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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.

Coefficient bounds for level 2 cusp forms

Paul Jenkins, Kyle Pratt (2015)

Acta Arithmetica

We give explicit upper bounds for the coefficients of arbitrary weight k, level 2 cusp forms, making Deligne’s well-known O ( n ( k - 1 ) / 2 + ϵ ) bound precise. We also derive asymptotic formulas and explicit upper bounds for the coefficients of certain level 2 modular functions.

Cohen-Kuznetsov liftings of quasimodular forms

Min Ho Lee (2015)

Acta Arithmetica

Jacobi-like forms for a discrete subgroup Γ of SL(2,ℝ) are formal power series which generalize Jacobi forms, and they correspond to certain sequences of modular forms for Γ. Given a modular form f, a Jacobi-like form can be constructed by using constant multiples of derivatives of f as coefficients, which is known as the Cohen-Kuznetsov lifting of f. We extend Cohen-Kuznetsov liftings to quasimodular forms by determining an explicit formula for a Jacobi-like form associated to a quasimodular form....

Coherent sheaves with parabolic structure and construction of Hecke eigensheaves for some ramified local systems

Jochen Heinloth (2004)

Annales de l'Institut Fourier

The aim of these notes is to generalize Laumon’s construction [20] of automorphic sheaves corresponding to local systems on a smooth, projective curve C to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article “ On the geometric Langlands conjecture” by Frenkel, Gaitsgory and Vilonen [11]...

Cohomology of Drinfeld symmetric spaces and Harmonic cochains

Yacine Aït Amrane (2006)

Annales de l’institut Fourier

Let K be a non-archimedean local field. This paper gives an explicit isomorphism between the dual of the special representation of G L n + 1 ( K ) and the space of harmonic cochains defined on the Bruhat-Tits building of G L n + 1 ( K ) , in the sense of E. de Shalit [11]. We deduce, applying the results of a paper of P. Schneider and U. Stuhler [9], that there exists a G L n + 1 ( K ) -equivariant isomorphism between the cohomology group of the Drinfeld symmetric space and the space of harmonic cochains.

Cohomology of the boundary of Siegel modular varieties of degree two, with applications

J. William Hoffman, Steven H. Weintraub (2003)

Fundamenta Mathematicae

Let 𝓐₂(n) = Γ₂(n)∖𝔖₂ be the quotient of Siegel's space of degree 2 by the principal congruence subgroup of level n in Sp(4,ℤ). This is the moduli space of principally polarized abelian surfaces with a level n structure. Let 𝓐₂(n)* denote the Igusa compactification of this space, and ∂𝓐₂(n)* = 𝓐₂(n)* - 𝓐₂(n) its "boundary". This is a divisor with normal crossings. The main result of this paper is the determination of H(∂𝓐₂(n)*) as a module over the finite group Γ₂(1)/Γ₂(n). As an application...

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