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Register a new userA Global Crystalline Period Map
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The crystalline period map is a tool for linearizing $p$-divisible groups. It has been applied to study the Langlands correspondences, and has possible applications to the homotopy groups of spheres. The original construction of the period map is inherently local. We present an alternative construction, giving a map on the entire moduli stack of $p$-divisbile groups, up to isogeny, which specializes to the original local construction.
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Let G be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible modules over the crystalline distribution algebra of G in terms of overconvergent isocrystals on locally closed subspaces in the (formal) flag variety of G. We treat the case of SL(2) as an example.
We consider stacks of filtered phi-modules over rigid analytic spaces and adic spaces. We show that these modules parametrize p-adic Galois representations of the absolute Galois group of a p-adic field with varying coefficients over an open substack containing all classical points. Further we study a period morphism (defined by Pappas and Rapoport) from a stack parametrizing integral data and determine the image of this morphism.
This is an improved version of the eprint previously entitled Unexpected isomorphisms between hyperkahler fourfolds. We study smooth projective hyperkahler fourfolds that are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. They are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksys Torelli theorem implies that their period map is an open embedding. Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We also prove that infinitely many Heegner divisors in a given period space have the property that their general points correspond to fourfolds which are isomorphic to Hilbert squares of a K3 surfaces, or to double EPW sextics. In two appendices, we determine the groups of biregular or birational automorphisms of various projective hyperkahler fourfolds with Picard number 1 or 2.
Let ${mathcal M}_{g,n}$ denote the moduli space of smooth, genus $ggeq 1$ curves with $ngeq 0$ marked points. Let ${mathcal A}_h$ denote the moduli space of $h$-dimensional, principally polarized abelian varieties. Let $ggeq 4$ and $hleq g$. If $F:{mathcal M}_{g,n}to{mathcal A}_h$ is a nonconstant holomorphic map then $h=g$ and $F$ is the classical period mapping, assigning to a Riemann surface $X$ its Jacobian.
We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.
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