A uniform Weyl bound for L-functions of Hilbert modular forms

Abstract

We establish a Weyl-type subconvex bound for L( 1 2 , f ), where f is a spherical Hilbert newform with level ideal N not divisible by p 4 for any prime ideal p. The proof exploits a distributional version of Motohashi's formula over number fields developed by the first author, as well as Katz's work on hypergeometric sums over finite fields in the language of ℓ-adic cohomology.

On Motohashi’s formula

“…• In Appendix A, we assume our local weight transform can pass through the inverse Bessel transform, apply the local weight transform formula in Theorem 1.2 to H(y) = j π (y) the Bessel function of a unitary principal series representation π of PGL 2 (R), and show that the kernel function coincides with the kernel function obtained in our previous work [1]. • In Appendix B, we apply the local weight transform formula in Theorem 1.2 to H(y) which selects the depth 0 supercuspidal representation of PGL 2 (F) over a non-archimedean field F considered in our previous work [43], and show that we get the same algebraic exponential sum.…”

“…It should be noted that neither version gives a form of the relevant exponential sums handy for estimation. Hence the relation with Katz's theory of hypergeometric sums, discovered in our previous works [43,44], is still worth deeper theoretic exploitation.…”

“…Nelson [29] addresses the convergence issue in the above formalism over any number field with a theory of regularized integrals. We fill a non-trivial gap in Nelson's treatment by giving a full analysis of the degenerate terms with the method of meromorphic continuation (see [42] and [43,Appendix]), which seems to be more suitable than the theory of regularized integrals for this purpose. Another interpretation via the Godement-Jacquet pre-trace formula, a type of pre-trace formula whose test functions are Schwartz functions on the 2 by 2 matrices M 2 instead of GL 2 is also given in [42].…”

“…We hope to generalize such construction in higher rank cases. Moreover, our method is based solely on meromorphic continuation, not regularized integrals (see [43,Appendix A]). Just as the distributions in [42], the main distributions in Theorem 1.1 admit decompositions as product of local terms.…”

Hybrid Weyl-type bound for p-power twisted mathrmGL2) L-functions

“…最后, 我们指出: 文献 [53,54,60] 中某些自对偶 (扭) L 函数也达到了 Weyl 型混合界, 但本文之前 所述的结果均未有自对偶假设的要求. 此外, 文献 [7,9,37,41,52,59]…”