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Subconvexity and equidistribution of Heegner points in the level aspect
Published online by Cambridge University Press: 17 June 2013
Abstract
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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s) 2013
References
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