number theory day

We are organizing a day of Number Theory talks in IISER Pune on February 17, 2020. The purpose of this symposium is to bring together members of the NT community for a day of conversation about their research.

Speakers

• Somnath Jha (IIT Kanpur)
• Matteo Longo (University of Padova)
• Marc-Hubert Nicole (Institut mathématique de Marseille)
• Vivek Rai (IISER Pune)

Location

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

Schedule

• 9:30-10:30 Somnath Jha
• 10:30-11:00 Coffee/Tea break
• 11:00-12:00 Matteo Longo
• 12:00-14:00 Lunch break
• 14:00-15:00 Vivek Rai
• 15:00-15:30 Coffee/Tea break
• 15:30-16:30 Marc-Hubert Nicole
• 19:00 Dinner (email mathoffice at iiserpune.ac.in if you plan to join the speakers for dinner)

Speaker: Matteo Longo

Title: The Kolyvagin Conjecture for modular forms

Abstract: Kolyvagin conjecture for elliptic curves was formulated by V. Kolyvagin in the nineties, and predicts the non-triviality of certain cohomology classes constructed from Heegner points. Consequences of this conjecture include the p-part of the Birch and Swinnerton-Dyer conjecture, parity results, and a precise description of the structure of the Tate-Shafarevich group of the elliptic curve. The original conjecture is now a theorem by the work of many people, including W. Zhang, C. Skinner and R. Venerucci. We investigate an analogue of Kolyvagin Conjecture for higher weight modular forms in which Heegner points are replaced by Heegner cycles on Kuga-Sato varieties. As a consequence, we obtain some results on the structure of the Tate-Shafarevich group attached to the modular form, and a p-part of a Bloch-Kato conjecture in analytic rank 1. This is a work in collaboration with Stefano Vigni.

Speaker: Vivek Rai

Title: Reductions of Galois representations at exceptional weights

Abstract: Let $p$ be an odd prime and $E$ be a finite extension of $\mathbb{Q}_p$. Given an integer $k \geq 2$ and $a_p \in E$ with $v(a_p)> 0$, there exists an irreducible two-dimensional crystalline representation $V_{k,a_p}$ of $\mathrm{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$ over $E$ of weight $k$ and slope $v(a_p)$. In this talk we would describe the behaviour of the semisimplification of the mod $p$ reduction of $V_{k,a_p}$ when $v(a_p) = \frac{3}{2}$ and $k\equiv 5\text{ mod }(p-1)$. We will show that the reduction depends on more parameters other than the congruence class of $k\text{ mod }(p-1)$. Such a behaviour is predicted by a Zig-zag conjecture made by Ghate for exceptional weights $k$ and half-integral slopes $v(a_p)$. Our result shows that zig-zag conjecture is true when $v(a_p) = \frac{3}{2}$. This is a joint work with Eknath Ghate.

Speaker: Marc-Hubert Nicole

Title: $\pi$-adic Drinfeld modular forms

Abstract: The theory of families of classical modular forms was developed by Hida, Coleman, Mazur et al., and includes p-adic modular forms e.g. overconvergent modular forms.

In this talk, we shall explain the key components of an analogous theory in the realm of function fields that is, for so-called Drinfeld modular forms.

In particular, there exists for GL(n) a theory of Hida families of ordinary modular forms, as well as a continuous analogue of the theory of Coleman families in finite slope. Further, a classicality theorem states that an overconvergent Drinfeld modular form with sufficient « small » slope is automatically classical.

At the end of the talk, we will sketch how Scholze’s theory of perfectoid Shimura varieties extends to Drinfeld modular varieties, at least for GL(2).

Joint work with G. Rosso (Montréal)