The Luxembourg Number Theory Day 2020 will be held online on **December 14, 2020**.

The videoconferencing will be with the program Webex. The local participants can follow the talks in MSA 3.100.

Everyone is more than welcome to join us! If you would like to participate, please contact Antonella Perucca (antonella.perucca@uni.lu).

Time | Speaker | Title |

10:00-10:40 | Gaetan Chenevier (Paris-Sud University) | Unimodular hunting |

11:00-11:40 | Judith Ludwig (University of Heidelberg) | p-adic automorphic forms for SL(2) |

13:00-13:40 | Jan Vonk (Leiden University) | Real quadratic singular moduli and Galois representations |

14:00-14:40 | Sarah Zerbes (University College London) | Euler systems and the Bloch-Kato conjecture for GSp(4) |

**Gaetan Chenevier (Paris-Sud University) ***Unimodular hunting*

I will explain how to classify the unimodular integral lattices of rank 26 and 27.

**Judith Ludwig (University of Heidelberg) ***p-adic automorphic forms for SL(2)*

The Coleman-Mazur eigencurve is a geometric object attached to the group GL(2) that interpolates modular forms *p*-adically. It is an important object in number theory. Most recently it has been used as a ping-pong table in the proof of the symmetric power functoriality conjectures. On the eigencurve there are certain points where the geometry is quite special, namely the points corresponding to (critically refined) modular forms with complex multiplication. In this talk we will study the analogous situation for SL(2). For SL(2) we also have an eigencurve and we have analogues of the CM points on them. From the perspective of Langlands-correspondences the group SL(2) is more complicated than GL(2) due to a phenomenon called endoscopy. I will explain how endoscopy accounts for a curious phenomenon at the "CM points" of the SL(2)-eigencurve, namely the existence of some interesting *p*-adic automorphic forms.

**Jan Vonk (Leiden University) ***Real quadratic singular moduli and Galois representations*

The theory of complex multiplication occupies an important place in number theory, an early manifestation of which was the use of special values of the j-function in explicit class field theory of imaginary quadratic fields, and the works of Eisenstein, Kronecker, Weber, Hilbert, and many others. In the early 20th century, Hecke studied the diagonal restrictions of Eisenstein series over real quadratic fields, which later lead to highly influential developments in the theory of complex multiplication initiated by Gross and Zagier in their famous work on Heegner points on elliptic curves. In this talk, we will explore what happens when we replace the imaginary quadratic fields in CM theory with real quadratic fields, and propose a framework for a conjectural 'RM theory', based on the notion of rigid meromorphic cocycles, introduced in joint work with Henri Darmon. I will discuss recent progress towards these conjectures.

**Sarah Zerbes (University College London) *** Euler systems and the Bloch-Kato conjecture for GSp(4)*

Euler systems are compatible families of Galois cohomology classes attached to a global Galois representations, and they play an important role in proving cases of the Bloch-Kato conjecture.

In my talk, I will review the construction of an Euler system attached to the spin representation of a genus 2 Siegel modular form. I will then sketch a relation between the Euler system and values of a p-adic L-function, which leads to new cases of the Bloch-Kato conjecture. This is joint work with David Loeffler.

Last modification: 10 November 2020.