# Iwasawa theory and $p$-adic $L$-functions

In this spring, we will talks every week. We will have decent mathematians from all over the world to talk about Iwasawa theory and $p$-adic $L$-functions.

The organiser and principal contact is Chao Qin.

Talks can be watched through Zoom ID: 894 6999 7231 Password: 2022
You can also enter the meeting room by clicking here.
The workshop is open to everyone and no registration is necessary.

## Title and Abstract

### Mar 2nd

Speaker: Jan Vonk
Title: Triple product periods in RM theory I (pdf)
Abstract: In these two talks, I will talk about recent progress on p-adic analogues of CM theory, for real quadratic fields. The emphasis will be on triple product periods, a set of invariants including (but not limited to) Gross-Stark units, Stark-Heegner points, and RM singular moduli.

### Mar 9th

Speaker: Jan Vonk
Title: Triple product periods in RM theory II (pdf)
Abstract: In these two talks, I will talk about recent progress on p-adic analogues of CM theory, for real quadratic fields. The emphasis will be on triple product periods, a set of invariants including (but not limited to) Gross-Stark units, Stark-Heegner points, and RM singular moduli.

### Mar 16th

Speaker: Chan-Ho Kim
Title: A structural refinement of Birch and Swinnerton-Dyer conjecture (pdf)
Abstract: We discuss how the structure of Selmer groups of elliptic curves can be described in terms of certain modular symbols from the viewpoint of refined Iwasawa theory.

### Mar 23th

Speaker: Andrei Jorza
Title: $p$-adic $L$-functions (pdf)
Abstract: $p$-adic $L$-functions have been essential, in the last decades, in proving instances of the Birch and Swinnerton-Dyer and Bloch-Kato conjectures. In this general talk, I will explain what $p$-adic $L$-functions are, and how they appear in connection with $p$-adic families of modular forms, focusing on the case of GL(2). The Taylor expansion of $p$-adic $L$-functions in $p$-adic families, was crucial in proving the trivial zero conjecture in Barrera-Dimitrov-Jorza, and we will explore a few such intriguing examples of Taylor expansions.

### Mar 30th

Speaker: Chung Pang Mok
Title: Pseudorandom Vectors Generation Using Elliptic Curves And Applications to Wiener Processes
Abstract: Using the arithmetic of elliptic curves over finite fields, we present an algorithm for the efficient generation of sequence of uniform pseudorandom vectors in high dimension with long period, that simulates sample sequence of a sequence of independent identically distributed random variables, with values in the hypercube $[0,1]^d$ with uniform distribution. As an application, we obtain, in the discrete time simulation, an efficient algorithm to simulate, uniformly distributed sample path sequence of a sequence of independent standard Wiener processes.

### Mar 31st

Speaker: Andrei Jorza
Title: $p$-adic $L$-functions for cuspidal representations of GL(2n) having Shalika models (pdf)
Abstract: In this second talk on $p$-adic $L$-functions we will discuss recent results on the construction of $p$-adic $L$-functions for cuspidal representations on GL(2n) which admit Shalika models. In ongoing work with Barrera, Dimitrov, Graham, and Williams, we have constructed such $p$-adic $L$-functions in $p$-adic families. These $p$-adic $L$-functions have recently been used by Loeffler and Zerbes to prove instances of Bloch-Kato.

### Apr 6th

Speaker: Meng-Fai Lim
Title: On growth of arithmetic objects in tower of number fields (pdf)
Abstract: The essence of Iwasawa theory is to study arithmetic objects via their variations in a tower of number fields. The theory was first initated by Iwasawa in the 1960s to study the growth of the Sylow p-subgroup of the class groups in the intermediate subfields of a Zp-extension of a number field F. The study has since been extended to considering even K-groups, Mordell-Weil groups, Tate-Shafarevich groups, fine Selmer groups, etale wild kernels and various arithmetic objects over a p-adic Lie extension. In this talk, we hope to give an overview and survey of these development.

### Apr 13th

Speaker: Xin Wan
Title: Iwasawa main conjecture for universal families(pdf)
Abstract: We formulate and prove an Iwasawa main conjecture for modular motives over the universal family of p-adic Langlands. From it we deduce Kato's Iwasawa main conjecture for modular forms without any assumption on the level group at p, and the BSD formula for rank 0 elliptic curves at primes of additive reduction. This is joint work with Olivier Fouquet.

### Apr 20th

Speaker: Hamish Gilmore(pdf)
Title: L-invariants attached to the symmetric square of an elliptic curve
Abstract: In this talk I will describe the algebraic and analytic $\mathcal{L}$-invariants attached to the symmetric square of and elliptic curve. I will also present an algorithm to compute the the analytic $\mathcal{L}$-invariant, and some computational results for elliptic curves of small conductor. This is joint work with Daniel Delbourgo.

### Apr 27th

Speaker: Takashi Hara
Title: On p-adic Artin L-functions for CM fields(pdf)
Abstract: We explain how to construct p-adic Artin L-functions for (p-ordinary) CM fields, which interpolate critical values of Hecke L-functions twisted by a fixed Artin representation. Our strategy is based upon Greenberg's patching construction of p-adic Artin L-functions for totally real fields, but one observes new phenomena and difficulties in the CM case. In this talk we would especially focus on differences between Greenberg's work and ours. This is joint work with Tadashi Ochiai.

### May 4th

Speaker: Chris Williams
Abstract: Let $\pi$ be a p-ordinary cohomological cuspidal automorphic representation of GL$(n,A_Q)$. A conjecture of Coates--Perrin-Riou predicts that the (twisted) critical values of its $L$-function $L(\pi x\chi,s)$, for Dirichlet characters $\chi$ of $p$-power conductor, satisfy systematic congruence properties modulo powers of $p$, captured in the existence of a $p$-adic $L$-function. For $n = 1,2$ this conjecture has been known for decades, but for $n > 2$ it is known only in special cases, e.g. symmetric squares of modular forms; and in all previously known cases, $\pi$ is a functorial transfer via a proper subgroup of GL($n$). In this talk, I will explain what a p-adic L-function is, state the conjecture more precisely, and then describe recent joint work with David Loeffler, in which we prove this conjecture for $n=3$ (without any transfer or self-duality assumptions).

### May 11th

Speaker: Dohyeong Kim
Title: Iwasawa theory and Selmer schemes I(pdf)
Abstract: Selmer schemes generalize Selmer groups by allowing non-abelian coefficients. Given the success of Iwasawa theory in the study of Selmer groups, it is natural to wonder whether its non-abelian analogue can be analyzed using similar tools. In the first talk, I will build upon Sakugawa's work on torsion Selmer pointed sets and extend his result. In the second talk, I will focus on the elliptic case of the non-abelian Chabauty method. I will explain how a p-adic L-function can help us verify new cases of the dimension hypothesis.

### May 18th

Speaker: Dohyeong Kim
Title: Iwasawa theory and Selmer schemes II(pdf)
Abstract: Selmer schemes generalize Selmer groups by allowing non-abelian coefficients. Given the success of Iwasawa theory in the study of Selmer groups, it is natural to wonder whether its non-abelian analogue can be analyzed using similar tools. In the first talk, I will build upon Sakugawa's work on torsion Selmer pointed sets and extend his result. In the second talk, I will focus on the elliptic case of the non-abelian Chabauty method. I will explain how a p-adic L-function can help us verify new cases of the dimension hypothesis.