Heegner Points and Representation Theory
نویسندگان
چکیده
Our aim in this paper is to present a framework in which the results of Waldspurger and Gross–Zagier can be viewed simultaneously. This framework may also be useful in understanding recent work of Zhang, Xue, Cornut, Vatsal, and Darmon. It involves a blending of techniques from representation theory and automorphic forms with those from the arithmetic of modular curves. I hope readers from one field will be encouraged to pursue the other.
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...
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We study the algebraicity of Stark-Heegner points on a modular elliptic curve E. These objects are p-adic points on E given by the values of certain p-adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field K. The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted b...
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Modular curves and their close relatives, Shimura curves attached to multiplicative subgroups of quaternion algebras, are equipped with a distinguished collection of points defined over class fields of imaginary quadratic fields and arising from the theory of complex multiplication: the so-called Heegner points. It is customary to use the same term to describe the images of degree zero divisors...
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Let j(z) = q−1 + 744 + 196884q + · · · denote the usual elliptic modular function on SL2(Z) (q := e throughout). We shall refer to a complex number τ of the form τ = −b+ √ b2−4ac 2a with a, b, c ∈ Z, gcd(a, b, c) = 1 and b −4ac < 0 as a Heegner point, and we denote its discriminant by the integer dτ := b − 4ac. The values of j at such points are known as singular moduli, and they play a substan...
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Introduction 1 1. Basic notions 6 1.1. Motives for rational and homological equivalence 6 1.2. Algebraic Hecke characters 7 1.3. The motive of a Hecke character 8 1.4. Deligne-Scholl motives 9 1.5. Modular parametrisations attached to CM forms 10 1.6. Generalised Heegner cycles and Chow-Heegner points 13 1.7. A special case 15 2. Chow-Heegner points over Cp 15 2.1. The p-adic Abel-Jacobi map 15...
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تاریخ انتشار 2004