On the modularity of the GL2-twisted spinor L-function

Abstract

In modern number theory there are famous theorems on the modularity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke's converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat's Last Theorem to name a few. In this article in the spirit of the Langlands philosophy we raise the question on the modularity of the GL 2 -twisted spinor L-function Z G⊗h (s) related to automorphic forms G, h on the symplectic group GSp 2 and GL 2 . This leads to several promising results and finally culminates into a precise very general conjecture. This gives new insights into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan's work on the modularity of the Rankin-Selberg L-series.

Towards Functoriality Of Spinor L-functions

“…Let the corresponding cuspidal Siegel modular forms of degree 1, 2, 3 have weights k 1 , k 2 , k 3 ∈ N. We prove that only the triple k − 2, k, k can occur. This conclusion includes Miyawaki's conjecture [32] of type II [23]. We also give a motivic interpretation of these lifts and deduce the same result.…”

“…The existence of such a map had been expected for many years, but several difficult auxiliary results were needed. Since we are in a parallel situation, we mention some essential auxiliary results still needed in order to prove the Main Conjecture in [23]. We recall briefly some of the main ingredients in the proof.…”

Applications of Arthur's multiplicity formula to Siegel modular forms

“…In the classical theory, there are several lifting conjectures. We recall the following conjectures predicted by Miyawaki [14], Heim [7], and Ibukiyama [8].…”

“…Miyawaki's conjecture of Type II: For any Hecke eigenforms f ∈ S 2k−2 (SL 2 (Z)) and g ∈ S k−2 (SL 2 (Z)), there should exist a Hecke eigenform F f,g ∈ S k (Sp 3 (Z)) such that In fact, Miyawaki [14] numerically computed the actions of Hecke operators on F 12 and F 14 which belong to the one dimensional vector spaces S 12 (Sp 3 (Z)) and S 14 (Sp 3 (Z)), respectively, and predicted the conjectures for them. The general forms of Miyawaki's conjectures were given by Heim [7]. Ibukiyama [8] also considered his lifting conjectures for the non-cuspidal cases in slightly wider situations.…”

Using Katsurada’s determination of the Eisenstein series to compute Siegel eigenforms

“…The apparent nonlifts are congruence neighbors of the Ikeda-Miyawaki lifts modulo a prime over 1753, and they are congruence neighbors of the apparent Miyawaki II lifts modulo two primes, one over 613 and the other over 677. Heim [8] has raised the question of whether GL(2)-twisted L-functions would appear as spinor L-functions of degree three Siegel cusp eigenforms. Our computations show that these L-functions do not arise in S k (Γ 3 ) for k ≤ 22, which suggests that Miyawaki found all the lifts that occur in degree three and level one.…”