Subconvexity Bounds for Automorphic L–functions

We break the convexity bound in the t–aspect for L–functions attached to cuspforms f for GL2(k) over arbitrary number fields k. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L(s, f ⊗χ) by grossencharacters χ, from our previous paper [Di-Ga]. §0. Introduction In many instances, for cuspidal automorphic forms f on reductive adele groups over number fields, the circle of ideas around the Phragmen-Lindelöf principle, together with the functional equation for L(s, f) and asymptotics for Γ(s), give an upper bound for L(s, f) on <(s) = 1 2 . These are convexity bounds, or trivial bounds. For example, for the standard L-functions for cuspforms for GLn, the convexity bound is known. The survey [Iw-Sa] gives a general formulation of the subconvexity problem. See also the survey [Mi2]. In particular, the convexity bound for the Riemann zeta function is ζ( 12 + it) ε (1 + |t|) 1 4 +ε (for all ε > 0) and for χ a primitive Dirichlet character of conductor q L( 12 , χ) ε q 1 4 +ε (for all ε > 0) A similar estimate holds with 12 replaced by 1 2 + it. The Generalized Lindelöf Hypothesis would replace the exponent 1 4 + ε by ε. The subconvexity problem for GL1 over Q asks for an estimate with exponent strictly below 14 . In [We] Weyl proved ζ( 12 + it) ε (1 + |t|) 1 6 +ε (for all ε > 0) and in [Bu] Burgess showed L(s, χ) ε q 3 16 +ε (fixed s with <(s) = 12 , for all ε > 0) 1991 Mathematics Subject Classification. 11R42, Secondary 11F66, 11F67, 11F70, 11M41, 11R47.