A formula for the central value of certain Hecke L-functions

Let N = 1 mod 4 be the negative of a prime, K=Q(sqrt{N}) and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters \psi_{\D} of K with conductor (\D) and infinite type $(1,0)$. Their L-series L(\psi_\D,s) are associated to a CM elliptic curve E(N,\D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(\psi_\D,s) of the form L(\psi_\D,1) = \Omega \sum_{[\A],I} r(\D,[\A],I) m_{[\A],I}([\D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [\A] are class group representatives of K$. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level$7|D|$over$K$.Comment: 43 page

Heegner points on Cartan non-split curves

Let $E$ be an elliptic curve of conductor $N$, and let $K$ be an imaginary quadratic field such that the root number of $E/K$ is $-1$. Let $O$ be an order in $K$ and assume that there exists an odd prime $p$, such that $p^2 \mid\mid N$, and $p$ is inert in $O$. Although there are no Heegner points on $X_0(N)$ attached to $O$, in this article we construct such points on Cartan non-split curves. In order to do that we give a method to compute Fourier expansions for forms in Cartan non-split curves, and prove that the constructed points form a Heegner system as in the classical case.Comment: 25 pages, revised versio

Computing ideal classes representatives in quaternion algebras

Let $K$ be a totally real number field and let $B$ be a totally definite quaternion algebra over $K$. In this article, given a set of representatives for ideal classes for a maximal order in $B$, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in $B$. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of level 30 in an algebra over the real quadratic field \Q[\sqrt{5}].Comment: Corrected version. Section 4 has been rewritten from scratc

Congruences between modular forms modulo prime powers

Given a prime $p \ge 5$ and an abstract odd representation $\rho_n$ with coefficients modulo $p^n$ (for some $n \ge 1$) and big image, we prove the existence of a lift of $\rho_n$ to characteristic $0$ whenever local lifts exist (under some technical conditions). Moreover, we can chose the inertial type of our lift at all primes but finitely many (where the lift is of Steinberg type). We apply this result to the realm of modular forms, proving a level lowering theorem modulo prime powers and providing examples of level raising. In particular, our method shows that given a modular eigenform $f$ without Complex Multiplication or inner twists, for all primes $p$ but finitely many, and for all positive integers $n$, there exists another eigenform $g\neq f$, which is congruent to $f$ modulo $p^n$.Comment: 22 pages; revised argument in section 5; hypotheses remove

Q-curves, Hecke characters and some Diophantine equations II

In the article [25] a general procedure to study solutions of the equations x4 − dy2 = z p was presented for negative values of d. The purpose of the present article is to extend our previous results to positive values of d. On doing so, we give a description of the extension Q(√d, √e)/Q(√d) (where e is a fundamental unit) needed to prove the existence of a Hecke character over Q(√d) with prescribed local conditions. We also extend some "large image" results due to Ellenberg regarding images of Galois representations coming from Q-curves from imaginary to real quadratic fields