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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 or 1 4 · 8 i + 4 5 : i , 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 ) .

On the spinor zeta functions problem: higher power moments of the Riesz mean

Haiyan Wang (2013)

Acta Arithmetica

Let F be a Siegel cusp form of integral weight k on the Siegel modular group Sp₂(ℤ) of genus 2. The coefficients of the spinor zeta function Z F ( s ) are denoted by cₙ. Let D ρ ( x ; Z F ) be the Riesz mean of cₙ. Kohnen and Wang obtained the truncated Voronoï-type formula for D ρ ( x ; Z F ) under the Ramanujan-Petersson conjecture. In this paper, we study the higher power moments of D ρ ( x ; Z F ) , and then derive an asymptotic formula for the hth (h=3,4,5) power moments of D ( x ; Z F ) by using Ivić’s large value arguments and other techniques.

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