Hodge Cycles on Abelian Varieties (Number Theory Seminar, Berkeley, Spring 2019)

HODGE CYCLES ON ABELIAN VARIETIES (NUMBER THEORY SEMINAR, BERKELEY, SPRING 2019)

SUG WOO SHIN AND KOJI SHIMIZU

The goal of this series of talks is to understand Hodge cycles on abelian varieties. We will read Deligne’s paper [Del82] carefully, discussing some of the key prerequisites. The paper uses various important and interesting topics in number theory and algebraic geometry, and the seminar will give you an opportunity to learn them. Every student is encouraged to give a talk. Note that Gross’ paper [Gro78] is related to our topic. He has a lecture note [Gro] with interesting historical remarks. The seminar website is available at https://math.berkeley.edu/~shimizu/seminar/2019S/ BNTS.html. In what follows, the symbol * indicates that the topic is accessible for beginners. Note that there is no meeting on March 6 due to the Arizona Winter School.

Lecture 1. Introduction and organization. Koji will outline the materials and distribute the weekly lectures to volunteers.

Lecture 2.* Review of cohomology. Review Betti cohomology (singular cohomology), de Rham cohomology and ´etalecohomology, and discuss the relations among them. Follow [Del82, pp.12-23] (until Proposition 1.5).

Lecture 3.* Absolute Hodge cycles. The main goal of this talk is to introduce absolute Hodge cycles. Start with the rest of [Del82, 1]. Then deﬁne absolute Hodge cycles and explain the Gauss–Manin connections, following [Del82, pp.27-32] and the references cited there.

Lecture 4. Main theorem and Principle B. Following [Del82, pp.32-39], state the main theorem on absolute Hodge cycles ([Del82, Main Theorem 2.1]) and prove one of the key ingredients called “Principle B”. Note that the proof of Theorem 2.15 uses Hodge structures, which will be explained later. So you may only explain the idea of the proof if you want.

Lecture 5.* Review of complex abelian varieties. Review abelian varieties over C and the Riemann form. Standard references are [Mum08, §1] and [BL04], but [GN09, §1] is probably shorter and easier to read. See also [Mil05, pp.317-319]. Since the symbol E is reserved for a CM ﬁeld in this seminar, use ψ for the Riemann form in your talk.

Lecture 6. Hodge structures and the Mumford–Tate groups. Introduce Hodge structures and mention the relation between complex abelian varieties and polarizable Hodge structures of type (−1, 0), (0, −1). Then discuss the Mumford–Tate groups, abelian varieties of CM-type and [Del82, Example 3.7]. The goal of this talk is to cover [Del82, pp.39-47], but the speaker may need to rearrange the order of materials so that the audience can follow the talk easily. See also [Mil05, pp.281-284, pp.319-320, pp.335-336].

Lecture 7. Absolute Hodge cycles on abelian varieties of CM-type I. Following [Del82, pp.47- 54], prove Principle A and then show that Hodge cycles are absolute Hodge cycles in a very special case ([Del82, Lemma 4.5]). 1 Lecture 8. Absolute Hodge cycles on abelian varieties of CM-type II. Following [Del82, pp.55- 62], discuss a moduli space of abelian varieties with additional structures and show that Hodge cycles are absolute Hodge cycles in a special case ([Del82, Theorem 4.8]). See also [Mil05, pp.291-294]. If you want, explain some ideas of Shimura varieties and/or the proof of the representability of the moduli space in the algebraic setting. For the latter, [MFK94, §6] is the standard reference. See also [GN09, 2.1-2.3] and [Hid04, §6] for a nice summary.

Lecture 9. Absolute Hodge cycles on abelian varieties of CM-type III. Following [Del82, §5], prove that Hodge cycles on abelian varieties of CM-type are absolute Hodge cycles.

Lecture 10. Absolute Hodge cycles on general abelian varieties. Following [Del82, §6], prove that Hodge cycles on general abelian varieties are absolute Hodge cycles. The proof uses a moduli space of abelian varieties with additional structures. Then discuss corollaries. If the time permits, feel free to discuss related topics.

Lecture 11.* Algebraicity of values of the Γ-function I. Following [Del82, pp.77-88], explain the calculation of the cohomologies of the Fermat hypersurfaces.

Lecture 12. Algebraicity of values of the Γ-function II. Following [Del82, pp.88-96], explain the algebraicity of products of special values of the Γ-functions.

Lecture 13. Related Topics. If the time permits, we will discuss related topics.

References [BL04] Christina Birkenhake and Herbert Lange, Complex abelian varieties, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. MR 2062673 [Del82] Pierre Deligne, Hodge cycles on abelian varieties, Hodge cycles, motives, and Shimura varieties (Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, eds.), Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982, pp. ii+414. MR 654325 [GN09] Alain Genestier and Bao ChâuNgô, Lectures on Shimura varieties, Autour des motifs—Ecole´ d’été Franco-Asiatique de GéométrieAlgébriqueet de Théoriedes Nombres/Asian-French Summer School on Algebraic Geometry and Number Theory. Volume I, Panor. Synthèses,vol. 29, Soc. Math. France, Paris, 2009, pp. 187–236. MR 2730658 [Gro] Benedict H. Gross, On the periods of abelian varieties. [Gro78] , On the periods of abelian integrals and a formula of Chowla and Selberg, Invent. Math. 45 (1978), no. 2, 193–211, With an appendix by David E. Rohrlich. MR 0480542 [Hid04] Haruzo Hida, p-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004. MR 2055355 [MFK94] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, third ed., Ergebnisse der Math- ematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer- Verlag, Berlin, 1994. MR 1304906 [Mil05] J. S. Milne, Introduction to Shimura varieties, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 265–378. MR 2192012 [Mum08] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008, With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition. MR 2514037