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Register a new userA local model for the trianguline variety and applications
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We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendiecks simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial description of the completed local rings of the fiber over the weight map, etc. Combined with the patched Hecke eigenvariety (under the usual Taylor-Wiles assumptions), these results in turn have several global consequences: classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, existence of singularities on global Hecke eigenvarieties.
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Let $Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $lambda$ of the field of algebraic numbers which is prime to p, consider the $lambda$-adic pro-semisimple completion of $Pi$ as an object of the groupoid whose objects are pro-semisimple groups and whose morphisms are isomorphisms up to conjugation by elements of the neutral connected component. We prove that this object does not depend on $lambda$. If dim X=1 we also prove a crystalline generalization of this fact. We deduce this from the Langlands conjecture for function fields (proved by L. Lafforgue) and its crystalline analog (proved by T. Abe) using a reconstruction theorem in the spirit of Kazhdan-Larsen-Varshavsky. We also formulate two related conjectures, each of which is a reciprocity law involving a sum over all $l$-adic cohomology theories (including the crystalline theory for $l=p$).
We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time--space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system (called its degree), which is always bounded by the Bezout number of the system. Our algorithm works for fields of any characteristic, but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety.
If $pi: Y to X$ is an unramified double cover of a smooth curve of genus $g$, then the Prym variety $P_pi$ is a principally polarized abelian variety of dimension $g-1$. When $X$ is defined over an algebraically closed field $k$ of characteristic $p$, it is not known in general which $p$-ranks can occur for $P_pi$ under restrictions on the $p$-rank of $X$. In this paper, when $X$ is a non-hyperelliptic curve of genus $g=3$, we analyze the relationship between the Hasse-Witt matrices of $X$ and $P_pi$. As an application, when $p equiv 5 bmod 6$, we prove that there exists a curve $X$ of genus $3$ and $p$-rank $f=3$ having an unramified double cover $pi:Y to X$ for which $P_pi$ has $p$-rank $0$ (and is thus supersingular); for $3 leq p leq 19$, we verify the same for each $0 leq f leq 3$. Using theoretical results about $p$-rank stratifications of moduli spaces, we prove, for small $p$ and arbitrary $g geq 3$, that there exists an unramified double cover $pi: Y to X$ such that both $X$ and $P_pi$ have small $p$-rank.
Let $F$ be a non-archimedean local field with residue field $mathbb{F}_q$ and let $G = GL_2/F$. Let $mathbf{q}$ be an indeterminate and let $H^{(1)}(mathbf{q})$ be Vigneras generic pro-p Iwahori-Hecke algebra of the p-adic group $G(F)$. Let $V_{widehat{G}}$ be the Vinberg monoid of the dual group of $G$. We establish a generic version for $H^{(1)}(mathbf{q})$ of the Kazhdan-Lusztig-Ginzburg antispherical representation, the Bernstein map and the Satake isomorphism. We define the flag variety for the monoid $V_{widehat{G}}$ and establish the characteristic map in its equivariant K-theory. These generic constructions recover the classical ones after the specialization $mathbf{q} = q in mathbb{C}$. At $mathbf{q} = q = 0 inoverline{mathbb{F}}_q$, the antispherical map provides a dual parametrization of all the irreducible $H^{(1)}_{overline{mathbb{F}}_q}(0)$-modules. This work supersedes our earlier work arXiv:1907.08808. We explain the relationship between the two articles in the introduction.
This is the sequel to arXiv:2007.01364v1. Let $F$ be any local field with residue characteristic $p>0$, and $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$ be the mod $p$ pro-$p$-Iwahori Hecke algebra of $mathbf{GL_2}(F)$. In arXiv:2007.01364v1 we have constructed a parametrization of the $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$-modules by certain $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Satake parameters, together with an antispherical family of $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$-modules. Here we let $F=mathbb{Q}_p$ (and $pgeq 5$) and construct a morphism from $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Satake parameters to $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Langlands parameters. As a result, we get a version in families of Breuils semisimple mod $p$ Langlands correspondence for $mathbf{GL_2}(mathbb{Q}_p)$ and of Pav{s}k={u}nas parametrization of blocks of the category of mod $p$ locally admissible smooth representations of $mathbf{GL_2}(mathbb{Q}_p)$ having a central character. The formulation of these results is possible thanks to the Emerton-Gee moduli space of semisimple $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-representations of the Galois group ${rm Gal}(overline{mathbb{Q}}_p/ mathbb{Q}_p)$.
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