number theory day

We are organizing a day of Number Theory talks in IISER Pune on November 21, 2019. The purpose of this symposium is to bring together members of the NT community for a day of conversation about their research.

Speakers

• Shaunak Deo (TIFR)
• Mladen Dimitrov (University of Lille)
• Rohit Joshi (Bhaskaracharya Pratishthana)
• A. Raghuram (IISER Pune)
• Kaneenika Sinha (IISER Pune)

Location

Madhava Hall, Main Building, IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune.

Schedule

10:15 - 11:15: Shaunak Deo

11:15 - 11:45: Coffee break

11:45 - 12:45: Kaneenika Sinha

12:45 - 2:00: Lunch

2:00 - 3:00: Rohit Joshi

3:15 - 4:15: Mladen Dimitrov

4:15 - 4:45: Tea break

4:45 - 5:45: A. Raghuram

7:00 onwards: Dinner

Titles and Abstracts

Speaker: Shaunak Deo

Title: Hilbert modular eigenvariety at exotic and CM classical points of parallel weight one

Abstract: We sketch our recent results about the geometry of the p-adic eigenvariety constructed by Andreatta-Iovita-Pilloni, which interpolates Hilbert modular eigenforms over a totally real field F, at classical, regular points of parallel weight one which either are CM or have exotic projective image. To prove these results, we assume the p-adic Schanuel conjecture in most of the cases. The key ingredient in our proof is the calculation of the dimension of the tangent spaces of some Galois deformation problems. This talk is based on joint work with A. Betina and F. Fite.

Speaker: A. Raghuram

Title: Special values of L-functions for GL(n) over a CM field.

Abstract: This talk will be an announcement of a recent rationality result on the ratios of special values of L-functions for GL(n) over a CM field. The proof uses (1) my work with Harder on GL(2n) over a totally real field, (2) period relations such as my results with Shahidi on behaviour of periods for GL(n) on twisting by characters, and (3) the results of Arthur and Clozel on automorphic induction.

Speaker: Rohit Joshi

Title: Spinorial representations of GL(2,q)

Abstract: A representation of a group G is orthogonal if it preserves a non-degenerate symmetric bilinear form. An orthogonal representation is called spinorial if it lifts to the double cover of the orthogonal group which is the spin group. We will discuss the spinorial representations of GL(2,q).

Speaker: Kaneenika Sinha.

Title: Central limit theorems in arithmetic geometry

Abstract: A classical theorem of Erdos and Kac states that for a large random positive integer n, the number of distinct prime factors of n is normally distributed with mean and variance equal to log log n. This theorem can be interpreted as a central limit theorem for the average of a sum of independent and identically distributed random variables. In turn, this leads to similar results for several types of arithmetic functions. In this talk, we describe central limit theorems for the error term in the Sato-Tate law for families of modular forms as well as elliptic curves. We also indicate how these theorems can be understood in the classical framework delineated by the work of Erdos and Kac. This is a report on joint work with Stephan Baier and Neha Prabhu.

Speaker: Mladen Dimitrov.

Title: Uniform irreducibility of Galois action on the l-primary part of Abelian 3-folds of Picard type

Abstract: Half a century ago Manin proved a uniform version of Serre's celebrated result on the openness of the Galois image in the automorphisms of the l-adic Tate module of any non-CM elliptic curve over a given number field. Recently in a series of papers Cadoret and Tamagawa established a definitive result regarding the uniform boundedness of the l-primary torsion for 1-dimensional abelian families. In a collaboration with D. Ramakrishnan we provide first evidence in higher dimension, in the case of abelian families parametrized by Picard modular surfaces over an imaginary quadratic field M. Namely, we establish a uniform irreducibility of Galois acting on the l-primary part of principally polarized Abelian 3-folds with multiplication by M, but without CM factors. In this talk we will present the strategy of the proof and emphasize some of its representation theoretic aspects.