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.

On the dynamics of quadratic polynomials

Let f_t(z)=z^2+t. For any z\in\mathbb{Q}, let S_z be the collection of t\in\mathbb{Q} such that z is preperiodic for f_t. In this talk, we will discuss a uniform result regarding the size of S_z over z\in\mathbb{Q}. This is a recent joint work with Michael Stoll (arXiv:2206.12154).

Cohomology and Geometry of Deligne—Lusztig varieties for $GL_n$

Deligne—Lusztig varieties associated to a connected reductive group $G$ in positive characteristic are originally constructed by Deligne and Lusztig (1976) for studying $\ell$-adic representations of $G(\mathbb{F}_q)$. In our work, we adapt the double induction strategy by Orlik (2018), originally developed for studying the individual $\ell$-adic cohomology groups of Deligne—Lusztig varieties, to the cohomology of the structure sheaf of smooth compactifications of Deligne—Lusztig varieties for $GL_n$. From our result we also used a closed version of the Mayer—Vietoris spectral sequence to obtain the compactly supported $p$-torsion and integral $p$-adic cohomology groups. After sketching the proof of the main theorem in this talk, we will discuss possible ways to generalise our result to the case of arbitrary connected reductive group $G$ and other quasi-coherent sheaves, as well as the difficulties.

Sachi Hashimoto (MPI MiS)

p-Adic Gross—Zagier and rational points on modular curves

Faltings' theorem states that there are finitely many rational points on a nice projective curve defined over the rationals of genus at least 2. The quadratic Chabauty method makes explicit some cases of Faltings' theorem. Quadratic Chabauty has recent notable success in determining the rational points of some modular curves. In this talk, I will explain how we can leverage information from p-adic Gross—Zagier formulas to give a new quadratic Chabauty method for certain modular curves. Gross—Zagier formulas relate analytic quantities (special values of p-adic L-functions) to invariants of algebraic cycles (the p-adic height and logarithm of Heegner points). By using p-adic Gross—Zagier formulas, this new quadratic Chabauty method makes essential use of modular forms to determine rational points.

Integral points on cubic surfaces via a Hardy–Littlewood heuristic

An affine cubic surface is defined by an irreducible cubic polynomial f with integer coefficients in three variables, and its integral points correspond to integral roots of this polynomial. These solutions tend to be sparser and much harder to find than in cases of higher dimension or lower degree, as witnessed by the search for representations of integers k as sums of three cubes — that is, the case f = x³ + y³ + z³ - k – whose existence or nonexistence has only recently been settled for the first one hundred integers.

To shed new light on this kind of problem, we develop a Hardy–Littlewood heuristic for the number of integral points on affine cubic surfaces, arriving at a prediction that looks similar to Manin's conjecture on rational points and related problems. We compare this heuristic to Heath-Brown's prediction for sums of three cubes, to Zagier's count on the Markoff surface and to Baragar's and Umeda's work on variants of it, as well as to numerical data.

This is joint work with Tim Browning.