Math 847: Topics in Algebra---Gross-Zagier formula

Prerequisites:

Basics on algebraic number theory, and modular forms. A little knowledge on modular curves, elliptic curves, and adeles will be helpful.

Instructor:

Tonghai Yang

Time and Place:

MWF 2:25-3:15pm

textbooks:

None

Course Content:

Gross and Zagier discovered about 30 years ago a deep and beautiful formula---the Gross-Zagier formula---which reveals a deep connection between the arithmetic on an elliptic curve and the central derivative of its L-function (analysis). It has far reaching applications, such as the Birch and Swinnerton-Dyer conjecture. A lot of progress and generalization has been made ever since, most notably by Shou-Wu Zhang (to totally real number fields), Bertolini and Darmon (p-adic version), Kudla, Rapoport, and myself (arithmetic Siegel-Weil formula). In this course, I will mainly discuss the original gross-Zagier formula and very recent development of Zhang and his students Xinyi Yuan and Wei Zhang on unifying the Gross-Zagier formula with a formula of Waldspurger on the central L-value, following Gross's suggestion. If times allows, we will discuss other development on Gross-Zagier formula too.

Basic References:

1. H. Darmon and S.W. Zhang (eds): Heegner points and Rankin $L$-series, 191--214, Math. Sci. Res. Inst. Publ., 49, Cambridge Univ. Press, Cambridge, 2004.

2. Gross, Benedict H. Heights and $L$-series. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 425--433, Amer. Math. Soc., Providence, RI, 1987. (Reviewer: Joseph H. Silverman) 11G40 (11G05)

3. Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of $L$-series. Invent. Math. 84 (1986), no. 2, 225--320.

4. X. Yuan, W. Zhang, and S.W. Zhang, Heights of CM points I, Gross–Zagier formula, preprint (2008)
http://www.math.columbia.edu/~szhang/papers/HCMI.pdf

5. Waldspurger, J.-L. Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie. (French) [Values of certain automorphic $L$-functions at their center of symmetry] Compositio Math. 54 (1985), no. 2, 173--242. (Reviewer: Stephen Gelbart) 11F70 (11F67 22E55).

6. Gross, Benedict H. Heegner points on $X\sb 0(N)$. Modular forms (Durham, 1983), 87--105, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984. (Reviewer: Loren D. Olson) 11G05 (11G16