OnL-functions of elliptic curves and cyclotomic towers

Abstract

We shall presuppose some familiarity with holomorphic modular forms on the complex upper half plane, and in particular with the theory of new forms as developed by Atkin-Lehner [1], Miyake [11], and Li [63. Let f be a normalized new form of weight 2, character 0, and level N, with Fourier expansion f(z)= y~ a(n)e 2~'"z.n>__l Let P be a finite set of primes not dividing N, and let X be the set of all primitive Dirichlet characters which are unramified outside P and infinity. Given Z in X, we write L(s,f, Z) for the L-function attached to f and ~(: in the region Re s > 3/2,The object of this paper is to prove the following theorem.Theorem. For all but finitely many Z in X, L(1, f, )0 4= 0.The theorem has various antecedents in the literature. The reader is referred in particular to Manin [7] Corollary. Let E be an elliptic curve defined over Q with complex multiplicationby the ring of integers of an imaginary quadratic field, and let P be a finite set of primes where E has good reduction. Let L be the maximal abelian extension of Q unramified outside P and infinity, and let E(L) be the group of points on E which are defined over L. Then E(L) is finitely generated.

Algebraicity of L-values for elliptic curves in a false Tate curve tower

“…To correct this, we replace L(E/K, s) in the above procedure by L(E/K, χ, s), for a suitable 1-dimensional Artin representation χ over K. Our proof uses the following well-known result of Rohrlich, Theorem 3.1. (Rohrlich [8,9]) Let E be an elliptic curve over Q. Let Σ be a finite set of primes.…”

The undecidability of cyclotomic towers

“…and the j-invariant is 12 3 . Hence E b also has complex multiplication by the ring of integers of Q( √ −1).…”

“…In general, we cannot expect E to have complex multiplication. However, as Rohrlich points in his paper ( [3], p. 422), if the Taniyama-Weil and the Birch-SwinnertonDyer conjectures are true, then his theorem holds for all elliptic curves over Q. Hence Q A would be undecidable for all finite sets of prime numbers.…”

Heegner points and $p$-adic $L$-functions for elliptic curves over certain totally real fields

“…For k D 2, then by the main result of [28], [29], (6.4) is satisfied for all but finitely many 2 T . Thus pick a non-trivial Q 2 T that satisfies (6.4) (so in particular p divides the conductor of Q ).…”

“…Indeed, the only place where this is essential is the proof of Lemma 6.1 and Proposition 6.4 below, where we make use of a non-vanishing result of Rohrlich [28], [29] on twisted L-values, which was proved only for the ground field Q. If the result of Rohrlich can be extended to general totally real fields, then everything in this section would work without the restriction to Q.…”