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Classical and overconvergent modular forms of higher level

Robert F. Coleman — 1997

Journal de théorie des nombres de Bordeaux

We define the notion overconvergent modular forms on Γ 1 ( N p n ) where p is a prime, N and n are positive integers and N is prime to p . We show that an overconvergent eigenform on Γ 1 ( N p n ) of weight k whose U p -eigenvalue has valuation strictly less than k - 1 is classical.

Duality for the de Rham cohomology of an abelian scheme

Robert F. Coleman — 1998

Annales de l'institut Fourier

In this paper the equality is established of three different pairings between the first de Rham cohomology group of an abelian scheme over a base flat over and that of its dual. These pairings have appeared and been used either explicitly or implicitly in the literature. In the last section we deduce a generalization to arbitrary characteristic of Serre’s formula for the Poincaré pairing on the first de Rham cohomology group of a curve over a field of characteristic zero.

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