Ramanujan-style congruences for prime level

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.