ICTS

In a joint work with H. Hida, we proved in the 90's that the anticyclotomic Katz p-adic L function associated to a p-adic CM type divides the characteristic power series of the Iwasawa module associated to this p-adic CM type. The goal of these two talks is to sketch this proof. Note that we couldn't treat the divisibility at the prime p of the Iwasawa algebra. This has been treated in subsequent works by H. Hida.

In a joint work with H. Hida, we proved in the 90's that the anticyclotomic Katz p-adic L function associated to a p-adic CM type divides the characteristic power series of the Iwasawa module associated to this p-adic CM type. The goal of these two talks is to sketch this proof. Note that we couldn't treat the divisibility at the prime p of the Iwasawa algebra. This has been treated in subsequent works by H. Hida.

The local epsilon conjecture is one of a series of Kato's conjectures on a generalization of the Iwasawa main conjecture to general families of p-adic Galois representations. It gives a precise description of a p-adic variation of the p-adic Hodge theoretic invariants, like local (L-, and epsilon) factors, Bloch-Kato's cohomologies, and Hodge-Tate weights which are only defined for de Rham representations, in p-adic families of local p-adic Galois representations. In my lectures, I will explain the formulation of this conjecture, the proof of the conjecture for the rank two case using the p-adic Langlands for GL_2(Q_p), and it's application to a generalization of Rubin's local sign decomposition conjecure.

The arithmetic Siegel-Weil formula, conjectured by Kudla-Rapoport and proved by Li-Zhang, expresses the degrees of certain 0-cycles on integral models of unitary Shimura varieties in terms of the nondegenerate Fourier coefficients of the central derivative of an Eisenstein series.
Feng-Yun-Zhang proved a higher derivative version of this arithmetic Siegel-Weil formula in the function field setting, now expressing degrees of 0-cycles on moduli spaces of unitary shtukas to the nondegenerate Fourier coefficients of higher central derivatives of an Eisenstein series.
The goal of my lecture series is (1) to explain all of this background, (2) extend the results of Feng-Yun-Zhang to include some degenerate coefficients, and (3) deduce from this extension an arithmetic application: the nonvanishing of higher central derivatives of certain Langlands L-functions implies the nonvanishing of classes in the Chow groups of moduli spaces of shtukas.
All of the new results are joint work with Tony Feng and Mikayel Mkrtchyan.

We first prove, for a prime p>3 unramified in a CM quadratic extension of a totally real field F, h(M/F)L(\chi)|H(\psi)|h(M/F)F(\chi) (h(M/F)=h(M)/h(F)) in \Lambda for the congruence power serie H(\psi) of \psi lifting a fixed anti-cyclotomic character \chi and anticyclotomic Katz p-adic L-function L(\chi) of branch character \chi, built on the lectures by Tilouine proving this over \Lambda[1/p]. Here \Lambda is the many variable Iwasawa algebra of M. In the second lecture, we give a sketch of the proof of the reverse divisibility: H(\psi)|h(M/F)L(\chi) resulting in the main conjecture, as H(\psi)=h(M/F)F(\chi) for the anticyclotomic Iwasawa power series F(\chi) by the “R=T”-theorem.

The local epsilon conjecture is one of a series of Kato's conjectures on a generalization of the Iwasawa main conjecture to general families of p-adic Galois representations. It gives a precise description of a p-adic variation of the p-adic Hodge theoretic invariants, like local (L-, and epsilon) factors, Bloch-Kato's cohomologies, and Hodge-Tate weights which are only defined for de Rham representations, in p-adic families of local p-adic Galois representations. In my lectures, I will explain the formulation of this conjecture, the proof of the conjecture for the rank two case using the p-adic Langlands for GL_2(Q_p), and it's application to a generalization of Rubin's local sign decomposition conjecure.

We first prove, for a prime p>3 unramified in a CM quadratic extension of a totally real field F, h(M/F)L(\chi)|H(\psi)|h(M/F)F(\chi) (h(M/F)=h(M)/h(F)) in \Lambda for the congruence power serie H(\psi) of \psi lifting a fixed anti-cyclotomic character \chi and anticyclotomic Katz p-adic L-function L(\chi) of branch character \chi, built on the lectures by Tilouine proving this over \Lambda[1/p]. Here \Lambda is the many variable Iwasawa algebra of M. In the second lecture, we give a sketch of the proof of the reverse divisibility: H(\psi)|h(M/F)L(\chi) resulting in the main conjecture, as H(\psi)=h(M/F)F(\chi) for the anticyclotomic Iwasawa power series F(\chi) by the “R=T”-theorem.

Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of L-functions. However, despite the key role they have played in making progress toward foundational conjectures in number theory like the Birch–Swinnerton-Dyer and Bloch– Kato Conjectures, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle to constructing Euler Systems lies in producing candidate Galois cohomology classes. This lecture series presents a method to overcome this obstacle that does not rely on rare (known) motivic classes. We will focus on building ´etale cohomology classes originating from automorphic data: Eisenstein series and Theta series. This framework not only retrieves most classical Euler systems but can also be applied to construct an Euler system for the adjoint of an elliptic modular form.
References:
• C. Skinner, L-values and nonsplit extensions: a simple case, https://msp.org/ent/2024/3-1/p03.xhtml
• H. Darmon etal, p-adic L-functions and Euler systems: a tale in two trilogies.

Spill-over from the last lecture, and finish with some low dimensional examples, including the fundamental work of Waldspurger; illustrative examples from finite fields.

Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of L-functions. However, despite the key role they have played in making progress toward foundational conjectures in number theory like the Birch–Swinnerton-Dyer and Bloch– Kato Conjectures, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle to constructing Euler Systems lies in producing candidate Galois cohomology classes. This lecture series presents a method to overcome this obstacle that does not rely on rare (known) motivic classes. We will focus on building ´etale cohomology classes originating from automorphic data: Eisenstein series and Theta series. This framework not only retrieves most classical Euler systems but can also be applied to construct an Euler system for the adjoint of an elliptic modular form.
References:
• C. Skinner, L-values and nonsplit extensions: a simple case, https://msp.org/ent/2024/3-1/p03.xhtml
• H. Darmon etal, p-adic L-functions and Euler systems: a tale in two trilogies.

Iwasawa theory of CM elliptic curves is well developed for primes p split in the CM field (good ordinary case), and has applications to the BSD conjecture. In contrast, for p inert (good supersingular) or ramified (bad additive), new phenomena occur and the theory is still fragmentary. For the anticyclotomic deformation at an inert prime, the last few years has seen a progress due to the work of Burungale-Kobayashi-Ota, following Rubin's pioneering work in the mid 80's. In this talk, we report on a similar progress for ramified primes. We first explain the root number and Mordell-Weil variations, followed by a local sign decomposition analogous to Rubin's conjecture in the inert case and an Iwasawa main conjecture. This is a joint work with A. Burungale, K. Nakamura and K. Ota.

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.

TBA

I will discuss on the construction and the reciprocity law of certain Euler systems via the study of congruences between automorphic forms.

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.

The Kronecker limit formula is an equality relating the Faltings height of an CM elliptic curve to the sub-leading term (at s=0) of the Dirichlet L-function of an imaginary quadratic character. Colmez conjectured a generalization relating the Faltings height of any CM abelian variety to the subleading terms of certain Artin L-functions. In this talk we will formulate a “non-Artinian” generalization of Colmez conjecture, relating the following two quantities: (1) the Faltings height of certain arithmetic Chow cycles on unitary Shimura varieties, and (2) the sub-leading terms of the adjoint L-functions of (cohomological) automorphic representations of unitary groups U(n). The $n=1$ case of our conjecture recovers the averaged Colmez conjecture. We are able to prove our conjecture when $n=2$, using a relative trace formula approach that is formulated for the general $n$. The “arithmetic relative Langlands” morally suggests that there should be a lot of other similar (conjectural) phenomena involving subleading terms of L-functions and Faltings-like heights of algebraic cycles on Shimura varieties, and I will give a few more examples. Joint work with Ryan Chen and Weixiao Lu.

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.

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.

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.

Let E be an elliptic curve over an imaginary quadratic extension K such that E has good ordinary reduction at the primes above a fixed odd prime $p$. We study the Iwasawa invariants of various Iwasawa modules associated to E over the anticyclotomic Z_p-extension.

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.

TBA

A famous question of Greenberg (which was also formulated independently by Coleman) asks the following:
Suppose p is a prime and f is a p-ordinary modular eigenform of weight at least 2. If the restriction of the p-adic Galois representation attached to f to the local Galois group at p splits into a direct sum of two characters, then does f have complex multiplication?
In this talk, we will explore an analogue of this question in the setting of Siegel modular forms of genus 2.
This talk is based on a joint work in progress with Bharathwaj Palvannan.