# Research Seminar Number Theory and Arithmetic Geometry

**Summer term 2022/23**

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

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

Victoria Cantoral Farfán, Ulrich Derenthal, Matthias Schütt, Ziyang Gao

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

Fr 14.4.2023 | Teppei Takamatsu (Kyoto University) | On Quasi-Frobenius-splitting |

Do 20.4.2023 10:15-11:15, Raum F428 | Adam Logan (Tutte Institute for Mathematics and Computing, Ottawa) |
Recent work has greatly enhanced our understanding of the ${\mathbb F}_p$-points on the Markoff surface defined by $x^2 + y^2 + z^2 = 3xyz$. In particular, we now know, thanks to work of Bourgain-Gamburd-Sarnak, that the action on a natural quotient is almost always transitive, and by Carmon-Meiri-Puder, that the action is almost always as large as possible. The goal of this talk is to prove a result analogous to that of Carmon-Meiri-Puder for the K3 surfaces of the form $(E x E)/\pm 1$, where $E$ is an elliptic curve defined over $\mathbb Q$. In order to do this we will prove a surprising property of points on certain rational curves on this surface. This is joint work with Owen Patashnick. |

Fr 5.5.2023 | Christian Bernert (Göttingen) |
Classical results of Heath-Brown and Wooley show that projective hypersurfaces defined by a cubic form in n variables contain a rational point if n is at least 14, and a rational line if n is at least 37. In this talk, I will report on two new results in this direction. i) If n is at least 10 (which is the conjectural threshold for the existence of rational points), the singular series of the cubic form converges absolutely. I will explain what the singular series is and why it is relevant to the existence of rational points (Keyword: Hardy-Littlewood Circle Method). ii) In joint work with Leonhard Hochfilzer, we show that 31 variables are sufficient to guarantee the existence of a rational line, improving on Wooley's result. The key ingredient is a generalization of Heath-Brown's 14-variable result to imaginary quadratic number fields. |

Mo 22.5.2023 11:00-12:00, room F128 | Timo Keller (LUH) |
We report on work in progress with Mulun Yin. Castella--Gross--Lee--Skinner recently proved Perrin-Riou's Heegner point main conjecture for RM abelian varieties at primes p of good reduction for which the mod-p Galois representation rho_p is reducible. They have the restriction that the characters in the semisimplification of rho_p are non-trivial. We work on removing this restriction and generalize the result to newforms of higher weight, allowing us to treat also bad multiplicative reduction. As a consequence, we get the p-part of the Birch--Swinnerton-Dyer conjecture for analytic rank 1 and a p-converse theorem. |

Fr 26.5.2023 | Victoria Cantoral Farfán (LUH/Göttingen) |
Let $A$ be an abelian variety over a number field. The connected monodromy field of $A$ is the minimal field over which the images of all the $\ell$-adic torsion representations have connected Zariski closure. During this talk, we will show that for all even $g\geq 4$, there exist infinitely many geometrically nonisogenous abelian varieties $A$ over $\mathbb Q$ of dimension $g$ where the connected monodromy field is strictly larger than the field of definition of the endomorphisms of $A$. Our construction arises from explicit families of hyperelliptic Jacobians with definite quaternionic multiplication. This is a joint work with Lombardo and Voight. |

Fr 9.6.2023 | Justin Uhlemann (LUH) | |

Fr 16.6.2023 | ||

Fr 23.6.2023
| Marta Pieropan (Utrecht) | |

Fr 23.6.2023
| Jérôme Poineau (Caen) | |

Fr 30.6.2023 | Carlo Pagano (Concordia) | |

Fr 7.7.2023 | ||

Fr 14.7.2023 | Shu Shen (Jussieu) | |

Fr 21.7.2023 | Florian Munkelt (Göttingen) |