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On computing quaternion quotient graphs for function fields

Gebhard Böckle, Ralf Butenuth (2012)

Journal de Théorie des Nombres de Bordeaux

Let Λ be a maximal 𝔽 q [ T ] -order in a division quaternion algebra over 𝔽 q ( T ) which is split at the place ∞ . The present article gives an algorithm to compute a fundamental domain for the action of the group of units Λ * on the Bruhat-Tits tree 𝒯 associated to PGL 2 ( 𝔽 q ( ( 1 / T ) ) ) . This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group Λ * in terms of generators and relations. Moreover we determine an upper bound...

On critical values of twisted Artin L -functions

Peng-Jie Wong (2017)

Czechoslovak Mathematical Journal

We give a simple proof that critical values of any Artin L -function attached to a representation ρ with character χ ρ are stable under twisting by a totally even character χ , up to the dim ρ -th power of the Gauss sum related to χ and an element in the field generated by the values of χ ρ and χ over ℚ . This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.

On Dirichlet Series and Petersson Products for Siegel Modular Forms

Siegfried Böcherer, Francesco Ludovico Chiera (2008)

Annales de l’institut Fourier

We prove that the Dirichlet series of Rankin–Selberg type associated with any pair of (not necessarily cuspidal) Siegel modular forms of degree n and weight k ≥ n / 2 has meromorphic continuation to ℂ . Moreover, we show that the Petersson product of any pair of square–integrable modular forms of weight k ≥ n / 2 may be expressed in terms of the residue at s = k of the associated Dirichlet series.

On effective determination of symmetric-square lifts

Qingfeng Sun (2014)

Open Mathematics

Let F be the symmetric-square lift with Laplace eigenvalue λ F (Δ) = 1+4µ2. Suppose that |µ| ≤ Λ. We show that F is uniquely determined by the central values of Rankin-Selberg L-functions L(s, F ⋇ h), where h runs over the set of holomorphic Hecke eigen cusp forms of weight κ ≡ 0 (mod 4) with κ≍ϱ+ɛ, t9 = max {4(1+4θ)/(1−18θ), 8(2−9θ)/3(1−18θ)} for any 0 ≤ θ < 1/18 and any ∈ > 0. Here θ is the exponent towards the Ramanujan conjecture for GL2 Maass forms.

On elliptic Galois representations and genus-zero modular units

Julio Fernández, Joan-C. Lario (2007)

Journal de Théorie des Nombres de Bordeaux

Given an odd prime   p   and a representation ϱ   of the absolute Galois group of a number field k onto PGL 2 ( 𝔽 p ) with cyclotomic determinant, the moduli space of elliptic curves defined over k with p -torsion giving rise to ϱ consists of two twists of the modular curve X ( p ) . We make here explicit the only genus-zero cases p = 3 and p = 5 , which are also the only symmetric cases: PGL 2 ( 𝔽 p ) ≃ 𝒮 n for n = 4 or n = 5 , respectively. This is done by studying the corresponding twisted Galois actions on the function field of the curve, for which...

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