Subconvexity and equidistribution of Heegner points in the level aspect
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Let q be a prime and - D<- 4 be an odd fundamental discriminant such that q splits in Q(√-D). For f a weight-zero Hecke-Maass newform of level q and Θχ the weight-one theta series of level D corresponding to an ideal class group character χ of Q(√-D), we establish a hybrid subconvexity bound for L(f× Θχ) at s= 1/2 when q⊗Dη for 0< η < 1. With this circle of ideas, we show that the Heegner points of level q and discriminant D become equidistributed, in a natural sense, as q, D→ ∞ for q≤D 1/20-ε. Our approach to these problems is connected to estimating the L2-restriction norm of a Maass form of large level q when restricted to the collection of Heegner points. We furthermore establish bounds for quadratic twists of Hecke-Maass L-functions with simultaneously large level and large quadratic twist, and hybrid bounds for quadratic Dirichlet L-functions in certain ranges. Copyright © The Author(s) 2013.
author list (cited authors)
Liu, S., Masri, R., & Young, M. P.