scholarly journals Subconvexity and equidistribution of Heegner points in the level aspect

2013 ◽  
Vol 149 (7) ◽  
pp. 1150-1174 ◽  
Author(s):  
Sheng-Chi Liu ◽  
Riad Masri ◽  
Matthew P. Young
Keyword(s):  
Theta Series ◽  
Ideal Class Group ◽  
Heegner Points ◽  
Maass Form ◽  
Quadratic Twists ◽  
Fundamental Discriminant ◽  
Weight One ◽  
Natural Sense ◽  
Large Level ◽  
Quadratic Twist

AbstractLet $q$ 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.

scholarly journals Hauteur asymptotique des points de Heegner

2008 ◽  
Vol 60 (6) ◽  
pp. 1406-1436 ◽  
Author(s):  
Guillaume Ricotta ◽  
Thomas Vidick
Keyword(s):  
Elliptic Curve ◽  
Quadratic Field ◽  
Theta Series ◽  
Dirichlet Character ◽  
Heegner Points ◽  
Quadratic Twists ◽  
Asymptotic Formulae ◽  
Weight One ◽  
Special Value ◽  
Geometric Intuition

AbstractGeometric intuition suggests that the Néron–Tate height of Heegner points on a rational elliptic curve E should be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross–Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin–Selberg convolution of E with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted L-function of E by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these L-series and L-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.

scholarly journals GOLDFELD’S CONJECTURE AND CONGRUENCES BETWEEN HEEGNER POINTS

2019 ◽  
Vol 7 ◽  
Author(s):  
DANIEL KRIZ ◽  
CHAO LI
Keyword(s):  
Elliptic Curve ◽  
Analogous Result ◽  
Heegner Points ◽  
Positive Proportion ◽  
Quadratic Twists ◽  
General Elliptic ◽  
Special Cases ◽  
General Bound

Given an elliptic curve$E$over$\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever$E$has a rational 3-isogeny. We also prove the analogous result for the sextic twists of$j$-invariant 0 curves. For a more general elliptic curve$E$, we show that the number of quadratic twists of$E$up to twisting discriminant$X$of analytic rank 0 (respectively 1) is$\gg X/\log ^{5/6}X$, improving the current best general bound toward Goldfeld’s conjecture due to Ono–Skinner (respectively Perelli–Pomykala). To prove these results, we establish a congruence formula between$p$-adic logarithms of Heegner points and apply it in the special cases$p=3$and$p=2$to construct the desired twists explicitly. As a by-product, we also prove the corresponding$p$-part of the Birch and Swinnerton–Dyer conjecture for these explicit twists.

scholarly journals ON SHIMURA'S DECOMPOSITION

2013 ◽  
Vol 09 (06) ◽  
pp. 1431-1445 ◽  
Author(s):  
SOMA PURKAIT
Keyword(s):  
Modular Forms ◽  
Theta Series ◽  
Dirichlet Character ◽  
Practical Applications ◽  
Quadratic Twists ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Sum Formula ◽  
Practical Algorithm ◽  
Definition Of

Let k be an odd integer ≥ 3 and N be a positive integer such that 4|N. Let χ be an even Dirichlet character modulo N. Shimura decomposes the space of half-integral weight cusp forms Sk/2(N,χ) as a direct sum [Formula: see text] where F runs through all newforms of weight k - 1, level dividing N/2 and character χ2, the space Sk/2(N,χ,F) is the subspace of forms that are "Shimura equivalent" to F, and the space S0(N,χ) is the subspace spanned by single-variable theta-series. The explicit computation of this decomposition is important for practical applications of a theorem of Waldspurger relating the critical values of L-functions of quadratic twists of newforms of even integral weight to coefficients of modular forms of half-integral weight. In this paper, we give a more precise definition of the summands Sk/2(N,χ,F) whilst proving that it is equivalent to Shimura's definition. We use our definition to give a practical algorithm for computing Shimura's decomposition, and illustrate this with some examples.

Irregular Weight One Points with Image

2019 ◽  
Vol 62 (1) ◽  
pp. 109-118
Author(s):  
Hao Lee
Keyword(s):  
Tangent Space ◽  
Fourier Coefficients ◽  
Theta Series ◽  
Algebraic Numbers ◽  
Weight One ◽  
Hida Families

AbstractDarmon, Lauder, and Rotger conjectured that the relative tangent space of an eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.

scholarly journals Sur le rang des courbes elliptiques sur les corps de classes de Hilbert

2011 ◽  
Vol 147 (4) ◽  
pp. 1087-1104 ◽  
Author(s):  
Nicolas Templier
Keyword(s):  
Elliptic Curve ◽  
Absolute Constant ◽  
Heegner Points ◽  
Fundamental Discriminant

AbstractLet E/ℚ be an elliptic curve and let D<0 be a sufficiently large fundamental discriminant. If $E(\overline {\BmQ })$ contains Heegner points of discriminant D, those points generate a subgroup of rank at least |D|δ, where δ>0 is an absolute constant. This result is compatible with the Birch and Swinnerton-Dyer conjecture.

scholarly journals Height pairings on orthogonal Shimura varieties

2017 ◽  
Vol 153 (3) ◽  
pp. 474-534 ◽  
Author(s):  
Fabrizio Andreatta ◽  
Eyal Z. Goren ◽  
Benjamin Howard ◽  
Keerthi Madapusi Pera
Keyword(s):  
Complex Multiplication ◽  
Theta Series ◽  
Positive Definite ◽  
Quadratic Space ◽  
Shimura Variety ◽  
Heegner Points ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Intersection Multiplicities ◽  
Derivatives Of

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and complex multiplication points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin–Selberg convolution associated with a cusp form of half-integral weight $n/2+1$, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross–Zagier theorem on heights of Heegner points.

scholarly journals Bounds for twisted symmetric square L-functions via half-integral weight periods

2020 ◽  
Vol 8 ◽  
Author(s):  
Paul D. Nelson
Keyword(s):  
Modular Forms ◽  
Triple Product ◽  
Quadratic Character ◽  
Maass Form ◽  
Half Integral Weight ◽  
Fundamental Discriminant ◽  
Integral Weight ◽  
Symmetric Square ◽  
Ramanujan Conjecture ◽  
First Moment

Abstract We establish the first moment bound $$\begin{align*}\sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_\varepsilon p^{5/4+\varepsilon} \end{align*}$$ for triple product L-functions, where $\Psi $ is a fixed Hecke–Maass form on $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and $\varphi $ runs over the Hecke–Maass newforms on $\Gamma _0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke–Maass newforms on $\Gamma _0(p) \backslash \mathbb {H}$ of bounded eigenvalue have very uniformly distributed mass after pushforward to $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathbb {H}$ . Our main result turns out to be closely related to estimates such as $$\begin{align*}\sum_{|n| < p} L(\Psi \otimes \chi_{n p},\tfrac{1}{2}) \ll p, \end{align*}$$ where the sum is over those n for which $n p$ is a fundamental discriminant and $\chi _{n p}$ denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke–Iwaniec.

scholarly journals Petersson inner products of weight-one modular forms

2019 ◽  
Vol 2019 (749) ◽  
pp. 133-159
Author(s):  
Maryna Viazovska
Keyword(s):  
Quadratic Form ◽  
Algebraic Number ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Theta Series ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Arithmetic Properties ◽  
Factorization Formula ◽  
Weight One

Abstract In this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that these Petersson products posses remarkable arithmetic properties. Namely, such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. A similar result was obtained independently using a different method by W. Duke and Y. Li [5]. The main result of this paper is an explicit factorization formula for the algebraic number obtained by exponentiating a Petersson product.

scholarly journals Rational points on elliptic K3 surfaces of quadratic twist type

2020 ◽  
Author(s):  
Zhizhong Huang
Keyword(s):  
Elliptic Curves ◽  
K3 Surfaces ◽  
Rational Points ◽  
Zariski Topology ◽  
Quadratic Twists ◽  
Quadratic Twist ◽  
Quartic Polynomials

Abstract In studying rational points on elliptic K3 surfaces of the form $$\begin{equation*} f(t)y^2=g(x), \end{equation*}$$ where f, g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell–Weil rank, and we relate it to the Hilbert property. Applying to surfaces of Cassels–Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.

scholarly journals Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jie Shu ◽  
Shuai Zhai
Keyword(s):  
Elliptic Curves ◽  
Large Class ◽  
Infinite Order ◽  
Rational Points ◽  
Heegner Points ◽  
Original Curve ◽  
Quadratic Twists

Abstract In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over ℚ {{\mathbb{Q}}} . We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.

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