Subconvexity for twisted L-functions over number fields via shifted convolution sums

Abstract

ABSTRACT. Assume that π is a cuspidal automorphic GL 2 representation over a number field F. Then for any Hecke character χ of conductor q, the subconvex boundholds for any ε > 0, where θ is any constant towards the Ramanujan-Petersson conjecture (θ = 7/64 is admissible). In these notes, we derive this bound from the spectral decomposition of shifted convolution sums worked out by the author in [Mag].

Eisenstein series and the cubic moment for PGL(2)

“…The above "subconvex" estimates improve upon the respective "trivial" or "convexity" bounds of C(χ) 1/2+ε and C(χ) 1/4+ε , and also upon earlier nontrivial subconvex bounds (see [7,62,61,37] and references). Via period formulas as in [51,15,3] (see also [59,31,56,30,32,32,34]) these estimates lead to improved bounds for the Fourier coefficients of half-integral weight modular forms over number fields (cf.…”

On the Waldspurger formula and the metaplectic Ramanujan conjecture over number fields

“…As an immediate application, we shall extend the Ramanujan-Lindelöf equivalence in (1) to arbitrary number fields. Moreover, combined with the GL 2ˆG L 1 subconvexity results over number fields in [MV,Wu,Mag2], we shall obtain the first nontrivial estimate towards the metaplectic Ramanujan conjecture for Ă SL 2 over arbitrary number fields. Recall that this is a key step in the settlement of Hilbert's eleventh problem in [CPSS], but the ground field therein is only needed to be totally real thanks to the work of Siegel.…”

“…The GL 2 -subconvexity problem is now completely solved by Michel and Venkatesh [MV] over arbitrary number fields. Later Han Wu [Wu] and Maga [Mag1,Mag2] proved by different methods the following subconvexity bound…”

BSD implies a non-trivial bound on 5-torsion in class groups of imaginary quadratic fields

“…Let E be an elliptic curve over Q. We have the exact sequences (9) for the fields F = Q and F = Q v for all completions Q v of Q. For each completion, we have a natural injection G Qv → G Q , which gives rise to the natural homomorphisms H i (Q, G) → H i (Q v , G) for any abelian group scheme G defined over Q.…”