# Research Seminar Number Theory and Arithmetic Geometry

**Winter term 2021/22**

**Fridays, 11:00-12:00**

**Seminar room G117 in Welfenschloss (main building, Welfengarten 1) or online (researchseminars.org)**

Datum | Vortragende/r | Vortragstitel |
---|---|---|

Fr 15.10.2021 | Brian Lawrence (UCLA) |
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 | Giacomo Mezzedimi (LUH) | Elliptic K3 surfaces and their moduli: dynamics, geometry and arithmetic |

Fr 19.11.2021 | Damián Gvirtz (University College London) | A Hilbert irreducibility theorem for K3 and Enriques surfaces |

Fr 26.11.2021 | Guy Fowler (LUH) |
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 x |

Fr 3.12.2021 | Alexei Skorobogatov (Imperial College London) |
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 10.12.2021 | Jinzhao Pan (LUH) |
A positive integer n is a non-tiling number if the quadratic twists E |

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