# Research Seminar Number Theory and Arithmetic Geometry

**Summer term 2022**

**Fridays, 10:45-11:45**

**Seminar room A410 in Welfenschloss (main building, Welfengarten 1) or online (researchseminars.org)**

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

Fr 22.4.2022 | John Voight (Dartmouth College) | Orthogonal modular forms and their L-functions |

Fr 29.4.2022 | Lars Kühne (LUH) |
I will talk about the deduction of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of elliptic families from my recent theorem on equidistribution in families of abelian varieties. This generalizes results of DeMarco and Mavraki and improves certain results of Manin-Mumford type proven by Masser and Zannier to results of Bogomolov type. |

Fr 13.5.2022 | North German Algebraic Geometry Seminar (Hannover) | |

Do 19.5.2022 (16:30, online) | Masahiro Nakahara (University of Washington) |
A variety satisfies potential density if it contains a dense subset of rational points after extending its ground field by a finite degree. A collection of varieties satisfies uniform potential density if that degree can be uniformly bounded. I will discuss this property for connected algebraic groups of a fixed dimension and elliptic K3 surfaces. This is joint work with Kuan-Wen Lai. |

Fr 20.5.2022 | Carlo Pagano (Glasgow University) |
In this talk we consider quantitative aspects of the question of the solvability of diophantine equations in (algebraic) families, namely for "how many" choices of the parameters the corresponding equation admits a solution. I will consider the case of integral solvability for one parameter family of conics: I will explain a recent joint work with Peter Koymans settling a conjecture of Stevenhagen for the, so called, negative Pell equation. I will also discuss a joint work in progress with Efthymios Sofos on the case of adelic/rational solvability (studied by Serre and Laughran-Smeets) and its application to Diophantine stability questions. Finally I will explain the (unexpected) relation(s) both works have with statistics of L-functions and Iwasawa towers. |

Fr 27.5.2022 | Yoto Maeda (University of Kyoto) | |

| Göttingen-Hannover Number Theory Workshop | |

Fr 3.6.2022 | Julian Demeio (MPI für Mathematik, Bonn) |
In joint work with Sam Streeter, we show that a surface X with two distinct conic bundles and Zariski-dense rational points satisfies, under some mild conditions, the weak weak approximation property. In particular, our result shows that a "generic" del Pezzo surface of degree 1 or 2 with Zariski-dense rational points and (even just) one conic fibration satisfies the weak weak approximation property. Our method is based on an application of a result of Denef on certain auxiliary rational higher-dimensional covers of X. |

Fr 17.6.2022 | Tobias Kreutz (HU Berlin) | On the Mumford-Tate conjecture for projective hypersurfaces |

Fr 24.6.2022 | Timo Keller (LUH) |
The strong Birch-Swinnerton-Dyer conjecture and in particular the exact order of the Shafarevich-Tate group for abelian varieties over the rationals has only been known for elliptic curves (dimension 1) or in higher dimension where the conjecture could be reduced to dimension 1. We give the first absolutely simple examples of dimension 2 where the conjecture can be verified: |

Fr 1.7.2022 (11:10) | Gregorio Baldi (IHES) |
I will report on a joint work with Klingler and Ullmo. Given a polarizable variation of Hodge structures on a smooth quasi-projective variety S (e.g. the one associated to a family of pure motives over S), Cattani, Deligne and Kaplan proved that its Hodge locus (the locus of closed points of S where exceptional Hodge tenors do appear) is a countable union of closed algebraic subvarieties of S. I will explain when this Hodge locus is actually algebraic. Depending on the interest of the audience, I will conclude the talk either by describing how such an algebraicity statement complements the Lawrence-Venkatesh method or by sketching how similar ideas guarantee the existence of infinitely many 4-dimensinal Jacobians (defined over Qbar) of "Mumford’s type". |

Fr 8.7.2022 | Gabriel Dill (LUH) |
I will report on joint work in progress with Francesco Campagna (MPIM Bonn). Let H |

Fr 15.7.2022 | Harry Schmidt (Universität Basel) |
In recent joint work with Myrto Mavraki we proved a uniform version of the Bogomolov conjecture for curves and split rational maps. I will discuss some of the ingredients of our proof and its relation to a relative version of the Bogomolov conjecture formulated by Lars Kühne. Time permitting I will talk about connections to problems in measure theory as well as related work in progress. |

Di 13.9.2022 (10:00, online) | Maximilian Rabeler (LUH) |