**MINI-COURSE**

**Francesc Castella**

**Title**: On the Iwasawa theory of elliptic curves at Eisenstein primes

**Abstract**: The goal of this mini-course is to explain some recent advances on the Iwasawa theory of rational elliptic curves at Eisenstein primes p, and their applications. Topics include the extension of the method of Greenberg-Vatsal to the anticyclotomic setting (joint with Grossi, Lee, and Skinner), and the related proof of Mazur's (cyclotomic) main conjecture for Eisenstein primes p (joint with Grossi and Skinner).

**Samit Dasgupta and Mahesh Kakde**

**Title**: On the Brumer-Stark Conjecture

**Abstract**:

**Lectures 1 and 2** : __(Samit Dasgupta)__ The Brumer-Stark conjecture is a generalisation of the classical theorem of Stickelberger on annihilation of class groups of abelian CM fields. We will first see a formulation of the Brumer-Stark conjecture. A consequence of the Brumer-Stark conjecture is existence of special elements called the Brumer-Stark units. These units generate abelian CM extensions of totally real number fields. A conjecture of Gross known as "the tower of fields conjecture" takes the first step towards giving an explicit description of the Brumer-Stark units. A precise conjectural p-adic analytic formula for the Brumer-Stark units was given by Dasgupta. We will review statement of the tower of fields conjecture as well as the explicit p-adic analytic formula and describe a precise relationship between the two. Furthermore, we will give an application towards explicit class field theory.

**Lectures 3, 4 and 5** : __(Mahesh Kakde)__ The main aim of these lectures will be to give a proof of the Brumer-Stark conjecture. We will start with a number of refinements of the Brumer-Stark conjecture. To this end we will see the definition of Ritter-Weiss module. We then formulate refinements of Brumer-Stark conjecture in terms of these Ritter-Weiss modules. We then describe our refinements of Ribet’s method to prove strengthening of the Brumer-Stark conjecture.

**Matthias Flach**

**Lecture 1**: Special values of Motivic L-functions I

**Abstract**: We discuss some history of special value conjectures with Dedekind Zeta functions as the main example. We give some algebraic background on determinant functors and introduce the concept of the fundamental line due to Fontaine and Perrin-Riou.

**Lecture 2**: Special values of Motivic L-functions II

**Abstract**: We give the general formulation of the Tamagawa number conjecture (of Bloch, Kato, Fontaine, Perrin-Riou) for motives over number fields. We discuss some proven cases with particular emphasis on Dirichlet L-functions.

**Lecture 3**: Special values of Zeta functions I

**Abstract**: This and the next talk cover joint work with B. Morin. We introduce a special value conjecture for the Zeta function of a proper, regular arithmetic scheme which has three equivalent formulations: In the style of the analytic class number formula, via an integral fundamental line and via Weil-Arakelov cohomology complexes.

**Lecture 4**: Special values of Zeta functions II

**Abstract**: We discuss compatibility of our special value conjecture with the functional equation of the Zeta function (the analogous compatibility for the Tamagawa number conjecture is not known in general). We also discuss compatibility of our conjecture with the conjecture of Birch and Swinnerton-Dyer.

**Haruzo Hida**

**Title**: NON-VANISHING MODULO p OF VALUES OF A MODULAR FORM AT CM POINTS

**Abstract:** LINK

**Reading materials**: https://www.math.ucla.edu/~

**Antonio Lei**

**Title**: Euler systems and Beilinson--Flach elements

**Abstract**: The first goal of this series of talks is to give a general introduction to the theory of Euler systems and provide certain motivations of the subject. In particular, we will review the definition and basic properties of Euler systems. We will study cyclotomic units as a first example of Euler systems. The second goal is to describe the construction of Beilinson--Flach elements and explain how they are used to define Euler systems for Rankin--Selberg convolutions of two modular forms. The third goal is to discuss arithmetic applications derived from Beilinson--Flach Euler systems, including Iwasawa main conjectures in both ordinary and non-ordinary settings. If time permits, we will also discuss some recent developments in the subject.

**Yifeng Liu**

**Title**: Recent advances on Beilinson-Bloch-Kato conjecture

**Abstract**: In this mini series of lectures, we will introduce the recent advances on the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives coming from automorphic representations of unitary groups, obtained by Liu-Tian-Xiao-Zhang-Zhu. We will tour the main path of the proof, using U(4) \times U(5) as a guiding example. In particular, we will explain how different ingredients like geometry of integral modules of Shimura varieties and period integrals of automorphic forms are used in the proof.

**Arul Shankar**

**Lecture 1**: Average rank of elliptic curves: overview of the results

**Abstract**: In this talk, I will describe the Goldfeld and Katz--Sarnak conjectures on the average rank of all elliptic curves, and the Poonen--Rains conjectures on the distribution of Selmer groups of elliptic curves. I will provide some intuitive reasons on why these conjectures are true, and how the two conjectures are related. Finally, I will detail recent progress towards these conjectures. This is joint work with Manjul Bhargava.

**Lecture 2**: Parametrization of the 2-, 3-, 4-, and 5-Selmer groups of elliptic curves

**Abstract**: For n=2, 3, 4, and 5, we will see how elements in the n-Selmer groups of elliptic curves can be parametrized in terms of the rational group orbits for the action of certain reductive groups G_n on coregular representations V_n. We will also collect important properties of these representations that will be necessary for the sequel. This is joint work with Manjul Bhargava.

**Lecture 3**: Geometry-of-numbers techniques in arithmetic statistics

**Abstract**: Let V be a coregular representation of a reductive group G. We will impose a natural height function on V(R) and see how analytic techniques can be used to count G(Z)-orbits on V(Z) having bounded height. We will apply this method to the representations discussed in the previous lecture, and obtain an explicit bound on the average rank of elliptic curves. This is joint work with Manjul Bhargava.

**Lecture 4**: Computations of local volumes and mass formulas

**Abstract**: In previous lectures, we had expressed the average size of the n-Selmer groups of elliptic curves in terms of a product of local volumes of sets in V(R) and V(\Z_p). In this lecture, we will see how the action of the reductive group G allows us to evaluate these volumes in terms of a product of mass formulas. We will also obtain improved bounds on the average ranks of elliptic curves by proving equidistribution results on the root numbers of elliptic curves, and combining this with certain "parity results" of Dokchitser and Dokchitser. This is joint work with Manjul Bharava.

**DISCUSSION MEETING**

**Debargha Banerjee**

**Title**: Ribet's conjecture for Eisenstein maximal ideals of cube-free level

**Abstract**: By Ogg's conjecture (or Mazur's Theorem), the cuspidal subgroup coincides with rational torsion subgroup of the Jacobian variety of modular curves X_0(N) for primes $N$. There is a recent interest to generalize this conjecture for arbitrary $N$, by Ribet, Ohta, and Yoo. In this direction, Ribet conjectured that all the Eisenstein maximal ideals are ''cuspidal" for general N. Yoo proved this conjecture, under a mild hypothesis, provided that those ideals are rational. In this talk, we show that (under certain hypothesis), Ribet's conjecture is true for non-rational Eisenstein maximal ideals of cube-free level. In this case, we provide several examples of Eisenstein congruences. This is a joint work with Narasimha Kumar and Dipramit Majumdar.

**Eknath Ghate**

**Title**: Semi-stable representations as limits of crystalline representations.

**Abstract**: We construct an explicit sequence of crystalline representations converging to a given irreducible two-dimensional semi-stable representation of the Gaois group of Q_p. The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier and is connected to a classical formula due to Greenberg and Stevens expressing the L-invariant as a logarithmic derivative.

Our convergence result can be used to compute the reductions of any irreducible two-dimensional semi-stable representation in terms of the reductions of certain nearby crystalline representations of exceptional weight.

In particular, this provides an alternative approach to computing the reductions of irreducible two-dimensional semi-stable representations that circumvents the somewhat technical machinery of integral p-adic Hodge theory. For instance, using our zig-zag conjecture on the reductions of crystalline representations of exceptional weights, we recover completely the work of Breuil-Mezard and Guerberoff-Park on the reductions of irreducible semi-stable representations of weights at most p+1, at least on the inertia subgroup. As new cases of the zig-zag conjecture are proved, we further obtain some new information about the reductions for small odd weights.

Finally, we use the above ideas to explain away some apparent violations to local constancy in the weight of the reductions of crystalline representations of small weight that were noted in our earlier work and which provided the initial impetus for this work.

This is joint work with Anand Chitrao and Seidai Yasuda.

**Ming-Lun Hsieh**

**Title**: On p-adic L-functions for U(3)xU(2)

**Abstract**: I will report on a joint work in progress with Michael Harris and Shunsuke Yamana on the construction of the five-variable p-adic L-functions for U(3)xU(2) via the Ichino-Ikeda conjecture.

**Yukako Kezuka**

**Title**: Non-vanishing theorems for the Gross family of elliptic curves

**Abstract**: The arithmetic of elliptic curves with complex multiplication has attracted many mathematicians. Amongst these curves, Gross has introduced a particularly nice elliptic curve with complex multiplication by the field $\mathbb{Q}(\sqrt{−q})$, where $q$ is any prime congruent to $7$ modulo $8$. I will report on recent joint work with Yong-Xiong Li proving non-vanishing theorems for the central values of $L$-series of certain quadratic twists of the Gross elliptic curve. This completes the remaining half of the non-vanishing theorems proven by Coates and Li in which the primes $q$ were taken to be congruent to $7$ modulo $16$.

**Chan-Ho Kim**

**Title**: The structure of Selmer groups of elliptic curves

**Abstract**: We discuss applications of (a part of) the Iwasawa main conjecture to the non-triviality of Kato's Kolyvagin systems and the structure of Selmer groups of elliptic curves over the rationals without any rank restriction.

**Shinichi Kobayashi**

**Title**: Rubin’s conjecture on local units in the anticyclotomic tower at inert primes

**Abstract**: We introduce a conjecture of Rubin on the structure of local units in the anticyclotomic Z_p-extension of the unramified quadratic extension of Q_p. Then we explain our proof of Rubin's conjecture and some applications. This is a joint work with Ashay Burungale and Kazuto Ota.

**Shilin Lai**

**Title**: Weight interlacing and Iwasawa theory

**Abstract**: For Iwasawa theory for Rankin-Selberg motives arising from a product of two unitary groups, the (archimedean) Gan-Gross-Prasad conjecture imposes a rich structure of multiple main conjectures indexed by archimedean weight interlacing. We will explain some features and how known constructions fit into this framework.

**Zheng Liu**

**Title**: p-adic L-functions for GSp(4)\times GL(2)

**Abstract**: For a cuspidal automorphic representation \Pi of GSp(4) and a cuspidal automorphic representation \pi of GL(2), Furusawa's formula can be used to study the special values of the degree-eight p-adic L-function L(s,\Pi\times\pi). In this talk, I will explain a construction of the p-adic L-function for \Pi\times\pi by using Furusawa's formula and a family of Eisenstein series. The construction includes choosing local test sections at p and computing the corresponding local zeta integrals.

**Ritabrata Munshi**

**Title**: Bounds for L-functions

**Abstract**: Estimating the size of L-functions on the critical line is a central problem in the analytic theory of L-functions. Often non-trivial bounds for L-functions have interesting consequences. In the last ten years we saw a number of breakthroughs in this field. Yet some of the most crucial cases still remain unsettled. This talk will be a survey of the present situation in the subject.

**Tadashi Ochiai**

**Title**: p-adic Artin L-function over a CM-field

**Abstract**: We will discuss a p-adic analog of the Artin L-function over a CM field. Let K be a CM field of degree 2d. For an Artin representation of the absolute Galois group of K, we will construct a ``d+1"-variable p-adic L-function assuming certain technical conditions and the validity of the ``d+1"-variable Iwasawa main conjecture for intermediate fields of K and the field of the kernel of the Artin representation. This is joint work with Takashi Hara.

**Kazuto Ota**

**Title**: On Kato’s epsilon conjecture for anticyclotomic CM deformations at inert primes.

**Abstract**: For families of p-adic Galois representations, Kato formulated the existence of local epsilon isomorphisms, which interpolate the local epsilon factors at p. In this talk, we construct the conjectural isomorphism for the anticyclotomic deformation of a Lubin-Tate formal group of height two in an explicit and elementary way, where Rubin’s conjecture on local units plays an essential role. We also explain a result on Kato’s global epsilon conjecture for the anticyclotomic deformation of CM elliptic curves at inert primes. This talk is based on a joint work with Ashay Burungale, Shinichi Kobayashi, and Seidai Yasuda.

**Aprameyo Pal**

**Title**: p-adic Artin formalism for the triple product of modular forms

**Abstract**: Via Artin formalism, if a complex Galois representation $V$ decomposes as a direct sum of $V_1$ and $V_2$, then the associated complex $L$-function also decomposes as a product of $L(V_1, s)$ and $L(V_2, s)$. Unfortunately, this can fail when one considers $p$-adic $L$-functions. I would recall some known cases where this holds and discuss what one can say about the case of the triple product of modular forms.

**Bharathwaj Palvannan**

**Title**: An ergodic approach towards an equidistribution result of Ferrero–Washington

**Abstract**: An important ingredient in the Ferrero--Washington proof of the vanishing of cyclotomic \mu-invariant for Kubota--Leopoldt p-adic L-functions is an equidistribution result which they established using the Weyl criterion. In joint work with Jungwon Lee, we provide an alternative proof by adopting a dynamical approach. We study an ergodic skew-product map on \Z_p * [0,1], which is then suitably identified as a factor of the 2-sided Bernoulli shift on the alphabet space {0,1,2,…,p-1}.

**Kartik Prasanna**

**Title**: Modular forms of weight one, motivic cohomology and the Jacquet-Langlands correspondence

**Abstract**: This is a report on joint work (in progress) with Ichino. In a previous paper, we showed that the Jacquet-Langlands correspondence for Hilbert modular forms, all of whose weights are at least two, preserves rational Hodge structures, as predicted by the Tate conjecture. In this talk, I will discuss a related result in the case of weight one forms. Since weight one forms are not cohomological, the Tate conjecture does not apply and thus it is not at all obvious what the content of such a result should be. I will motivate and explain the statement, which is suggested by another recent development, namely the conjectural connection between motivic cohomology and the cohomology of locally symmetric spaces.

**A Raghuram**

**Title**: The special values of Rankin-Selberg L-functions over a totally imaginary field.

**Abstract**: Building on my previous work with Günter Harder, by studying rank-one Eisenstein cohomology for GL(N) over a totally imaginary base field F, new rationality results are obtained for the ratios of critical values for Rankin-Selberg L-functions for GL(n) x GL(n') over F. Whether F contains a CM subfield or not has a delicate bearing on the structure of cohomology and the shape of the rationality results.

**C S Rajan**

**Title**: Locally potentially equivalent Galois representations

**Abstract**: We will discuss two results on locally potentially equivalent Galois representations of a number field (carried out respectively with Vijay Patankar and Plawan Das). One is about a refinement of strong multiplicity one type property for such representations; the other is about an extension of Faltings' finiteness criterion for equivalence of Galois representations. We will also discuss a character theory for potential equivalence and some applications.

**Sujatha Ramdorai**

**Title**: Iwasawa invariants for elliptic curves in a family.

**Abstract**: Let E be an elliptic curve defined over the rationals with good ordinary reduction at a (fixed) odd prime p. Let K be a totally imaginary number field and let $K_infty$ be a ${\mathbb Z}_p^2$-extension of K containing the cyclotomic ${\mathbb Z}_p$-extension of $K$. Using the knowledge of Iwasawa invariants over intermediary ${\mathbb Z}_p$-extensions of $K$, we obtain results on the Iwasawa module structure of the Selmer group of E as a $\Lambda(H)$_module as $H$ varies over subextensions ${\mathcal L} of $K_infty$ with $Gal(K_{infty}/L)={\mathbb Z}_p.$

**Sudhanshu Shekhar**

**Title**: Iwasawa theory for rankin-selberg convolution.

**Abstract**: Fix an odd prime $p$. Let f be a p-ordinary newform of weight k and h be a normalized cuspidal p-ordinary Hecke eigenform of weight l < k. In this talk, we will discuss the $p$-adic L function and the structure of the p∞-Greenberg Selmer group of the Rankin-Selberg convolution of f and h. In the special cases when the residual representation of $h$ at p is reducible, we also discuss certain congruences between the associated characteristic ideal and the $p$-adic Rankin-Selberg L-function. This is a Joint work with Somnath Jha and RaviTeja Vangala.

**Pratiksha Shingavekar**

**Title**: 3 Selmer group, ideal class group and cube sum problem

**Abstract**: Let E be an elliptic curve over \mathbb{Q} with a rational 3-isogeny \phi. We discuss the bounds on the \phi-Selmer group of E in terms of ideal class groups of certain number fields. We also give some applications of these bounds towards Sylvester's conjecture on the cube sum problem.

**Ari Shnidman**

**Title**: Sums of two cubes

**Abstract**: I'll discuss recent joint work with Alpöge and Bhargava proving that a positive proportion (around one tenth) of integers n can be written as a sum of two rational numbers and a positive proportion (at least one sixth) cannot. More generally, we prove that in any cubic twist family of elliptic curves, at least one sixth of twists have rank 0 and at least one sixth of twists with good reduction at 2 have rank 1. The proof involves viewing 2-Selmer elements in such families as orbits of pairs of binary cubic forms living in an invariant quadric. We then combine Bhargava's averaging method with the circle method to compute the average size of the 2-Selmer group. Finally, we analyze root numbers and invoke p-parity results (Dokchitser-Dokchitser and Nekovář) as well as a recent p-converse theorem of Burungale-Skinner.

**Christopher Skinner**

**Title**: Elliptic curves of ranks zero and one.

**Abstract**: I will review some results on the refined arithmetic of elliptic curves, with an emphasis on recent results for curves of ranks zero and one, and especially the ideas underlying their proofs.

**Matteo Tamiozzo**

**Title**: Special values of L-functions and Ihara’s lemma for quaternionic Shimura varieties

**Abstract**: I will explain a proof of one inequality in the Bloch-Kato special value formula in analytic rank at most one for (certain) Hilbert modular forms of parallel weight two, highlighting the role played by Ihara’s lemma for Shimura curves. I will then discuss an extension of Ihara’s lemma to arbitrary quaternionic Shimura varieties. The latter result is joint work in progress with Ana Caraiani.

**Otmar Venjakob**

**Title**: Serre duality on character varieties and explicit reciprocity Laws

**Abstract**: Classically explicit reciprocity laws or formulas usually mean an explicit computation of Hilbert symbols or (local) cup products using e.g. differential forms, (Coleman) power series etc. In the same spirit Perrin-Riou's reciprocity law gives an explicit calculation of the Iwasawa cohomology pairing in terms of big exponential and regulator maps for crystalline representations of $G_{\mathbb{Q}_p}$; more precisely, the latter maps are adjoint to each other when also involving the crystalline duality paring after base change to the distribution algebra. In this talk I shall try to explain how - in the Lubin-Tate setting generalising the cyclotomic situation - such reciprocity laws are a consequence of Serre duality on certain rigid analytic character varieties combined with comparison isomorphisms in $p$-adic Hodge theory expressing $D_{cris}$ in terms of $(\varphi,\Gamma)$-modules.

**Xin Wan**

**Title**: Iwasawa main conjecture for universal families

**Abstract**: Let p be an odd prime. We formulate and prove the Iwasawa main conjecture for the universal family in the p-adic Langlands program for elliptic modular forms. As a consequence we prove Kato's Iwasawa main conjecture for elliptic modular forms with no assumption on the ramification type at p, and prove the rank 0 BSD formula for elliptic curves at additive primes. This is joint work with Olivier Fouquet.

**Gergely Zábrádi **

**Title**: Multivariate (φ,Γ)-modules

**Abstract**: Classical (φ,Γ)-modules of Fontaine classify p-adic local Galois representations. Through certain refinements they have become a very powerful tool in p-adic Hodge theory and related topics. Originally motivated by the p-adic Langlands programme, the goal of this talk is to explain how to relate p-adic representations of products of local Galois groups to multivariate (φ,Γ)-modules using a version of Drinfeld's lemma for perfectoid spaces. The overconvergent refinement hopefully opens up the way to possible applications in Iwasawa theory. Partly joint work with A. Carter and K. Kedlaya, resp. with A. Pal.

**Wei Zhang**

**Title**: p-adic Heights of the arithmetic diagonal cycles

**Abstract**: This is a work in progress joint with Daniel Disegni. We formulate a p-adic analogue of the Arithmetic Gan–Gross–Prasad conjecture for unitary groups, relating the p-adic height pairing of the arithmetic diagonal cycles to the first central derivative (along the cyclotomic direction) of a p-adic Rankin–Selberg L-function associated to cohomological cuspidal automorphic representations on unitary groups. When all the p-adic places are good ordinary and are split in the quadratic extension, we are able to prove the conjecture, at least when the ramifications are mild at inert primes. We deduce some applications to Fontaine--Perrin-Riou's p-adic version of Bloch-Kato conjecture.