scholarly journals Horizontal non-vanishing of Heegner points and toric periods

2020 ◽  
Vol 362 ◽  
pp. 106938
Author(s):  
Ashay A. Burungale ◽  
Ye Tian
Keyword(s):  
Heegner Points

scholarly journals Vertical Distribution Relations for Special Cycles on Unitary Shimura Varieties

2018 ◽  
Vol 2020 (13) ◽  
pp. 3902-3926
Author(s):  
Réda Boumasmoud ◽  
Ernest Hunter Brooks ◽  
Dimitar P Jetchev
Keyword(s):  
Vertical Distribution ◽  
Three Dimensional ◽  
Complex Multiplication ◽  
Horizontal Distribution ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Heegner Points ◽  
Universal Norm

Abstract We consider cycles on three-dimensional Shimura varieties attached to unitary groups, defined over extensions of a complex multiplication (CM) field $E$, which appear in the context of the conjectures of Gan et al. [6]. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of [8], and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda $-module constructed from Heegner points.

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

Local units, elliptic units, Heegner points and elliptic curves

1987 ◽  
Vol 88 (2) ◽  
pp. 405-422 ◽  
Author(s):  
Karl Rubin
Keyword(s):  
Elliptic Curves ◽  
Heegner Points ◽  
Elliptic Units

Chow–Heegner Points on CM Elliptic Curves and Values of p-adic L-functions

2012 ◽  
Vol 2014 (3) ◽  
pp. 745-793 ◽  
Author(s):  
Massimo Bertolini ◽  
Henri Darmon ◽  
Kartik Prasanna
Keyword(s):  
Elliptic Curves ◽  
Heegner Points

Stark–Heegner points and special values of L-series

2010 ◽  
pp. 1-23 ◽  
Author(s):  
Massimo Bertolini ◽  
Henri Darmon ◽  
Samit Dasgupta
Keyword(s):  
Heegner Points ◽  
Special Values

scholarly journals Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes

2018 ◽  
Vol 12 (3) ◽  
pp. 315-347 ◽  
Author(s):  
Matteo Longo ◽  
Stefano Vigni
Keyword(s):  
Elliptic Curves ◽  
Iwasawa Theory ◽  
Heegner Points ◽  
Supersingular Primes

Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields

2008 ◽  
Vol 60 (3) ◽  
pp. 481
Author(s):  
Florian Breuer ◽  
Bo-Hae Im
Keyword(s):  
Elliptic Curves ◽  
Heegner Points ◽  
Global Fields

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 Heights of Heegner Points on Shimura Curves

2001 ◽  
Vol 153 (1) ◽  
pp. 27 ◽  
Author(s):  
Shouwu Zhang
Keyword(s):  
Shimura Curves ◽  
Heegner Points

A NOTE ON THE VARIANCE OF THE DISTRIBUTION OF HEEGNER POINTS

2012 ◽  
Vol 08 (02) ◽  
pp. 525-529
Author(s):  
SHENG-CHI LIU
Keyword(s):  
Cusp Forms ◽  
Heegner Points ◽  
Explicit Constant

In this note, we evaluate the variance of the distribution of Heegner points asymptotically. More precisely, we show [Formula: see text] for any Hecke–Maass cusp forms ϕ, ψ for SL 2(ℤ), where c is an explicit constant which depends on the eigenvalue of ϕ.

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