Research Seminar Number Theory and Arithmetic Geometry

Datum Vortragende/r Vortragstitel
Fr 14.4.2023 Teppei Takamatsu (Kyoto University) On Quasi-Frobenius-splitting

Do 20.4.2023

10:15-11:15, Raum F428

Adam Logan (Tutte Institute for Mathematics and Computing, Ottawa)

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.

Fr 5.5.2023 Christian Bernert (Göttingen)

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

Timo Keller (LUH)

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.

Fr 26.5.2023 Victoria Cantoral Farfán (LUH/Göttingen)

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 9.6.2023 Justin Uhlemann (LUH)  
Fr 16.6.2023    

Fr 23.6.2023


Marta Pieropan (Utrecht)  

Fr 23.6.2023


Jérôme Poineau (Caen)  
Fr 30.6.2023 Carlo Pagano (Concordia)  
Fr 7.7.2023    
Fr 14.7.2023 Shu Shen (Jussieu)  
Fr 21.7.2023 Florian Munkelt (Göttingen)