Explicit Gross–Zagier and Waldspurger formulae

Quadratic twists of elliptic curves

“…, p k be distinct primes which all split completely in the field H defined by (1.4) Note that, in the special case when k = 0 but r is arbitrary, no hypothesis about the ideal class group is needed for the statement of the theorem, since Q( √ −l 0 ) has odd class number. We also remark that, in the paper [2], the assertions of Theorems 1.3 and 1.4, are strengthened to show that both theorems hold under the weaker hypothesis that the primes p 1 , ..., p k split completely in the subfield Q(A [4], √ R) of H. Unfortunately, we still do not know enough at present to prove that the orders of the TateShafarevich group of the twists of A in Theorems 1.3 and 1.4 are as predicted by the conjecture of Birch and Swinnerton-Dyer.…”

“…shows that S (φ) (A ′(M) ) has order 2, whence the assertion follows from the exact sequence (3.14), and the fact that S (2) …”

“…The Heegner points P M are related to the value at s = 1 of the derivative of the L-function by the following generalized Gross-Zagier formula, which is proven in general by Yuan-Zhang-Zhang in [19], and its explicit form used here in [2]. If χ denotes an abelian character of K, we write L(E/K, χ, s) for the complex L-series of E/K twisted by χ.…”

“…Let E be an elliptic curve over Q of conductor C = C E , and let f : X 0 (C) → E be a modular parametrization as in (1.1). Assume that (1) f ([0]) / ∈ 2E(Q), (2) there is a good supersingular prime q 1 for E, with q 1 ≡ 1 mod 4, and C a square modulo q 1 . If k is any integer ≥ 1, there are infinitely many square free integers M , having exactly k prime factors, such that L(E (M) , s) has a zero at s = 1 of order 1.…”

“…Let c ∞ (E) denote the number of connected components of E(R), and for each prime q dividing C, let c q (E) = [E(Q q ) : E 0 (Q q )], where Q q is the q-adic completion of Q, and E 0 (Q q ) is the subgroup of points with non-singular reduction modulo q. Then the full Birch-Swinnerton-Dyer conjecture asserts that, under the assumption that L(E, 1) = 0, we have (1.3) L(E, 1)/ω(E) = c ∞ (E) q|C c q (E)#(X(E))/#(E(Q)) 2 .…”