Periods of Hilbert Modular Surfaces

Description

The author investigates Hodge structures of Hilbert modular surfaces and shows that the Hodge structure attached to a Hilbert modular surface is a direct sum of tensor products of two Hodge structures of weight 1. A corollary of this result allows the determination of algebraic cycles on a Hilbert modular surface by determining the endomorphism rings of abelian varieties. Using the formalism and the lifting of modular forms one can calculate the Picard number of certain Hilbert modular surfaces and thereby check Tate's conjecture in these cases. This monograph establishes a generalization of the theory of modular symbols and leads to an investiga- tion of the Hodge structures attached to primitive forms on higher dimensional modular varieties. Table of contents 0. Historical background, motivations and outline of the contents. I. Hodge structures attached to primitive forms of weight Oefinitions and notations. Hodge structures attached to Hilbert modular surfaces. Hodge structures attached to primitive forms of weight 2. Nonholomorphic involutive automorphisms of Hilbert modular surfaces. Period relation of Riemann-Hodge Chapter II. Abelian varieties attached to primitive forms. 5. Abelian varieties attached to Clifford algebras. 6. The period moduli of the isogeny classes A}e(f) and ACII(f). 7. Main theorem A and its corollaries. 8. Selfconjugate forms and algebraic cycles. 9. Abelian varieties attached to non-selfconjugate forms. Chapter III.Correspondence between real Nebentype elliptic modular forms and Hilbert modular forms. 10. Weil representation and theta series. 11. Construction of real Nebentype elliptic modular forms. 12. Construction of Hilbert modular forms. 13. The adjointness formula. Chapter IV. Period relation for the lifting of modular forms transcendental cycles. 14. Hodge structures attached to real Nebentype elliptic forms of weight 2. 15. Cons tructi on of 2-cyc 1 es. 16. Arithmetic index theorems. 17. Period relation for the Ooi-Naqanuma lifting and Main Theorem B. 18. Selfconju9ate forms and transcendental cycles. 19. Notes on l-adic cohomology groups of certain Hilbert modular surfaces. 20. Remarks. Bibliography