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.

Let \(K\) be a finite unramified extension of \(\mathbb{Q}_p\). Let \(\pi\) be an admissible smooth mod \(p\) representation of \(\mathrm{GL}_2(K)\) occurring in some Hecke eigenspaces of the mod \(p\) cohomology of a Shimura curve, and \(r\) be its underlying global \(2\)-dimensional Galois representation. When \(r\) is sufficiently generic, we prove that the associated multivariable \((\varphi,\Gamma)\)-module \(D_A(\pi)\) defined by Breuil–Herzig–Hu–Morra–Schraen is completely determined by the restriction of \(r\) to the decomposition group at \(p\) in an explicit way, generalizing their results.

The classical Dieudonne theory classifies \(p\)-divisible groups over a perfect field \(k\) in terms of semi-linear algebra over \(W(k)\). In this talk, I will explain a conjectural generalization - due to Drinfeld - of Dieudonne's classification to a broader class of rings, whose spectra form a basis for the fpqc topology on \(p\)-nilpotent schemes. A key input in Drinfeld's approach is a certain square-zero extension of the ring of Witt vectors, known as the ring of sheared Witt vectors.

In 1972, Katz proved the Grothendieck-Katz \(p\)-curvature conjecture for linear differential equations arising from algebraic geometry—that is, Gauss-Manin connections. His proof made use of the structures and properties of the cohomology of a family of varieties: for example, the Hodge and conjugate filtrations, the Hodge index theorem, etc. I'll explain analogues of these structures and properties for non-abelian cohomology (that is, the moduli of representations of \(\pi_1\)) and how to use them to prove a version of the Grothendieck-Katz \(p\)-curvature conjecture in the non-abelian setting.

A classical theorem of Birch and Merriman states that, for fixed \(n\), the set of integral binary \(n\)-ic forms with fixed nonzero discriminant breaks into finitely many \(\mathrm{GL}{_2}(\mathbb{Z})\) orbits. In this talk, I will present several extensions of this finiteness result from both the representation theoretic and the geometric perspectives, along with an application of these generalizations.

On the representation theoretic side, we study ternary \(n\)-ic forms and prove analogous finiteness theorems for \(\mathrm{GL}{_3}(\mathbb{Z})\) orbits with fixed nonzero discriminant. We also establish a corresponding result for a 27 dimensional representation associated with a family of K3 surfaces (joint with A. Shankar).

On the geometric side, we prove a finiteness theorem for Galois invariant point configurations on arbitrary smooth curves with controlled reduction, unifying the classical finiteness theorems of Birch and Merriman, Siegel, and Faltings (joint with F. Sajadi).

As an application, we consider families of pencils of curves with fixed discriminant, where the representation theoretic and geometric extensions, together with the Birch and Merriman theorem, play a central role in establishing finiteness.

Hecke operators play a fundamental role in understanding the arithmetic properties of modular and automorphic forms. Since the advent of the original Eichler-Shimura relation, it has been clear that the mod-\(p\) behavior of Hecke correspondences is crucial for such applications. However, one could argue a truly robust theory of such correspondences yielding convenient access to their mod-\(p\) reductions has so far been elusive, especially when dealing with higher rank groups.

In this talk, I will present a new approach to these matters, using recent advances in \(p\)-adic geometry and \(p\)-adic cohomology, building on work of Drinfeld and Bhatt-Lurie, and combining them with a tool familiar to the geometric Langlands and representation theory community: the Vinberg monoid. In particular, this approach yields direct access to geometric incarnations of the 'standard' basis elements of the spherical Hecke algebra.

For another application, this approach also gives the first general construction of Rapoport–Zink spaces associated with exceptional groups.

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The study of special values and derivatives of automorphic L-functions reveals deep connections between arithmetic geometry and harmonic analysis. A central theme is the Arithmetic Fundamental Lemma (AFL), which predicts precise identities between orbital integrals and intersection numbers of cycles. While methods based on perverse sheaves have achieved remarkable results in the function field case, they often obscure the underlying local geometry.

In this talk I will present recent progress on the higher linear AFL through the framework of the Relative Trace Formula (RTF). This approach provides explicit structural links between analytic orbital integrals and local intersection theory, enabling direct local proofs beyond global sheaf-theoretic methods. I will also outline several new directions: extending the AFL by adding conductors, and exploring their applications to global conjectures on derivatives of L-functions, including variants of Gross–Zagier type formulas.

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We seek a \(p\)-adic Langlands correspondence between a Galois representation \(\mathrm{Gal}(\overline{K}/K) \to \mathrm{GL}{_n}(\overline{\mathbb{Q}{_p}})\) and an admissible unitary representation of \(\mathrm{GL}{_n}(K)\) over a \(p\)-adic Banach space. This correspondence is known when \(\mathrm{GL}{_n}(K) = \mathrm{GL}{_2}(\mathbb{Q}{_p})\), but remains unknown even for \(\mathrm{GL}{_2}(\mathbb{Q}{_{p^f}})\). That said, given a \(p\)-adic Galois representation \(\mathrm{Gal}(\overline{K}/K) \to \mathrm{GL}{_2}(\overline{\mathbb{Q}{_p}})\), one can construct an admissible unitary representation of \(\mathrm{GL}{_2}(\mathbb{Q}{_{p^f}})\) using a global setup. However, it is unclear whether this construction is independent of the global setting. Breuil's lattice conjecture provides evidence for such a claim. Proving the conjecture shows strong local-global compatibility. In the talk, I will explain the motivation behind the conjecture and, time permitting, briefly sketch the proof.

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