Research Seminar Number Theory and Arithmetic Geometry

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 RM abelian varieties at primes p of good reduction for which the mod-p Galois representation rho_p is reducible. They have the restriction that the characters in the semisimplification of rho_p are non-trivial. We work on removing this restriction and generalize the result to newforms of higher weight, allowing us to treat also bad multiplicative reduction. As a consequence, we get the p-part of the Birch--Swinnerton-Dyer conjecture for analytic rank 1 and a p-converse theorem.

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.

Fr 23.6.2023

9:55-10:55

Fr 23.6.2023

11:10-12:10