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Nonanalytic automorphic integrals on the Hecke groups

Paul C. Pasles (1999)

Acta Arithmetica

1. Introduction. Since its genesis over a century ago in work of Jacobi, Riemann, Poincar ́e and Klein [Ja29, Ri53, Le64], the theory of automorphic forms has burgeoned from a branch of analytic number theory into an industry all its own. Natural extensions of the theory are to integrals [Ei57, Kn94a, KS96, Sh94], thereby encompassing Hurwitz’s prototype, the analytic weight 2 Eisenstein series [Hu81], and to nonanalytic forms [He59, Ma64, Sel56, ER74, Fr85]. A generalization in both directions...

On arbitrary products of eigenforms

Arvind Kumar, Jaban Meher (2016)

Acta Arithmetica

We characterize all the cases in which products of arbitrary numbers of nearly holomorphic eigenforms and products of arbitrary numbers of quasimodular eigenforms for the full modular group SL₂(ℤ) are again eigenforms.

On the generation of the coefficient field of a newform by a single Hecke eigenvalue

Koopa Tak-Lun Koo, William Stein, Gabor Wiese (2008)

Journal de Théorie des Nombres de Bordeaux

Let f be a non-CM newform of weight k ≥ 2 . Let L be a subfield of the coefficient field of  f . We completely settle the question of the density of the set of primes p such that the p -th coefficient of  f generates the field  L . This density is determined by the inner twists of  f . As a particular case, we obtain that in the absence of nontrivial inner twists, the density is  1 for L equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions...

On the slopes of the  U 5 operator acting on overconvergent modular forms

L. J. P Kilford (2008)

Journal de Théorie des Nombres de Bordeaux

We show that the slopes of the  U 5 operator acting on 5-adic overconvergent modular forms of weight  k with primitive Dirichlet character  χ of conductor 25 are given by either 1 4 · 8 i 5 : i ∈ ℕ or 1 4 · 8 i + 4 5 : i ∈ ℕ , depending on  k and  χ .We also prove that the space of classical cusp forms of weight  k and character  χ has a basis of eigenforms for the Hecke operators  T p and  U 5 which is defined over  Q 5 ( 5 4 , 3 ) .

Opérateurs de Hecke pour Γ 0 ( N ) et fractions continues

Loïc Merel (1991)

Annales de l'institut Fourier

Nous rappelons que Manin décrit l’homologie singulière relative aux pointes de la courbe modulaire X 0 ( N ) comme un quotient du groupe Z ( P 1 ( Z / N Z ) ) . En s’appuyant sur des techniques de fractions continues, nous donnons une expression indépendante de N d’un relèvement de l’action des opérateurs de Hecke de H 1 ( X 0 ( N ) , p t e s , Z ) sur Z ( P 1 ( Z / N Z ) ) .

Oscillations of Hecke eigenvalues at shifted primes.

Liangyi Zhao (2006)

Revista Matemática Iberoamericana

In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is √n at prime arguments.

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