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

Abstract

Let E/Q be an elliptic curve and let D < 0 be a sufficiently large fundamental discriminant. If E(Q) 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. RésuméSoit E/Q une courbe elliptique. Soit D < 0 un discriminant fondamental suffisamment grand. Si E(Q) contient des points de Heegner de discriminant D, ces points engendrent un sous-groupe dont le rang est supérieurà |D| δ , où δ > 0 est une constante absolue. Ce résultat est en accord avec la conjecture de Birch et Swinnerton-Dyer.

Mordell-Weil growth for GL2-type abelian varieties over Hilbert class fields of CM fields

“…• When F = Q and K d is the Hilbert class field of an imaginary quadratic field Q( √ −d), Templier shows that the rank of E over K d is at least ≫ d δ for some small but positive fixed δ, provided that ε(E) = −ε(E ⊗ χ −d ). In fact, Templier has given two distinct proofs of this theorem: a short proof [Te2] built on the Gross-Zagier theorem and equidistribution theorems for Galois orbits, and an analytic proof [Te1] which analyzes an average value of L-functions directly, using tools from analytic number theory.…”

Horizontal non-vanishing of Heegner points and toric periods

“…In the split case, the geometric argument only uses Burgess bound for character sums which controls the average of toric period against Eisenstein series. It seems suggestive to compare our approach with the one in [27]. Here we only mention the following.…”

“…The non-vanishing over the family of twists by class group characters has been considered in the literature in the case of root number −1. Here we mention [19], [26], [27] and refer to these articles for a quantitative version of Theorem 1.3 in the case F = Q and c = 1 under certain hypotheses. The results in [27] are perhaps closest to our study of the Heegner points.…”

“…When F = Q and c = 1, the non-vanishing in Theorem 1.3 goes back to Templier. For the results and various approaches to the non-vanishing, we refer to Templier's articles [26], [27] along with references therein. Under the existence of enough small split primes in the imaginary quadratic fields, Thm.…”

“…Here we mention [19], [26], [27] and refer to these articles for a quantitative version of Theorem 1.3 in the case F = Q and c = 1 under certain hypotheses. The results in [27] are perhaps closest to our study of the Heegner points. In these articles, the approach perhaps has more analytic/ ergodic flavour.…”