Henri Darmon

Rational Points on Modular Elliptic Curves

Elliptic curves Modular forms Heegner points on

A rigid analytic Gross-Zagier formula and arithmetic applications

X_0(N)

Generalized Heegner cycles and

Heegner points on Shimura curves Rigid analytic modular forms Rigid analytic modular parametrisations Totally real fields ATR points Integration on

p

\mathcal{H}_p\times\mathcal{H}

-adic Rankin

Kolyvagins theorem Bibliography.

L

Let f be a newform of weight 2 and squarefree level N. Its Fourier coefficients generate a ring Of whose fraction field Kf has finite degree over Q. Fix an imaginary quadratic field K of discriminant prime to N, corresponding to a Dirichlet character E. The L-series L(f /K, s) = L(f, s)L(f 0 E, s) of f over K has an analytic continuation to the whole complex plane and a functional equation relating L(f/K, s) to L(f/K, 2 s). Assume that the sign of this functional equation is 1, so that L(f/K, s) vanishes to even order at s = 1. This is equivalent to saying that the number of prime factors of N which are inert in K is odd. Fix any such prime, say p. The field K determines a factorization N = N+Nof N by taking N+, resp. Nto be the product of all the prime factors of N which are split, resp.

-series

This article studies a distinguished collection of so-called generalized Heegner cycles in the product of a Kuga–Sato variety with a power of a CM elliptic curve. Its main result is a p-adic analogue of the Gross–Zagier formula which relates the images of generalized Heegner cycles under the p-adic Abel–Jacobi map to the special values of certain p-adic Rankin L-series at critical points that lie outside their range of classical interpolation.

EFFICIENT CALCULATION OF STARK-HEEGNER POINTS VIA OVERCONVERGENT MODULAR SYMBOLS

This note presents a qualitative improvement to the algorithm presented in [DG] for computing Stark-Heegner points attached to an elliptic curve and a real quadratic field. This algorithm computes the Stark-Heegner point with ap-adic accuracy ofM significant digits in time which is polynomial inM, the primep being treated as a constant, rather than theO(pM) operations required for the more naive approach taken in [DG]. The key to this improvement lies in the theory of overconvergent modular symbols developed in [PS1] and [PS2].

Periods of Hilbert modular forms and rational points on elliptic curves

Let E be a modular elliptic curve over a totally real field. Chapter 8 of [Dar2] formulates a conjecture allowing the construction of canonical algebraic points on E by suitably integrating the associated Hilbert modular form. The main goal of the present work is to obtain numerical evidence for this conjecture in the first case where it asserts something nontrivial, namely, when E has everywhere good reduction over a real quadratic field.

Beilinson-Flach elements and Euler systems I: syntomic regulators and p-adic Rankin L-series

This article is the first in a series devoted to the Euler system arising from p-adic families of Beilinson-Flach elements in the first K-group of the product of two modular curves. It relates the image of these elements under the p-adic syntomic regulator (as described by Besser (2012)) to the special values at the near-central point of Hidas p-adic Rankin L-function attached to two Hida families of cusp forms.