Ebook: Periods of Hilbert Modular Surfaces

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