# Ramanujan-style congruences for prime level

Posted in
Speaker:
Moni Kumari
Affiliation:
Bar-Ilan University
Date:
Wed, 02/02/2022 - 14:30 - 15:30
Parent event:
Number theory lunch seminar
For zoom details please contact Pieter Moree (moree@mpim-bonn.mpg.de)

Ramanujan in 1916 proved the following notable congruence
$$\tau(n)\equiv \sigma_{11}(n) \pmod{691}, \forall~ n\ge 1$$
between the two important arithmetic functions $\tau(n)$ and
$\sigma_{11}(n)$. In other words, this says that there is a congruence
between the cuspidal Hekce eigenform $\Delta(z)$ and the non-cuspidal
eigenform $E_{12}(z)$ modulo the prime $691$. Existence of such congruences
opened the door for many modern developments in the theory of modular forms.

There are several well-known ways to prove, interpret, and generalize
Ramanujan's congruence. For newforms of prime level, some partial results
about the existence of such congruences are known. Recently, using the
theory of period polynomials, Gaba-Popa (under some technical assumptions)
extended these results by determining also the Atkin-Lehner eigenvalue of
the newform involved. In this talk, we refine the result of Gaba-Popa under
a mild assumption by using completely different ideas. More precisely, we
establish congruences modulo certain primes between a cuspidal newform and
an Eisenstein series of weight k and prime level. The main ingredients to
establish our result are some classical theorems from the theory of Galois
representations attached to newforms. As an application, we derive a lower
bound for the largest degree of the coefficients field among Hecke
eigenforms.
This is joint work with A. Kumar, P. Moree and S. K. Singh.

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