Subconvexity and equidistribution of Heegner points in the level aspect

Abstract

Abstract. 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 Θ χ 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 ×Θ χ , s) at s = 1/2 when q ≍ D η 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 L 2 -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.

Rankin–Selberg L-functions and the reduction of CM elliptic curves

“…We need to more carefully treat the case p = 2 in the proof of Lemma 10.5 of [29]. By a factorization argument, it suffices to consider the case where c and ±D are both powers of 2.…”

“…By a factorization argument, it suffices to consider the case where c and ±D are both powers of 2. We have, for any p, We need to understand the analytic properties of r(m; c, D) where this calculation is different from that of [29]. We claim that for c ≫ √ N , we have while for c ≪ √ N we have…”

“…2 below for a more thorough explanation). This is an analog of [29,Proposition 1.7] which is a bound on the L 4 norm of a Maass form in the level aspect.…”

“…Lastly, we take this opportunity to make a correction and improvement to [29]. In [29, Corollary 1.3] the word "effective" should be replaced with "ineffective. "…”

$$L^4$$ L 4 -norms of Hecke newforms of large level

“…Thus Theorem 1.1 follows from Theorem 1.2. The same strategy was used in [4] to study the L 4 -norm in terms of the weight k. Notice how this differs from the approach in [16]: there, the Cauchy-Schwarz inequality is applied to (3.2) and the problem is reduced to studying the second power moments of L( 1 2 , g) and L( 1 2 , sym 2 f × g), for Maass forms. We remark that the Lindelöf bound for the triple product L-function above would imply the best expected bound f 4 ≪ q −1/4+ǫ .…”