Oberseminar Zahlentheorie und Arithmetische Geometrie

Fr 28.10.2022

Counting torsion points on subvarieties of the algebraic torus

We estimate the growth rate of the function which counts the number of torsion points of order at most T on an algebraic subvariety of the algebraic torus G_m over some algebraically closed field. 

For torsion cosets we get an asymptotic formula and for irreducible subvarieties not equal to a torsion coset we get a power saving bound compared to the one for torsion cosets of the same dimension.

In characteristic zero the Manin-Mumford Conjecture reduces the problem to torsion cosets. If K is the algebraic closure of a finite field we have lower bounds.

Deligne and Birch-Swinnerton-Dyer volumes in hypergeometric Calabi-Yau families

I will discuss the variation of the Birch-Swinnerton-Dyer volumes in hypergeometric Calabi-Yau families. I'll compare them to the first derivative of the L-functions of the respective motives at s=2. This work connects to a recent paper by Bloch, de Jong, and Sertöz.

André-Pink-Zannier conjecture in abelian type Shimura varieties

We will discuss a recent work with A. Yafaev on the André-Pink-Zanier conjecture. It concerns Hecke orbits in Shimura varieties, and predict the Zariski closure of a Subset contained in a single Hecke orbit is a union of weakly special subvariety. Joint with Andrei Yafaev.

Spreading out atypical intersections

Recent developments in Hodge theory and functional transcendence have significantly strengthened our understanding "atypical" algebraic cycle loci in moduli spaces. Such methods, however, only work in characteristic zero. In this talk we describe a new technique for "spreading out" atypical algebraic cycle loci within a moduli space over a ring of integers, providing a way of controlling atypical algebraic cycle loci at all but finitely many primes. As an application, we explain how this allows us to characterize positive-dimensional atypical intersections with certain Ekedahl-Oort strata (Bruhat strata) in abelian-type Shimura varieties.

Samuel Le Fourn (Grenoble)

Torsion growth in extensions of number fields for a fixed abelian variety

Let A be an abelian variety over a number field K. Masser proved in a letter that for finite extensions L/K, the order of the torsion group of A(L) is bounded by C(A) [L:K]^{dim A} for some constant C(A) depending only on A. Later, Hindry and Ratazzi conjectured that the optimal exponent (instead of \dim A) for such a polynomial bound is some \gamma_A defined in terms of the Mumford-Tate group of A. In this talk, I will explain how with Davide Lombardo and David Zywina, we proved Hindry-Ratazzi's conjecture assuming Mumford-Tate conjecture for A, including an unconditional result relating the optimal exponent to the monodromy groups of A.

Reduction modulo p of the Noether's problem

Let k be an algebraically closed field of characteristic p≥0 and V a faithful k-rational representation of an l-group G. The Noether's problem asks whether V/G is (stably) birational to a point. If l is equal to p, then Kuniyoshi proved that this is true, while, if l is different from p, Saltman constructed l-groups for which V/G is not stably rational. Hence, the geometry of V/G depends heavily on the characteristic of the field. We show that for all the groups G constructed by Saltman, one cannot interpolate between the Noether problem in characteristic 0 and p. More precisely, we show that it does not exist a complete valuation ring R of mixed characteristic (0,p) and a smooth proper R-scheme X---->Spec(R) whose special fiber and generic fiber are both stably birational to V/G. The proof combines the integral p-adic Hodge theoretic results of Bhatt-Morrow-Scholze, with the study of the Cartier operator on differential forms in positive characteristic. This is a work in progress with Domenico Valloni.

The S-unit equation and non-abelian Chabauty in depth 2

The S-unit equation is a classical and well-studied Diophantine equation, with numerous connections to other Diophantine problems. Recent work of Kim and refinements due to Betts-Dogra have suggested new cohomological strategies to find rational and integral points on curves, based on but massively extending the classical method of Chabauty. At present, these methods are only conjecturally guaranteed to succeed in general, but they promise several applications in arithmetic geometry if they could be proved to always work.

In order to better understand the conjectures of Kim that suggest that this method should work, we consider the case of the thrice punctured projective line, in "depth 2", the "smallest" non-trivial extension of the classical method. In doing so we get very explicit results for some S-unit equations, demonstrating the usability of the aforementioned cohomological methods in this setting. To do this we determine explicitly equations for (maps between) the (refined) Selmer schemes defined by Kim, and Betts-Dogra, which turn out to have some particularly simple forms.

This is joint work with Alexander Betts, Theresa Kumpitsch, Martin Lüdtke, Angus McAndrew, Lie Qian, Elie Studnia, and Yujie Xu.

Sachi Hashimoto (MPI MiS)