Speaker: Matteo Longo (Padova)
Title: On the p-adic variation of the Gross-Kohnen-Zagier theorem.
Abstract: Given an elliptic curve defined over the field of rational numbers, and given an imaginary quadratic field K, one may define (using the theory of complex multiplication) a K-rational point of the elliptic curve, called Heegner point. Heegner points are crucial tools for studying the arithmetic of elliptic curves; in particular, the celebrated theorem of Gross and Zagier relates, under suitable arithmetic assumptions, the Neron-Tate height of Heegner points and the leading term of the complex L-function of E over K. The Gross-Kohnen-Zagier theorem (GKZ), complementary to the Gross-Zagier theorem mentioned above, shows that, under suitable arithmetic assumptions, the relative positions of Heegner points, as the imaginary quadratic field varies while the elliptic curve stay fixed, are encoded by Fourier coefficients of a Jacobi form. Briefly, Heegner points are generating series for Jacobi forms. Several generalizations of the GKZ theorem are available in the literature, by Kudla (putting things in a general perspective by the formulation of a series of conjectures, known as Kudla program), Borcherds (using singular theta liftings) and Yuan-Zhang-Zhang (in the automorphic representation setting). In this seminar I will try to explore a further possible direction suggested by the GKZ theorem, where we make all objects vary in p-adic analytic families. This is a joint work with M.-H. Nicole.