Subconvexity and equidistribution of Heegner points in the level aspect
Subconvexity and equidistribution of Heegner points in the level aspect
复制标题
水平方面Heegner点的次凸性和均分布
作者：
Sheng;R. Masri;M. Young
Abstract Let $q$ be a prime and $- Dlt - 4$ 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 $L(f imes {Theta }_{chi } , s)$ at $s= 1/ 2$ when $qasymp {D}^{eta } $ for $0lt eta lt 1$. 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
ightarrow infty $ for $qleq {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.
Abstract Let $q$ be a prime and $- Dlt - 4$ 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 $L(f imes {Theta }_{chi } , s)$ at $s= 1/ 2$ when $qasymp {D}^{eta } $ for $0lt eta lt 1$. 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
ightarrow infty $ for $qleq {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.