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 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 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.

scholarly journals Heegner Points on Cartan Non-split Curves

2016 ◽  
Vol 68 (2) ◽  
pp. 422-444 ◽  
Author(s):  
Daniel Kohen ◽  
Ariel Pacetti
Keyword(s):  
Elliptic Curve ◽  
Quadratic Field ◽  
Classical Case ◽  
Imaginary Quadratic Field ◽  
Root Number ◽  
Heegner Points ◽  
Fourier Expansions

AbstractLet E/ℚ be an elliptic curve of conductor N, and let K be an imaginary quadratic field such that the root number of E/K is −1. Let be an order in K and assume that there exists an odd prime p such that p2 ∥ N, and p is inert in . Although there are no Heegner points on X0(N) attached to , in this article we construct such points on Cartan non-split curves. In order to do that, we give a method to compute Fourier expansions for forms on Cartan non-split curves, and prove that the constructed points form a Heegner system as in the classical case.

On elliptic curves with complex multiplication and root numbers

2020 ◽  
pp. 1-16
Author(s):  
Wan Lee ◽  
Myungjun Yu
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Root Number ◽  
Root Numbers ◽  
Quadratic Twists

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text]. Suppose that [Formula: see text] has complex multiplication over [Formula: see text], i.e. [Formula: see text] is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a constant root number.

scholarly journals On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

2020 ◽  
Author(s):  
Joachim Petit
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Asymptotic Estimate ◽  
Rational Point ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Quadratic Twists ◽  
The Family ◽  
Minimal Height ◽  
Real Quadratic

Abstract We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

scholarly journals Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation?

2020 ◽  
Vol 14 (1) ◽  
pp. 293-306
Author(s):  
Claire Delaplace ◽  
Alexander May
Keyword(s):  
Elliptic Curve ◽  
Quadratic Field ◽  
Discrete Logarithm ◽  
Field Size ◽  
Square Root ◽  
Multivariate Polynomial ◽  
Bivariate Polynomial ◽  
Testing Problem ◽  
Polynomial Zero ◽  
Representation Technique

AbstractWe give a 4-list algorithm for solving the Elliptic Curve Discrete Logarithm (ECDLP) over some quadratic field 𝔽p2. Using the representation technique, we reduce ECDLP to a multivariate polynomial zero testing problem. Our solution of this problem using bivariate polynomial multi-evaluation yields a p1.314-algorithm for ECDLP. While this is inferior to Pollard’s Rho algorithm with square root (in the field size) complexity 𝓞(p), it still has the potential to open a path to an o(p)-algorithm for ECDLP, since all involved lists are of size as small as $\begin{array}{} p^{\frac 3 4}, \end{array}$ only their computation is yet too costly.

A certain eta-quotient and the class number of an imaginary quadratic field

2014 ◽  
Vol 10 (06) ◽  
pp. 1485-1499
Author(s):  
Takeshi Ogasawara
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Theta Series ◽  
Imaginary Quadratic Field

We prove that the dimension of the Hecke module generated by a certain eta-quotient is equal to the class number of an imaginary quadratic field. To do this, we relate the eta-quotient to the Hecke theta series attached to a ray class character of the imaginary quadratic field.

scholarly journals A Markov model for Selmer ranks in families of twists

2014 ◽  
Vol 150 (7) ◽  
pp. 1077-1106 ◽  
Author(s):  
Zev Klagsbrun ◽  
Barry Mazur ◽  
Karl Rubin
Keyword(s):  
Markov Process ◽  
Markov Model ◽  
Elliptic Curve ◽  
Arbitrary Number ◽  
Equilibrium Distribution ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Quadratic Twists ◽  
The Family ◽  
Positive Integers

AbstractWe study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

scholarly journals p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

2015 ◽  
Vol 219 ◽  
pp. 269-302
Author(s):  
Kenichi Bannai ◽  
Hidekazu Furusho ◽  
Shinichi Kobayashi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Theta Function ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Good Reduction ◽  
Ring Of Integers

AbstractConsider an elliptic curve defined over an imaginary quadratic fieldKwith good reduction at the primes abovep≥ 5 and with complex multiplication by the full ring of integersof K. In this paper, we constructp-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then provep-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

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