Research Seminar Number Theory and Arithmetic Geometry

Fr 15.10.2021Brian Lawrence (UCLA)

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) 

Giacomo Mezzedimi (LUH)Elliptic K3 surfaces and their moduli: dynamics, geometry and arithmetic
Fr 19.11.2021Damián Gvirtz (University College London)A Hilbert irreducibility theorem for K3 and Enriques surfaces
Fr 26.11.2021Guy Fowler (LUH)

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,...,xn 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)

Alexei Skorobogatov (Imperial College London)

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.

Fr 10.12.2021Jinzhao Pan (LUH)

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.

Fr 25.3.2022 (online)Lucas Surmann (LUH)Ideal class groups and integral points on conics