U of T number theory seminar (2025–26)

Let \( \mathcal{O}_K \) be a mixed characteristic complete DVR with residue field \( k \) of characteristic \( p \), and \( X/\mathcal{O}_K \) a smooth proper formal scheme. A celebrated result of Berthelot–Ogus shows that the de Rham cohomology groups \( H^i_\mathrm{dR}(X/\mathcal{O}_K) \) and crystalline cohomology groups \( H^i_\mathrm{crys}(X_k/W(k))\) agree after tensoring to \(K\). Using recent advances in prismatic cohomology we show that an integral version of this statement holds using appropriate notions of 'twisted' de Rham and crystalline cohomology. This is a prismatic manifestation of Dwork's trick. This further gives an integral comparison even for non-trivial (prismatic) coefficients, and allows one to initiate a prismatic study of the relationship between crystalline and de Rham torsion. This is based off joint work with Abhinandan.

For a connected reductive group scheme over \(\mathbb{Z}\), it is interesting to study its level one automorphic forms, for example, give a dimension formula or some explicit construction. For a classical group with small rank or a compact \(G_2\)-type group, a computable dimension formula is given by Chenevier–Taïbi–Renard, and it is known that some families of automorphic forms can be constructed via theta correspondences. In this talk, I will present my recent work on the automorphic forms of a compact \(F_4\)-type group, emphasizing an "exceptional" analogue of the classical theta series on Euclidean spaces.

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