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Subconvexity and equidistribution of Heegner points in the level aspect

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  17 June 2013

Sheng-Chi Liu ,
Riad Masri  and
Matthew P. Young
Sheng-Chi Liu
Affiliation:
Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368, USA email scliu@math.tamu.edumasri@math.tamu.edumyoung@math.tamu.edu
Riad Masri
Affiliation:
Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368, USA email scliu@math.tamu.edumasri@math.tamu.edumyoung@math.tamu.edu
Matthew P. Young
Affiliation:
Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368, USA email scliu@math.tamu.edumasri@math.tamu.edumyoung@math.tamu.edu
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Abstract

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Let $q$ be a prime and $- D\lt - 4$ be an odd fundamental discriminant such that $q$ splits in $ \mathbb{Q} ( \sqrt{- D} )$. For $f$ a weight-zero Hecke–Maass newform of level $q$ and ${\Theta }_{\chi } $ the weight-one theta series of level $D$ corresponding to an ideal class group character $\chi $ of $ \mathbb{Q} ( \sqrt{- D} )$, we establish a hybrid subconvexity bound for $L(f\times {\Theta }_{\chi } , s)$ at $s= 1/ 2$ when $q\asymp {D}^{\eta } $ for $0\lt \eta \lt 1$. 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\rightarrow \infty $ for $q\leq {D}^{1/ 20- \varepsilon } $. 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.

Keywords

subconvexityequidistributionHeegner pointsL4 norm

MSC classification

Secondary: 11M41: Other Dirichlet series and zeta functions
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 7 , July 2013 , pp. 1150 - 1174
Copyright
© The Author(s) 2013 

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