Congruences for modular forms of weights two and four

Abstract

We prove a conjecture of Calegari and Stein regarding mod p congruences between modular forms of weight four and the derivatives of modular forms of weight two.

Congruences for newforms and the index of the Hecke algebra

“…Ahlgren and Barcau [2] have proved the following, which is Conjecture 4 of [6] (this result has been generalized to higher level in [4]). Let p be the maximal ideal in Z p .…”

On mod $\mathfrak{p}$ congruences for Drinfeld modular forms of level $\mathfrak{p}\mathfrak{m}$

“…[CS04, Conjecture 4]). In [AB07], Ahlgren and Barcau settled this conjecture affirmatively. More precisely, they prove: Theorem 1.1.…”

“…In the next section, we shall state the above theorems for p-new forms which are natural generalizations of the results of [AB07] and [BP11].…”

“…Finally we note that, the theorems in [AB07], [BP11] were proved only for smaller weights. We are not aware of existence of those results for other weights in the literature.…”

“…One might wonder if one can formulate conjectures, which are similar to the conjectures of Calegari and Stein, for Drinfeld modular forms. Before formulating all those conjectures and the inter-relations among them, we wish to understand if the results of [AB07] and [BP11] continue hold for Drinfeld modular forms or not. If true, this gives us a hope to formulate the other conjectures and check the validity of those for Drinfeld modular forms.…”