منابع مشابه
Heegner points, Stark-Heegner points, and values of L-series
Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1...
متن کاملHeegner points, Heegner cycles, and congruences
We define certain objects associated to a modular elliptic curve E and a discriminant D satisfying suitable conditions. These objects interpolate special values of the complex L-functions associated to E over the quadratic field Q( √ D), in the same way that Bernouilli numbers interpolate special values of Dirichlet L-series. Following an approach of Mazur and Tate [MT], one can make conjecture...
متن کاملHigher-Weight Heegner Points
In this paper we formulate a conjecture which partially generalizes the Gross-Kohnen-Zagier theorem to higher weight modular forms. For f ∈ S2k(N) satisfying certain conditions, we construct a map from the Heegner points of level N to a complex torus, C/Lf , defined by f . We define higher weight analogues of Heegner divisors on C/Lf . We conjecture they all lie on a line, and their positions a...
متن کاملHeegner Points: The Beginnings
Dick Gross and I were invited to talk about Heegner points from a historical point of view, and we agreed that I should talk first, dealing with the period before they became well known. I felt encouraged to indulge in some personal reminiscence of that period, particularly where I can support it by documentary evidence. I was fortunate enough to be working on the arithmetic of elliptic curves ...
متن کاملHeegner points and Sylvester’s conjecture
We consider the classical Diophantine problem of writing positive integers n as the sum of two rational cubes, i.e. n = x3 + y3 for x, y ∈ Q. A conjecture attributed to Sylvester asserts that a rational prime p > 3 can be so expressed if p ≡ 4, 7, 8 (mod 9). The theory of mock Heegner points gives a method for exhibiting such a pair (x, y) in certain cases. In this article, we give an expositor...
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ژورنال
عنوان ژورنال: Proceedings of the London Mathematical Society
سال: 2020
ISSN: 0024-6115,1460-244X
DOI: 10.1112/plms.12363