On the independence of Heegner points on CM elliptic curves associated to distinct quadratic imaginary fields (original) (raw)
Stark-Heegner points over real quadratic fields
Contemporary Mathematics, 1998
Motivated by the conjectures of "Mazur-Tate-Teitelbaum type" formulated in and by the main result of [BD3], we describe a conjectural construction of a global point P K ∈ E(K), where E is a (modular) elliptic curve over Q of prime conductor p, and K is a real quadratic field satisfying suitable conditions. The point P K is constructed by applying the Tate p-adic uniformization of E to an explicit expression involving geodesic cycles on the modular curve X 0 (p). These geodesic cycles are a natural generalization of the modular symbols of Birch and Manin, and interpolate the special values of the Hasse-Weil L-function of E/K twisted by certain abelian characters of K. In the analogy between Heegner points and circular units, the point P K is analogous to a Stark unit, since it has a purely conjectural definition in terms of special values of L-functions, but no natural "independent" construction of it seems to be known. We call the conjectural point P K a "Stark-Heegner point" to emphasize this analogy. The conjectures of section 4 are inspired by the main result of [BD3], in which the real quadratic field is replaced by an imaginary quadratic field. The methods of [BD3], which rely crucially on the theory of complex multiplication and on the Cerednik-Drinfeld theory of p-adic uniformization of Shimura curves, do not seem to extend to the real quadratic situation. One must therefore content oneself with numerical evidence for the conjectures.
Heegner Points and Elliptic Curves of Large Rank over Function Fields
Heegner Points and Rankin L-Series, 2004
This note presents a connection between Ulmer's construction [Ulm02] of non-isotrivial elliptic curves over F p (t) with arbitrarily large rank, and the theory of Heegner points (attached to parametrisations by Drinfeld modular curves, as sketched in Section 3 of Ulmer's article (see page ??). This ties in the topics in Section 4 of that article more closely to the main theme of this volume.
On the index of the Heegner subgroup of elliptic curves
2007
Let E be an elliptic curve of conductor N and rank one over Q. So there is a non-constant morphism X+0(N) --> E defined over Q, where X+0(N) = X0(N)/wN and wN is the Fricke involution of the modular curve X+0(N). Under this morphism the traces of the Heegner points of X+0(N) map to rational points on E. In this paper we study the index I of the subgroup generated by all these traces on E(Q). We propose and also discuss a conjecture that says that if N is prime and I > 1, then either the number of connected components of the real locus X+0(N)(R) is greater than 1 or (less likely) the order S of the Tate-Safarevich group is non-trivial. This conjecture is backed by computations performed on each E that satisfies the above hypothesis in the range N < 129999. This paper was prepared for the proceedings of the Conference on Algorithmic Number Theory, Turku, May 8-11, 2007. We tried to make the paper as self contained as possible.
Elliptic curves and class field theory
Arxiv preprint math/0304235, 2003
Suppose E is an elliptic curve defined over Q. At the 1983 ICM the first author formulated some conjectures that propose a close relationship between the explicit class field theory construction of certain abelian extensions of imaginary quadratic fields and an explicit construction that (conjecturally) produces almost all of the rational points on E over those fields. Those conjectures are to a large extent settled by recent work of Vatsal and of Cornut, building on work of Kolyvagin and others. In this paper we describe a collection of interrelated conjectures still open regarding the variation of Mordell-Weil groups of E over abelian extensions of imaginary quadratic fields, and suggest a possible algebraic framework to organize them.
Heegner points, Heegner cycles, and congruences
We dene 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 eld Q( p 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 conjectures about congruences satised by these objects which are resonant with the usual Birch and Swinnerton-Dyer conjectures. These conjectures exhibit some surprising features not apparent in the classical case. 1 Heegner objects 1.1 Modular elliptic curves Let E be an elliptic curve dened over Q by the Weierstrass equation y 2 = 4x 3 g 2 x g 3 ; g 2 ; g 3 2 Z; and let N denote the arithmetic conductor of E, which can be computed from g 2 and g 3 by Tate's algorithm [Ta]. To simplify the discussion, let us assume that N is odd. The group E(C) is isomorphic to the complex torus C=, where is a free Z-l...
Journal of Number Theory, 1994
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz 138 (2013) MATHEMATICA BOHEMICA No. 2, 149-163
Multiples of integral points on elliptic curves
Journal of Number Theory, 2009
If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the global Tamagawa number of E, such that for any point P ∈ E(Q), nP is integral for at most one value of n > C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E ′ of E of sufficient height, and all torsion-free, rank-one subgroups Γ ⊆ E ′ (Q), Γ contains at most 6 integral points. Explicit computations for congruent number curves are included.