theory have led to conjectures predicting that the vanishing of the twisted $L$-functions $L(E,\chi,s)$ at $s=1$ is a very rare event (David-Fearnley-Kisilevsky and Mazur-Rubin). In particular, it is conjectured that there are only finitely many characters of order $\ell>5$ such that $L(E,\chi,1)=0$ for a fixed curve $E$. We investigate in this talk the case of elliptic curves over function fields. For Dirichlet $L$-functions over function fields, Li and Donepudi-Li have shown how to use the geometry to produce infinitely many characters of order $\ell\geq 2$ such that the Dirichlet $L$-function $L(\chi,s)$ vanishes at $s=1/2$, contradicting (the function field analogue of) Chowla's conjecture. We show that their work can be generalized to isotrivial curves $E/\mathbb F_q(t)$, and we show that if there is one Dirichlet character $\chi$ of order $\ell$ such that $L(E,\chi,1)=0$, then there are infinitely many, leading to some specific examples contradicting (the function field analogue of) the number field conjectures on the vanishing of twisted $L$-functions. Such a dichotomy does not seem to exist for general (non-isotrivial) curves over $\mathbb F_q(t)$, and we produce empirical evidence which suggests that the conjectures over number fields also hold over function fields for non-isotrivial $E/\mathbb F_q(t)$.

• David Kohel (Aix-Marseille Université, France)

On Sato-Tate groups ${\rm SO}(2n+1)$ and the exceptional group ${\rm UG}_2$
Abstract: The character method, developed by Yih-Dar Shieh in his thesis, recognizes a Sato-Tate from an associated Frobenius distribution. Previous algorithms used moments of coefficients of a characteristic polynomial of Frobenius. The higher moments are degrees of the tensor product characters, which are direct sums with high multiplicities, hence the moment sequences converge (slowly, with sufficient precision) to large integers. The character method replaces the moments with a precomputed list of irreducible characters. From the orthogonality relations of characters, a Sato-Tate group $G$ is recognized by inner products yielding 0 or 1 (for which only one bit of precision is required to determine its value). We describe the character theory of the orthogonal groups ${\rm SO}(2n+1)$, with a view to characterizing orthogonal Sato-Tate groups. In particular, we specialize the character theory method to ${\rm SO}(7)$ and its subgroup ${\rm UG}_2$, the unitary subgroup ${\rm UG}_2$, of the exceptional Lie group $G_2$. In particular, we demonstrate its effectiveness with certain character sums associated to abelian factors of families of Jacobians known to give rise to the Sato-Tate group ${\rm UG}_2$.

• Peter Koymans (University of Michigan, USA)

The negative Pell equation and applications
Abstract: In this talk we will study the negative Pell equation, which is the conic $C_D : x^2 - D y^2 = -1$ to be solved in integers $x, y \in \mathbb{Z}$. We shall be concerned with the following question: as we vary over squarefree integers $D$, how often is $C_D$ soluble? Stevenhagen conjectured an asymptotic formula for such $D$. Fouvry and Klüners gave upper and lower bounds of the correct order of magnitude. We will discuss a proof of Stevenhagen's conjecture, and potential applications of the new proof techniques. This is joint work with Carlo Pagano.

• Wanlin Li (Washington University in St Louis, USA)

A generalization of Elkies's theorem
Abstract: Elkies proved that for a fixed elliptic curve $E$ defined over $\mathbb Q$, there exist infinitely many primes at which the reductions of $E$ are supersingular. In this talk, we give the first generalization of Elkies's theorem to curves of genus $>2$. We consider families of cyclic covers of the projective line ramified at $4$ points parametrized by a Shimura curve. This is joint work in progress with Elena Mantovan, Rachel Pries, and Yunqing Tang.

• Philippe Michel (Ecole Polytechnique Fédérale de Lausanne, Switzerland)

Equidistribution of CM points on products
Abstract: In this talk we will discuss several results concerning the equidistribution of CM elliptic curves mapped on various product of arithmetic quotients. The main ingredient towards the proofs is a special case of a general classification theorem for joinings on product of locally homogeneous spaces due to Einsiedler and Lindenstrauss. We will explain which additional information are needed to verify the assumptions of the EL classification theorem.

• Peter Sarnak (Institute for Advanced Study and Princeton University, USA)

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• Kaneenika Sinha (Indian Institute of Science, Education, and Research - Pune, India)

Questions about error terms in Sato-Tate distributions
Abstract A sequence that is equidistributed with respect to a probability measure such as the Sato-Tate measure often motivates us to ask finer questions. Can we find explicit bounds for the discrepancy in these sequences? By varying the sequences over a suitable family, is the discrepancy estimate better upon averaging? How do the discrepancies fluctuate? What do we know about the small scale statistics and spacing statistics of these sequences? We explore these questions in the context of the Sato-Tate distribution law for the Hecke eigenvalues with respect to modular cusp forms.

• Yunqing Tang (University of California at Berkeley, USA)

Reductions of abelian varieties and $K3$ surfaces
Abstract: For a $K3$ surface $X$ over a number field, we prove that there are infinitely many primes modulo which the reduction of $X$ has larger geometric Picard rank than that of the generic fiber $X$. There is also analogous result for $K3$ surface over global function field (under certain assumptions). In this talk, I will sketch the ideas in the proofs via the (arithmetic) intersection theory on good integral models of ${\rm GSpin}$ Shimura varieties and its consequences on certain abelian varieties. This talk is based on joint work with Davesh Maulik, Ananth Shankar, Arul Shankar, and Salim Tayou, and also the work of Tayou removing the good reduction assumptions.

• Jesse Thorner (University of Illinois at Urbana-Champaign, USA)

Extremal class groups
Abstract: Fix an integer $n \geq 2$, and let $K_n$ be the set of number fields $F$ with $[F:\mathbb{Q}]=n$ whose Galois closure (over $\mathbb{Q}$) has as its Galois group the full symmetric group $S_n$. Conditional on the generalized Riemann hypothesis and Artin's holomorphy conjecture, Duke proved that there are infinitely many number fields $F\in K_n$ whose ideal class group has maximal order (as a function of the absolute discriminant). The result is now known unconditionally for $n=2,3,4$, and it is known conditionally on the strong Artin conjecture for $n\geq 5$. I will report on joint work with Robert Lemke Oliver and Asif Zaman wherein we prove Duke's theorem for all $n\geq 2$ without any unproven hypotheses.

• David Zywina (Cornell University, USA)

Computing images of Galois representations for elliptic curves over $\mathbb{Q}$.
Abstract: Consider a non-CM elliptic curve $E/\mathbb{Q}$. The natural Galois action on the torsion points of $E(\overline{\mathbb{Q}})$ can be encoded by a Galois representation $\rho_E : Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$. A famous theorem of Serre says that the image of $\rho_E$ is an open, and hence finite index, subgroup of $GL_2(\widehat{\mathbb{Z}})$. The image of $\rho_E$ is an important invariant for studying the distribution of the traces of Frobenius $a_p(E)$ for a fixed $E/\mathbb{Q}$ and varying prime $p$.
We shall describe recent results that allow us to actually compute the image of $\rho_E$. As an application, we explain how to compute the constants occurring in the conjecture of Lang and Trotter on the distribution of primes $p$ for which $a_p(E)$ is equal to a fixed integer.