Modular Forms and Calabi-Yau Varieties

CRM proceedings & lecture notes, 2000

Regulators and characteristic classes of flat bundles Harris, Bruno and Wang, Bin, Height pairing asymptotics and Bott-Chem forms Kato, Kazuya and Usui, Sampei, Logarithmic Hodge structures and classifying spaces Chow Groups and Motives Asakura, Masanori, Motives and algebraic deRham cohomology Burgos Gil, Jose Ignacio, Hermitian vector bundles and characteristic classes de Jeu, Rob, A counterexample to a conjecture of Beilinson Hanamura, Masaki, The mixed motive of a projective variety Pedrini, Claudio, Bloch's conjecture and the K-theory of projective spaces Ramachandran, Niranjan, Jirom Jacobians to one-motives: Exposition of a conjecture of Deligne Saito, Shuji, Motives, algebraic cycles and Hodge theory Arithmetic Methods Doran, Charles, Picard-Fuchs uniformization: Modularity of the mifTOr maps and mifTOr-moonshine Goren, Eyal, Hilbert modular varieties in positive characteristic Goto, Yasuhiro, On the Neron-Severi groups of some K3 sur/aces van Hamel, Joost, Torsion zero-cycles and the Abel-Jacobi map over the real numbers Kimura, Ken-Ichiro, A remark on the Griffiths group of certain product varieties Nekovar, Jan, p-adic Abel-Jacobi maps and p-adic heights Shiho, Atsushi, Crystalline fundamental groups and p-adic Hodge theory vii Vlll Verrill, Helena and Yui, Noriko, Thompson series, and the mirror maps of certain pencils of K3 surfaces

1986

2016

Let F be a totally real field and χ an abelian totally odd character of F. In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt's conjecture holds, assuming that either there are at least two primes above p in F , or that a certain condition relating the L-invariants of χ and χ −1 holds. This condition on L-invariants is always satisfied when χ is quadratic. Contents S. DASGUPTA, H. DARMON, and R. POLLACK 4. Galois representations 477 4.1. Representations attached to ordinary eigenforms 477 4.2. Construction of a cocycle 480 References 482

2012

We begin with some general remarks. Let X be a smooth projective variety of dimension n over a field k. For any positive integer p < n, it is of

Journal of Number Theory, 2010

Let f be a weight two newform for Γ 1 (N) without complex multiplication. In this article we study the conductor of the absolutely simple factors B of the variety A f over certain number fields L. The strategy we follow is to compute the restriction of scalars Res L/Q (B), and then to apply Milne's formula for the conductor of the restriction of scalars. In this way we obtain an expression for the local exponents of the conductor N L (B). Under some hypothesis it is possible to give global formulas relating this conductor with N. For instance, if N is squarefree we find that N L (B) belongs to Z and N L (B) f dim B L

Pure and Applied Mathematics Quarterly, 2009

Fields Institute Communications, 2013

This paper presents the current status on modularity of Calabi-Yau varieties since the last update in 2003. We will focus on Calabi-Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic modularity and geometric modularity. These will include: (1) the modularity (automorphy) of Galois representations of Calabi-Yau varieties (or motives) defined over Q or number fields, (2) the modularity of solutions of Picard-Fuchs differential equations of families of Calabi-Yau varieties, and mirror maps (mirror moonshine), (3) the modularity of generating functions of invariants counting certain quantities on Calabi-Yau varieties, and (4) the modularity of moduli for families of Calabi-Yau varieties. The topic (4) is commonly known as geometric modularity.

We prove the isogeny property for special fibres of integral canonical models of compact Shimura varieties of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type. The approach used also shows that many crystalline cycles on abelian varieties over finite fields which are specializations of Hodge cycles, are algebraic. These two results have many applications. First, we prove a variant of the conditional Langlands--Rapoport conjecture for these special fibres. Second, for certain isogeny sets we prove a variant of the unconditional Langlands--Rapoport conjecture (like for many basic loci). Third, we prove that integral canonical models of compact Shimura varieties of Hodge type that are of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type, are closed subschemes of integral canonical models of Siegel modular varieties.

2002

1. (note. This conjecture was originally formulated for weight 2 newforms associated to modular elliptic curves). Obviously, this conjecture implies the weaker statement

Journal of the Institute of Mathematics of Jussieu, 2002

An exact control theorem is proven for nearly ordinary p-adic automorphic forms on symplectic and unitary groups over totally real fields if the algebraic group is split at p. In particular, a given nearly ordinary holomorphic Hecke eigenform can be lifted to a family of holomorphic Hecke eigenforms indexed by weights of the standard maximal split torus of the group. Their q-expansion coefficients are Iwasawa functions on the Iwasawa algebra of Zp-points of the split torus. The method is applicable to any reductive algebraic groups yielding Shimura varieties of PEL type under mild assumptions on the existence of integral toroidal compactification of the variety. Even in the Hilbert modular case, the result contains something new: freeness of the universal nearly ordinary Hecke algebra over the Iwasawa algebra, which eluded my scrutiny when I studied general theory of the p-adic Hecke algebra in the 1980s.