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Register a new userA note on Shimura subvarieties in the hyperelliptic Torelli locus
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We prove the non-existence of Shimura subvarieties of positive dimension contained generically in the hyperelliptic Torelli locus for curves of genus at least 8, which is an analogue of Oorts conjecture in the hyperelliptic case.
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By constructing suitable Borcherds forms on Shimura curves and using Schofers formula for norms of values of Borcherds forms at CM-points, we determine all the equations of hyperelliptic Shimura curves $X_0^D(N)$. As a byproduct, we also address the problem of whether a modular form on Shimura curves $X_0^D(N)/W_{D,N}$ with a divisor supported on CM-divisors can be realized as a Borcherds form, where $X_0^D(N)/W_{D,N}$ denotes the quotient of $X_0^D(N)$ by all the Atkin-Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.
We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or when the base field is C_1. In general, we show that a large (but in general proper) subgroup of the 2-torsion classes are given by the construction. Examples demonstrating applications to the arithmetic of hyperelliptic curves defined over number fields are given.
The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a Shimura variety is dense in the central leaf containing it. In this paper, we prove the conjecture for certain irreducible components of Newton strata in Shimura varieties of PEL type A and C, when $p$ is an unramified prime of good reduction. Our approach generalizes Chai and Oorts method for Siegel modular varieties.
We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.
We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovar and Scholl. This is achieved with the help of Morels work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.
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