Antisymmetry in the theory of rigid meromorphic cocycles

Around six years ago, Darmon and Vonk initiated the theory of p-adic singu- lar moduli for real quadratic fields. The central object in their theory are rigid meromorphic cocycles. These are elements of the first cohomology group of Ihara’s group SL2(Z[1/p]) with values in the multiplicative group of non-zero rigid mero- morphic functions on Drinfeld’s upper half plane. Such classes can be evaluated at RM points, i.e. at points in the p-adic upper half plane that satisfy a rational quadratic equation with positive discriminant. Conjecturally, these values are alge- braic and belong to certain abelian extensions of real quadratic fields. Using rigid meromorphic cocycles, Darmon and Vonk assign to each pair of RM points a p-adic number. The two RM points play vastly different roles in the construction of this assignment. However, it is expected to behave like the difference of two classical singular moduli. In particular, it should be antisymmetric in the argument. In this thesis we use recent work of Darmon, Gehrmann and Lipnowski on rigid meromor- phic cocycles for higher dimensional orthogonal groups to give a new construction of this assignment, which leads to a proof of the expected antisymmetry. Furthermore, we use the setup of Darmon, Gehrmann and Lipnowski for the qua- dratic space (M2(Q), det) of signature (2, 2) to prove the modularity of a generating series made up from so-called Kudla–Millson divisors. In our theorem, these divi- sors play the role of special cycles on orthogonal Shimura varieties in the classical Kudla program. A crucial ingredient for this result is the observation that every invertible rigid analytic function on the product of two copies of the p-adic upper half plane is the product of two invertible rigid analytic functions on one copy. We prove this theorem in the appendix.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Bielefeld

Year

2025

Author dc:creator

• Sprehe, Sören

Repository record source_url

https://pub.uni-bielefeld.de/record/3003482

OAI identifier oai:identifier

oai:pub.uni-bielefeld.de:3003482