In the first part, we revisit Drinfeld modular curves associated to GL(2) from perfectoid point of view, and show how recover (a perfectized) part theory overconvergent π-adic forms. second review open problems for families forms GL(n).

Abstract A (folklore?) conjecture states that no holomorphic modular form $F(\tau )=\sum _{n=1}^{\infty } a_nq^n\in q\mathbb Z[[q]]$ exists, where $q=e^{2\pi i\tau }$ , such its anti-derivative $\sum a_nq^n/n$ has integral coefficients in the q -expansion. recent observation of Broadhurst and Zudilin, rigorously accomplished by Li Neururer, led to examples meromorphic forms possessing integrali...

We discuss practical and some theoretical aspects of computing a database classical modular forms in the L-functions Modular Forms Database (LMFDB).

In his letter [Ser96], J.-P. Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions A×B/B × → F̄p, where B is the endomorphism algebra of a supersingular elliptic curve. After giving a detailed exposition of Serre’s result, we prove that the systems of Hecke eigenvalues given by Siegel modular forms (mod p) o...