2013 Theses Doctoral

# p-adic Heights of Heegner points on Shimura curves

Let f be a primitive Hilbert modular form of weight 2 and level N for the totally real field F, and let p be an odd rational prime such that f is ordinary at all primes dividing p. When E is a CM extension of F of relative discriminant prime to Np, we give an explicit construction of the p-adic Rankin-Selberg L-function L_p(f_E,-) and prove that when the sign of its functional equation is -1, its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated to f. This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when F=Q and N satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.

## Subjects

## Files

- Disegni_columbia_0054D_11281.pdf application/pdf 522 KB Download File

## More About This Work

- Academic Units
- Mathematics
- Thesis Advisors
- Zhang, Shou-Wu
- Degree
- Ph.D., Columbia University
- Published Here
- May 1, 2013