Oberseminar Zahlentheorie und Arithmetische Geometrie

Do 20.4.2023

10:15-11:15, Raum F428

The action of automorphisms on the Kummer surface of the square of an elliptic curve

Recent work has greatly enhanced our understanding of the ${\mathbb F}_p$-points on the Markoff surface defined by $x^2 + y^2 + z^2 = 3xyz$.  In particular, we now know, thanks to work of Bourgain-Gamburd-Sarnak, that the action on a natural quotient is almost always transitive, and by Carmon-Meiri-Puder, that the action is almost always as large as possible.  The goal of this talk is to prove a result analogous to that of Carmon-Meiri-Puder for the K3 surfaces of the form $(E x E)/\pm 1$, where $E$ is an elliptic curve defined over $\mathbb Q$.  In order to do this we will prove a surprising property of points on certain rational curves on this surface.  This is joint work with Owen Patashnick.

Rational Points and Lines on Cubic Hypersurfaces

Classical results of Heath-Brown and Wooley show that projective hypersurfaces defined by a cubic form in n variables contain a rational point if n is at least 14, and a rational line if n is at least 37.

In this talk, I will report on two new results in this direction.

i) If n is at least 10 (which is the conjectural threshold for the existence of rational points), the singular series of the cubic form converges absolutely. I will explain what the singular series is and why it is relevant to the existence of rational points (Keyword: Hardy-Littlewood Circle Method).

ii) In joint work with Leonhard Hochfilzer, we show that 31 variables are sufficient to guarantee the existence of a rational line, improving on Wooley's result. The key ingredient is a generalization of Heath-Brown's 14-variable result to imaginary quadratic number fields.

Mo 22.5.2023

11:00-12:00, room F128

On the Birch--Swinnerton-Dyer conjecture at reducible primes of good and bad multiplicative reduction

We report on work in progress with Mulun Yin. Castella--Gross--Lee--Skinner recently proved Perrin-Riou's Heegner point main conjecture for modular abelian varieties at odd primes p of good reduction for which the mod-p Galois representation rho_p is reducible ("Eisenstein primes"). They have the restriction that the characters in the semisimplification of rho_p are non-trivial on Gal_{Q_p}. For example this excludes the case when there is a non-trivial p-torsion point.

We are working on removing this restriction and generalize the result to newforms of higher weight, allowing us to also treat bad multiplicative reduction using Hida theory. As a consequence, we get the p-part of the Birch-Swinnerton-Dyer conjecture for analytic rank 1 (and 0 with work in progress by Castella-Grossi-Skinner) and a p-converse theorem.

Combining this with previous results of Skinner-Urban, Skinner, Castella-Çiperiani--Skinner-Sprung, we get the strong BSD conjecture in analytic rank 0 and 1 for squarefree level N under a mild condition on the discriminant except maybe for the 2-part.

Monodromy groups of Jacobians with definite quaternionic multiplication

Let $A$ be an abelian variety over a number field. The connected monodromy field of $A$ is the minimal field over which the images of all the $\ell$-adic torsion representations have connected Zariski closure. During this talk, we will show that for all even $g\geq 4$, there exist infinitely many geometrically nonisogenous abelian varieties $A$ over $\mathbb Q$ of dimension $g$ where the connected monodromy field is strictly larger than the field of definition of the endomorphisms of $A$. Our construction arises from explicit families of hyperelliptic Jacobians with definite quaternionic multiplication. This is a joint work with Lombardo and Voight.

Symmetric Chabauty and points of low degree on modular curves

​​​​​​​In joint work with Josha Box and Stevan Gajović, we determined cubic and quartic points on a collection of modular curves whose Jacobians have positive rank. To do this we further developed the symmetric Chabauty method used previously by Siksek. In this talk, I will outline how this method works, how to apply it to explicit examples and point out complications that may arise when trying to do so.

Semi-integral points on Markoff surfaces

Semi-integral points of a variety are rational points that satisfy local intersection conditions with respect to some weighted boundary divisor. Mitankin--Nakahara--Streeter developed a semi-integral version of the Brauer-Manin obstruction to study local-global principles of such points. We apply their construction to show that in most cases there is no Brauer-Manin obstruction to the existence of semi-integral points on a family of projective Markoff surfaces. Additionally, we use results from the theory of binary quadratic forms over the integers to give six explicit sub families with semi-integral points for arbitrary weights.

Fr 23.6.2023

9:55-10:55

The circle method for Campana points

Campana points on a variety over a number field are rational points that have prescribed intersection with a fixed boundary divisor on an given integral model. In this talk, I will describe the set of rational points on diagonal hypersurfaces of projective space for the boundary given by the coordinate hyperplanes. I will also explain how the circle method can be used to estimate the cardinality of sets of Campana points of bounded height on diagonal hypersurfaces over the rational numbers and how this computation is affected by working with S-integral models for a finite set of places S. This is joint work with Balestrieri, Brandes, Kaesberg, Ortmann, and Winter.

Fr 23.6.2023

11:10-12:10

Torsion points of elliptic curves via Berkovich spaces over Z

Berkovich spaces over Z may be seen as fibrations containing complex analytic spaces as well as p-adic analytic spaces, for every prime number p. We will give an introduction to those spaces and explain how they may be used in an arithmetic context to prove height inequalities. As an application, following a strategy by DeMarco-Krieger-Ye, we will give a proof of a conjecture of Bogomolov-Fu-Tschinkel on uniform bounds on the number of common images on P^1 of torsion points of two elliptic curves.

Chowla's conjecture over function fields

A conjecture of Chowla postulates that no L-function of Dirichlet characters over the rationals vanishes at s=1/2. Soundararajan has proved non-vanishing for a positive proportion of quadratic characters. Over function fields Li has discovered that Chowla's conjecture fails for infinitely many distinct quadratic characters. However, on the basis of the Katz--Sarnak heuristics, it is still widely believed that one should have non-vanishing for 100% of the characters in natural families (such as the family of quadratic characters). Works of Bui--Florea, David--Florea--Lalin, Ellenberg--Li--Shusterman, among others, provided evidence giving a positive proportion of non-vanishing in several such families. I will present an upcoming joint work with Peter Koymans and Mark Shusterman, where we prove that for each fixed q congruent to 3 modulo 4 one has 100% non-vanishing in the family of imaginary quadratic function fields.

Arithmetic converse theorems for L-functions of GL(2) type

I will explain a new Hecke--Weil type of converse theorem for elliptic modular forms which works for any level and does not involve any character twists. We call this result an 'integral converse theorem' due to involving only a single L-function under the Diophantine hypothesis that the Dirichlet series coefficients are rational integers. The techniques are based on a joint work with Frank Calegari and Yunqing Tang. As an application, we prove that if A/Q is an abelian surface whose L-function L(s,A) is entire holomorphic (as opposed to the entire meromorphy presently known after Boxer, Calegari, Gee, and Pilloni), then L(s,A) has a zero of odd multiplicity, unless A is isogenous to the square of an elliptic curve.

Coherent sheaves, superconnection, and the Riemann-Roch-Grothendieck formula

In this talk, I will explain a construction of Chern character for coherent sheaves on a closed complex manifold with values in Bott-Chern cohomology. I will also show a corresponding Riemann-Roch-Grothendieck formula, which holds for general holomorphic maps between closed non-Kahler manifolds. Our proof is based on two fundamental objects: the superconnection and the hypoelliptic deformations. This is a joint work with J.-M. Bismut and Z. Wei arXiv:2102.08129.

On the number of rational points close to a compact manifold

The fundamental question of number theory to count the rational points on an algebraic variety can be approached from the perspective of studying rational points close to manifolds. Groundbreaking work for the case of hypersurfaces has been done by Huang and was generalized by Schindler and Yamagishi for manifolds in arbitrary codimension. The validity of these results rely on certain curvature conditions imposed on the manifold. We establish an asymptotic formula for the number of rational points within a given distance to a manifold and with bounded denominators under a relaxed curvature condition, generalizing previous results.  We are also able to recover a slightly weaker analogue of Serre's dimension growth conjecture for compact submanifolds.