arXiv subjects

Richard Pink

Publications and source records attributed to Richard Pink.

26 records · Page 2Linked to original sources

arXiv2013

Finiteness and liftability of postcritically finite quadratic morphisms in arbitrary characteristic

We show that for any integer n and any field k of characteristic different from 2 there are at most finitely many isomorphism classes of quadratic morphisms from the projective line over k to itself with a finite postcritical orbit of size n. As a consequence we prove that every postcritically finite quadratic morphism over a field of positive characteristic can be lifted to characteristic zero with the same combinatorial type of postcritical orbit. The associated profinite geometric monodromy group is therefore the same as in characteristic zero, where it can be described explicitly by generators as a self-similar group acting on a regular rooted binary tree.

math.AG↗

arXiv2012

Compactification of Drinfeld modular varieties and Drinfeld Modular Forms of Arbitrary Rank

We give an abstract characterization of the Satake compactification of a general Drinfeld modular variety. We prove that it exists and is unique up to unique isomorphism, though we do not give an explicit stratification by Drinfeld modular varieties of smaller rank which is also expected. We construct a natural ample invertible sheaf on it, such that the global sections of its $k$-th power form the space of (algebraic) Drinfeld modular forms of weight $k$. We show how the Satake compactification and modular forms behave under all natural morphisms between Drinfeld modular varieties; in particular we define Hecke operators. We give explicit results in some special cases.

math.AG↗

arXiv2012

Kummer Theory for Drinfeld Modules

Let ϕ be a Drinfeld A-module of characteristic p0 over a finitely generated field K. Previous articles determined the image of the absolute Galois group of K up to commensurability in its action on all prime-to-p0 torsion points of ϕ, or equivalently, on the prime-to-p0 adelic Tate module of ϕ. In this article we consider in addition a finitely generated torsion free A-submodule M of K for the action of A through ϕ. We determine the image of the absolute Galois group of K up to commensurability in its action on the prime-to-p0 division hull of M, or equivalently, on the extended prime-to-p0 adelic Tate module associated to ϕ and M.

math.NT↗

arXiv2012

Adelic Openness for Drinfeld Modules in Special Characteristic

For any Drinfeld module of special characteristic p0 over a finitely generated field, we study the associated adelic Galois representation at all places different from p0 and \infty, and determine the image of the geometric Galois group up to commensurability.

math.NT↗

arXiv2011

Compactification of a Drinfeld Period Domain over a Finite Field

We study a certain compactification of the Drinfeld period domain over a finite field which arises naturally in the context of Drinfeld moduli spaces. Its boundary is a disjoint union of period domains of smaller rank, but these are glued together in a way that is dual to how they are glued in the compactification by projective space. This compactification is normal and singular along all boundary strata of codimension $\ge2$. We study its geometry from various angles including the projective coordinate ring with its Hilbert function, the cohomology of twisting sheaves, the dualizing sheaf, and give a modular interpretation for it. We construct a natural desingularization which is smooth projective and whose boundary is a divisor with normal crossings. We also study its quotients by certain finite groups.

math.AG↗

arXiv2011

Algebraic zip data

An algebraic zip datum is a tuple $\CZ := (G,P,Q,ϕ)$ consisting of a reductive group $G$ together with parabolic subgroups $P$ and $Q$ and an isogeny $ϕ\colon P/R_uP\to Q/R_uQ$. We study the action of the group $E := \{(p,q)\in P{\times}Q | ϕ(π_{P}(p)) =π_Q(q)\}$ on $G$ given by $((p,q),g)\mapsto pgq^{-1}$. We define certain smooth $E$-invariant subvarieties of $G$, show that they define a stratification of $G$. We determine their dimensions and their closures and give a description of the stabilizers of the $E$-action on $G$. We also generalize all results to non-connected groups. We show that for special choices of $\CZ$ the algebraic quotient stack $[E \backslash G]$ is isomorphic to $[G \backslash Z]$ or to $[G \backslash Z']$, where $Z$ is a $G$-variety studied by Lusztig and He in the theory of character sheaves on spherical compactifications of $G$ and where $Z'$ has been defined by Moonen and the second author in their classification of $F$-zips. In these cases the $E$-invariant subvarieties correspond to the so-called "$G$-stable pieces" of $Z$ defined by Lusztig (resp. the $G$-orbits of $Z'$).

math.RT↗

arXiv2004

Monodromy Groups associated to Non-Isotrivial Drinfeld Modules in Generic Characteristic

Let $ϕ$ be a non-isotrivial family of Drinfeld A-modules of rank r in generic characteristic with a suitable level structure over a connected smooth algebraic variety X. Suppose that the endomorphism ring of $ϕ$ is equal to A. Then we show that the closure of the analytic fundamental group of X in $SL_r(\mathbb{A}_F^f)$ is open, where $\mathbb{A}_F^f$ denotes the ring of finite adeles of the quotient field F of A. From this we deduce two further results: (1) If X is defined over a finitely generated field extension of F, the image of the arithmetic étale fundamental group of X on the adelic Tate module of $ϕ$ is open in $GL_r(\mathbb{A}_F^f)$. (2) Let $ψ$ be a Drinfeld A-module of rank r defined over a finitely generated field extension of F, and suppose that $ψ$ cannot be defined over a finite extension of F. Suppose again that the endomorphism ring of $ψ$ is A. Then the image of the Galois representation on the adelic Tate module of $ψ$ is open in $GL_r(\mathbb{A}_F^f)$. Finally, we extend the above results to the case of arbitrary endomorphism rings.

math.NT↗

arXiv2002

On Hrushovski's proof of the Manin-Mumford conjecture

The Manin-Mumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski gave a different proof using model theory. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. In this paper we prove that intervening result using classical algebraic geometry alone. Altogether, this yields a new proof of the Manin-Mumford conjecture using only classical algebraic geometry.

math.NT↗