Search Talks

We discuss new applications of some "zeta elements" which form Kolyvagin systems. A portion of this talk originated from discussions with Minhyong Kim, and another portion is joint work in progress with Gyujin Oh.

I will introduce via examples, and then state and prove generalizations of Chen's cross ratios theorem on the special values of Rankin-Selberg L-functions for GL(n) x GL(m). I will then discuss applications of this theorem to Deligne's conjecture on the special values of various automorphic L-functions. This talk is a report of an ongoing collaboration with Harald Grobner, Michael Harris, and Jie Lin.

Let p \geq 5 be a prime. We construct a lattice in a self-dual modular Galois representation for which the p-torsion of the corresponding Bloch-Kato Selmer group is arbitrarily large. This extends the work of Matsuno for elliptic curves and small primes. Our representation is constructed by appropriately modifying an argument of Hamblen and Ramakrishna which allows one to lift reducible mod p Galois representations to characteristic zero with local conditions at a prescribed set of primes. This talk is based on recent joint work with Anwesh Ray (https://arxiv.org/pdf/2504.16287).

Let E/F be an elliptic curve with CM by an imaginary quadratic field K, and assume that the extension of F generated by the torsion points of E is abelian over K. In this talk I will outline the proof of the p-part of the Birch-Swinnerton-Dyer formula for E in analytic rank 1 for primes p>3 of ordinary reduction. For F=Q, this was originally proved by Rubin in 1991 as a consequence of his proof of the Iwasawa main conjecture for K. In contrast, our approach to the problem is based on the study of an auxiliary Rankin-Selberg convolution, and extends to CM abelian varieties A/K and higher weight CM modular forms.

I will explain the theory of Kolyvagin systems of Gauss sum type (or of rank 0) for certain self-dual representations over a number field. I especially discuss a remarkable, decisive property of the systems for the structure of Selmer groups, based on joint work with R. Sakamoto.

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.