Research Seminar Number Theory and Arithmetic Geometry

Sparsity of Integral Points on Moduli Spaces of Varieties

Interesting moduli spaces don't have many integral points.  More precisely, if X is a variety over a number field, admitting a variation of Hodge structure whose associate period map is injective, then the number of S-integral points on X of height at most H grows more slowly than H^{\epsilon}, for any positive \epsilon.  This is a sort of weak generalization of the Shafarevich conjecture; it is a consequence of a point-counting theorem of Broberg, and the largeness of the fundamental group of X. Joint with Ellenberg and Venkatesh.

Fr 29.10.2021 11:15 (F303) 

Multiplicative relations among special points of modular and Shimura curves

Let Y be a modular or Shimura curve. Then Y comes with a (countably infinite) collection of so-called special points. I will outline a result describing when special points x₁,...,x_n in Y are multiplicatively dependent and also explain some conditions under which one can show this happens only finitely often (for fixed n). These results are closely connected to the Zilber-Pink conjecture on unlikely intersections.

Fr 3.12.2021 (online)

Enriques quotients of K3 surfaces and associated Brauer classes

This is a joint work in progress with Domenico Valloni. Let X be a complex K3 surface with an Enriques quotient S. It is known that the Brauer group of S has a unique non-zero element. Beauville gave a criterion for the natural map from Br(S) to Br(X) to be injective. Extending a result of Keum, who proved that every Kummer surface has an Enriques quotient, we show for an arbitrary Kummer surface X that every element of Br(X) of order 2 comes from an Enriques quotient of X. Work of Ohashi implies that in some `generic' cases this gives a bijection between the set of elements of order 2 in Br(X) and the set of Enriques quotients of X.

Toric peroids, 2-Selmer groups and non-tiling numbers

A positive integer n is a non-tiling number if the quadratic twists E⁽ⁿ⁾ and E^(-n) of E:y²=x(x-1)(x+3) have both rank zero. We studied the 2-divisibility of algebraic L-values of them using Waldspurger formula and an induction method, and studied the 2-Selmer groups of them using 2-descent. We proved that their algebraic L-values being odd is equivalent to that their 2-Selmer groups being minimal, and which have a positive explicit density. If time permits, we will introduce an ongoing work of a linear algebra framework which allows us to study the distributions of 2-Selmer groups for a general elliptic curve defined over rationals, and more.