241256 Rigid meromorphic cocycles III (S) (WiSe 2022/2023)

Short comment

Recently H.Darmon and J.Vonk initiated the theory of p-adic singular moduli for real quadratic fields. In this theory classical modular functions such as the j-invariant are replaced by so-called rigid meromorphic cocycles. These are SL2(Z[1/p])-invariant modular symbols with values in rigid meromorphic functions on Drinfeld’s p-adic upper half plane. One of their first results states that the divisor of a rigid meromorphic cocycle is supported on finitely many SL2(Z[1/p])-orbits of real quadratic points, i.e. points which generate real quadratic extensions of Q. This highly suggests that rigid meromorphic cocyles are a real quadratic analogue of Borcherds’ singular theta lifts of modular forms of weight 1/2. This approach does not generalize easily to a more general setup. The aim of this seminar is to follow L.Gehrmann's work "On Quaterionic Rigid Meromorphic", where he proves the algebraicity of divisors in a more general situation by purely cohomological methods.

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