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Title: The correlation coefficient in representation theory Abstract: Given a group G and two Gelfand subgroups H and K of G, associated to an irreducible representation \pi of G, there is a notion of H and K being correlated with respect to \pi in G. We discuss this theme in the context of toric periods for GL(2) over a finite field.

The Harris–Venkatesh plus Stark conjecture says that the action of the derived Hecke algebra on weight-1 cusp forms describes Stark units modulo p for all but finitely many primes p. These derived Hecke operators H^0 → H^1 on the cohomology of modular curves are defined by Shimura classes arising from the cover of X_1(p) over X_0(p). I will report on in-progress work to describe p-adic Shimura classes, define derived Hecke operators on completed cohomology, and formulate a similar conjecture for p-adic regulators of Stark units.

Certain issues arise while considering Local Tate duality over characteristics $p>0$, when the Galois module has $p$-torsion. A solution was given by Shatz, where he considered finite flat group schemes instead of Galois modules and the dual group is Cartier dual. The duality theorem is then a topological duality of the cohomology groups. We give a more natural construction and proof of the topological aspects of the duality theorem. This is joint work with Manodeep Raha.

We describe how the relative Satake isomorphism due to Sakellaridis gives a conceptual way of choosing test vectors when constructing the tame part of an Euler system. This is joint work with Li Cai and Yangyu Fan.

Let p be an odd prime. In this talk, I will explain some recent progress towards Mazur's conjecture on the growth of the Mordell-Weil ranks of an elliptic curve E/Q over Zp-extensions of an imaginary quadratic field, where p is a prime of good reduction for E. If time permits, I will also discuss results towards generalization of this conjecture for abelian varieties.

Let p and N be two odd primes >=5 such that N is congruent to 1 mod p. While studying Eisenstein congruences at prime level N extends from Mazur's work on the Eisenstein ideal (1977) continuing on to more recent work of Wake--Wang-Erickson (2020), we study Eisenstein congruences at prime square level N^2. We prove precise R = T theorems identifying suitable universal pseudo-deformation rings with Hecke algebras, both at level Gamma0(N^2) and Gamma1(N^2). Our study requires working with (cyclic p-group) group ring valued Eisenstein series, which in turn necessitates us to establish a new module-theoretic criterion to prove an R = T theorem. This is joint work with Jaclyn Lang and Katharina Mueller.

We describe how Euler added up all positive integers into a mysterious fraction when he was 28 years old, and I try to legitimize his method “p-adically”. This is a story of Number Theory from the 17th century on. We only need some knowledge of polynomials and fractions of polynomials and very basics of differentiation. If time allows, I enter into some results related to Ramanujan I found when I was 28 years old. For the results exposed here, a detailed proof can be found in my book: “Elementary Theory of L-functions and Eisenstein Series,” LMSST vol. 26, 1993, Cambridge U. Press.

For a normalized newform g in S_{k}(\Gamma_{0}(N)) with complex multiplication by an imaginary quadratic field K, there is a mock modular form f^{+} corresponding to g. K. Bringmann, P. Guerzhoy, and B. Kane modified f^{+} to obtain the p-adic modular form by a certain p-adic constant \alpha_{g}. In addition, they showed that \alpha_{g}=0 if p is split in K and does not divide N. On the other hand, the speaker showed that \alpha_{g} is a p-adic unit for an inert prime p that does not divide 2N when \dim S_{k}(\Gamma_{0}(N))=1. In this talk, the speaker determines the p-adic valuation of \alpha_{g} for an inert prime p under a mild condition, when g has weight 2 and rational Fourier coefficients.

The celebrated BDP formula evaluates Rankin–Selberg p-adic L-functions at points outside their interpolation range in terms of Generalised Heegner cycles (a phenomenon referred to as wall-crossing). This principle has been extended to triple products by Bertolini–Seveso–Venerucci and Darmon–Rotger, who relate values of Hsieh’s unbalanced p-adic L-functions on the balanced range to diagonal cycles. I will report on a result where wall-crossing is used to factor a triple product p-adic L-function with an empty interpolation range, to yield a p-adic Artin formalism for families of the form f × g × g. The key input is the arithmetic Gan–Gross–Prasad (Gross–Kudla) conjecture, linking central derivatives of (complex) triple product L-functions to Bloch–Beilinson heights of diagonal cycles and their comparison with their GL(2) counterpart (Gross–Zagier formulae). I will also discuss an extension to families on GSp(4) × GL(2) × GL(2), where a new double wall-crossing phenomenon arises and is required to explain a p-adic Artin formalism for families of the form F x g x g. This suggests a higher BDP/arithmetic GGP formula concerning second-order derivatives.

I will present a construction of anticyclotomic Euler systems, for those Galois representations of a CM field that are conjugate-symplectic, automorphic, and of regular, "balanced" Hodge-Tate type. Its main ingredients are variants of the generating series of special cycles on unitary Shimura varieties studied by Kudla and Liu, and the construction is conditional on a conjecture on their modularity. The relevant notion of Euler system is the one studied by Jetchev-Nekovar-Skinner. Combining with their work and with a height formula obtained with Liu yields (unconditionally) some new cases of the p-adic Beilinson-Bloch-Kato conjecture in analytic rank one.