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Let p‚ 5 be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of difierent weights. We show how to produce such… (More)

We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is… (More)

In this paper, we study the structure of the local components of the (shallow, i.e. without $U_{p}$
) Hecke algebras acting on the space of modular forms modulo $p$
of level $1$
, and relate them… (More)

We prove the existence in many cases of minimally ramified p-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod p representations of the absolute Galois group of Q. It is… (More)

We formulate for function fields an analog of Serre's conjecture on the modularity of 2-dimensional irreducible mod l representations of the absolute Galois group of Q: our analog is not restricted… (More)

We prove the level one case of Serre’s conjecture. Namely, we prove that any continuous, odd, irreducible representation ρ̄ : Gal(Q̄/Q) → GL2(Fp) which is unramified outside p arises from a cuspidal… (More)

DFG grants FG 1920 and SPP 1489
ERC Grant 290766 (AAMOT)
NSF Grant DMS-1404769.
NSF Grant DMS-1161671
Humboldt Research Award
Clay Mathematics Institute
ERC Grant no. 714405 (GMLP)

Let F be a totally real extension of Q and ρF : Gal (F/F ) −→ GL2(k) be an absolutely irreducible, continuous, and odd representation, with k a finite field of characteristic p > 2, and where GF :=… (More)