Stark-Heegner points and generalised Kato classes (Henri Darmon)
We describe some relations between Stark-Heegner points, which are conjecturally defined over ring class fields of real quadratic fields, and generalised Kato classes, which are canonical global Selmer classes defined over the same ring class fields. This relationship lends some theoretical support for the conjectured algebraicity properties of Stark-Heegner points. This is a report on ongoing joint work with Victor Rotger.
On a universal Torelli theorem for elliptic surfaces (CS Rajan)
Given two elliptic surfaces over a base curve C, we show that any compatible family of effective isometries of the N´eron-Severi lattices of the base changed elliptic surfaces for all finite maps B → C arises from an isomorphism of the elliptic surfaces upto a finite base change. Without the effectivity hypothesis, we can conclude that the two elliptic fibrations are isomorphic.
Root numbers and parity of local Iwasawa invariants (Suman Ahmed)
Given two elliptic curves E1 and E2 defined over the field of rational numbers, Q, with good reduction at an odd prime p and equivalent mod p Galois representation, we compare the p-Selmer rank, global and local root numbers of E1 and E2 over number fields.
Torsion points of the Jacobian of modular curves X0(p2 ) and non-holomorphic Eisenstein series (Debargha Banerjee)
In the BSD conjecture, we relate the rank of elliptic curves to the analytic data, namely the special values of the associated L-functions. In a similar theme, we study the torsion points of the families of abelian varieties J0(p2) using analytic data, namely suitable non-holomorphic Eisenstein series. The constant terms of these Eisenstein series encode information about the special values of L-functions. By using an explicit description of the special fibers of the modular curves X0(p2) due to Edixhoven and Manin-Drinfeld theorem, we give an effective bound to the number of torsion points of the abelian varieties J0(p2). The above mentioned bound is an effective version of Bogomolov’s conjecture for X0(p2). This is a joint work with Chitrabhanu Chowdhuri and Diganta Borah.
Rigidity of p-adic local systems and Abapplications to Shimura varieties (Rouchan Liu)
I'll talk about recent progress on de Rham rigidity of p-adic local systems as well as its applications to Shimura varieties.
Towards a p-adic Asai L-function (Eknath Ghate)
In his thesis, the speaker proved a rationality result for the Asai L-function attached to a Bianchi cusp form in the style of Deligne's general conjecture on critical values of L-functions. This was soon after generalized from the case of imaginary quadratic fields to the case of CM fields.
Since then he has wondered if it possible to p-adically interpolate these special values. This talk will describe some preliminary work with Balasubramanyam and Ravitheja on this question and some of the obstacles remaining.
Comparing the corank of fine Selmer group and Selmer group of elliptic curves (Sudhanshu Shekhar)
Let p be an odd prime, K∞ be a pro-p, p-adic Lie extension of K = Q(µp) of dimension two containing the cyclotomic Zp-extension Kcyc of K and H be the Galois group of K∞/Kcyc. Let Λ(H) be the Iwasawa algebra over H. Given an elliptic curve E defined over Q with good and supersingular reduction at p, we compare the Λ(H)-corank of the fine Selmer group of E over K∞ with the Iwasawa λ-invariant of the ±-Selmer group of E over Kcyc. Using this, we find examples of elliptic curves defined over Q with good and supersingular reduction at p satisfying pseudo nullity conjecture over K∞.
p-adic Asai transfer (Baskar Balasubramanyam)
Let F be a real quadratic extension. The Asai transfer is an instance of Langlands principle of functoriality that takes automorphic representations over GL2/F to automorphic representations over GL4/Q. In this talk, I will describe the construction of a p-adic version of this Langlands transfer. We use the universal eigenvarieties and comparison theorem developed by Hansen to obtain our construction. This is joint work with Dipramit Majumdar.
Horizontal variation of the arithmetic of elliptic curves (Ashay Burungale)
Let E be an elliptic curve over the rationals. Let K be an imaginary quadratic field and H_K the corresponding Hilbert class fields. We discuss recent results on the arithmetic of E over H_K as K varies (joint with H. Hida and Y. Tian).
A twisting result in non-commutative Iwasawa theory (Somnath Jha)
We will discuss a well-known twisting lemma in classical Iwasawa Theory. We will then discuss a non-commutative generalization of this twisting lemma. This talk is based on a joint work with T. Ochiai and G. Zabradi.
On the Fourier coefficients of a Cohen-Eisenstein series (Srilakshmi Krishnamoorthy)
In the first part of this talk, we present a formula for the coefficients of a weight 3/2 Cohen-Eisenstein series of squarefree level N. This formula generalizes a result of Gross and it proves in particular a conjecture of Quattrini. In the second part, we explain the proof of this formula and as an application, we discuss the distribution of the order of the Tate-Shafarevich groups of quadratic twists of elliptic curves.
On the gaps between non-zero Fourier coefficients of cusp forms of higher weight and level (Narasimha Kumar)
We show that if a modular cuspidal eigenform f of weight 2k > 2 and level N > 1 is 2-adically close to an elliptic curve E/Q, which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval (X, X + cX1/4) for all X>>0 and for some c > 0. We use this fact to produce non-CM cuspidal eigenforms and infinitely many CM cuspidal eigenforms, both of which are of level N > 1 and weight k > 2.
Simultaneous non-vanishing of L-values (Soumya Das)
We would discuss the question of simultaneous non-vanishing of central critical values of L-functions corresponding to modular forms, and in particular talk about one result on this theme, which is a joint work with R. Khan.
On the 2-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication (Zhibin Liang)
Given an elliptic curve E over Q with complex multiplication having good reduction at 2, we investigate the 2-adic valuation of the algebraic part of the L-value at 1 for a family of quadratic twists. In particular, we prove a lower bound for this valuation in terms of the Tamagawa number in a form predicted by the conjecture of Birch and Swinnerton-Dyer.
K-groups and Global Fields (Haiyan Zhou)
In this talk, we will introduce some results of K-groups of global fields and some relations between K-groups of global fields and some L-functions.
two variables p-adic L function (Shanwen Wang)
In this talk, we will present a construction of two variables p-adic L function using the universal Kato's Euler system.