Richard Pink
Publications and Preprints
Abstract: We give an effective
algorithm to determine the endomorphism ring of a Drinfeld
module, both over its field of definition and over a separable
or algebraic closure thereof. Using previous results we deduce
an effective description of the image of the adelic Galois
representation associated to the Drinfeld module, up to
commensurability. We also give an effective algorithm to
decide whether two Drinfeld modules are isogenous, again both
over their field of definition and over a separable or
algebraic closure thereof.
Abstract: Consider the polynomial ring in any finite number of variables over the complex numbers, endowed with the l1-norm on the system of coefficients. Its completion is the Banach algebra of power series that converge absolutely on the closed polydisc. Whereas the strong Hilbert Nullstellensatz does not hold for Banach algebras in general, we show that it holds for ideals in the polynomial ring that are closed for the indicated norm. Thus the corresponding statement holds at least partially for the associated Banach algebra. We also describe the closure of an ideal in small cases.
Abstract: To every automorphism w of an infinite rooted regular binary tree we associate a two variable generating function Φ_w that encodes information on the orbit structure of w. We prove that this is a rational function if w can be described by finitely many recursion relations of a particular form. We show that this condition is satisfied for all elements of the discrete iterated monodromy group Γ associated to a postcritically finite quadratic polynomial over C. For such Γ we also prove that there are only finitely many possibilities for the denominator of Φ_w, and we describe a procedure to determine their lowest common denominator.
Abstract: We study in detail the
profinite group G
arising as geometric étale iterated monodromy group of an
arbitrary quadratic morphism f with an infinite postcritical orbit over a
field of characteristic different from two. This is a
self-similar closed subgroup of the group of automorphisms of
a regular rooted binary tree. In many cases it is equal to the
automorphism group of the tree, but there remain some
interesting cases where it is not. In these cases we prove
that the conjugacy class of G
depends only on the combinatorial type of the postcritical
orbit of f. We also
determine the Hausdorff dimension and the normalizer of G. This result is then
used to describe the arithmetic étale iterated monodromy group
of f.
The methods used mostly group theoretical and of the same type
as in a previous article of the same author dealing with
quadratic polynomials with a finite postcritical orbit. The
results on abstract self-similar profinite groups acting on a
regular rooted binary tree may be of independent interest.
Abstract: We study in detail the
profinite group G
arising as geometric étale iterated monodromy group of an
arbitrary quadratic polynomial over a field of characteristic
different from two. This is a self-similar closed subgroup of
the group of automorphisms of a regular rooted binary tree.
(When the base field is C
it is the closure of the finitely generated iterated monodromy
group for the usual topology which is also often studied.)
Among other things we prove that the conjugacy class and hence
the isomorphism class of G
depends only on the combinatorial type of the post-critical
orbit of the polynomial.
We represent a chosen instance of G by explicit recursively defined
generators. The uniqueness up to conjugacy depends on a
certain semirigidity property, which ensures that arbitrary
conjugates of these generators under the automorphism group of
the tree always generate a subgroup that is conjugate to G. We determine the
Hausdorff dimension, the maximal abelian factor group, and the
normalizer of G
using further explicit generators. The description of the
normalizer is then used to describe the arithmetic étale
iterated monodromy group of the quadratic polynomial.
The methods used are purely group theoretical and do not
involve fundamental groups over C at all.
Abstract: 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.
Abstract: A basic problem of algebra is the determination of the Galois group of a separable polynomial in one variable. If the coefficients of the polynomial lie in a finite field of cardinality q^n, this Galois group is generated by the image of the Frobenius automorphism x\mapsto x^{q^n}. If, in addition, the polynomial has the special form a_0X+a_1X^q+...+a_nX^{q^n} with a_0,a_d\not=0, the action of Frobenius is represented by a matrix in GL_d(F_q). The current article determines which matrices can appear in this way for given q, n, and d. In some sense this solves a variant of the ``inverse problem of Galois theory'' over finite fields.
Abstract: Let F_q be a fixed finite field
of cardinality q. An F-zip over a scheme S over F_q is a
certain object of semi-linear algebra consisting of a locally
free sheaf of O_S-modules with a descending filtration and an
ascending filtration and a Frob_q-twisted isomorphism between
the respective graded sheaves. In this article we define and
systematically investigate what might be called “F-zips with a
G-structure”, for an arbitrary reductive linear algebraic
group G over F_q.
These objects come in two incarnations. One incarnation is an
exact F_q-linear tensor functor from the category of finite
dimensional representations of G over F_q to the category of
F-zips over S. Locally any such functor has a type χ, which is
a cocharacter of G_k for a finite extension k of F_q that
determines the ranks of the graded pieces of the filtrations.
The other incarnation is a certain G-torsor analogue of the
notion of F-zips. We prove that both incarnations define
stacks that are naturally equivalent to a quotient stack of
the form [E_{G,χ}\G_k] that was studied in our earlier paper
[19]. By the results of [19] they are therefore smooth
algebraic stacks of dimension 0 over k. Using [19] we can also
classify the isomorphism classes of such objects over an
algebraically closed field, describe their automorphism
groups, and determine which isomorphism classes can degenerate
into which others.
For classical groups we can deduce the corresponding results
for twisted or untwisted symplectic, orthogonal, or unitary
F-zips, a part of which has been described before in [17]. The
results can be applied to the algebraic de Rham cohomology of
smooth projective varieties (or generalizations thereof) and
to truncated Barsotti-Tate groups of level 1. In addition, we
hope that our systematic group theoretical approach will help
to understand the analogue of the Ekedahl-Oort stratification
of the special fibers of arbitrary Shimura varieties.
Abstract: 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.
Abstract: 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 ∞, and determine the image of the geometric Galois
group up to commensurability.
Abstract: 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.
Abstract: An algebraic zip datum is a tuple Z = (G, P, Q, φ) consisting of a reductive group G together with parabolic subgroups P and Q and an isogeny φ: P/RuP → Q/RuQ. We study the action of the group EZ := {(p,q) ∈ P×Q | φ(πP (p)) = πQ(q)} on G given by ((p, q), g) → pgq^{−1}. We define certain smooth EZ -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 EZ -action on G. We also generalize all results to non-connected groups. We show that for special choices of Z the algebraic quotient stack [EZ \G] is isomorphic to [G\Z] or to [G\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 EZ-invariant subvarieties correspond to the so-called “G-stable pieces” of Z defined by Lusztig (resp. the G-orbits of Z′).
Abstract: 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.
Abstract: We
prove the isogeny conjecture for A-motives over finitely generated fields K of transcendence degree
\le 1. This conjecture says that for any semisimple A-motive M over K, there exist only
finitely many isomorphism classes of A-motives M'
over K for which
there exists a separable isogeny M' \to M. The result is in precise analogy
to known results for abelian varieties and for Drinfeld
modules and will have strong consequences for the ℘-adic and
adelic Galois representations associated to M. The method makes
essential use of the Harder-Narasimhan filtration for locally
free coherent sheaves on an algebraic curve.
Abstract: Let φ be a Drinfeld A-module of arbitrary rank and generic characteristic over a finitely generated field K. If the endomorphism ring of φ over an algebraic closure of K is equal to A, we prove that the image of the adelic Galois representation associated to φ is open.
The Isogeny Conjecture for
t-Motives Associated to Direct Sums of Drinfeld Modules
(with Matthias Traulsen)
J.
Number Theory 117 (2006), no.
2, 355-375.
Abstract: Let
K be a finitely
generated field of transcendence degree 1 over a finite field.
Let M be a t-motive over K of characteristic ℘_{0},
which ist semisimple up to isogeny. The isogeny conjecture for M says that there are
only finitely many isomorphism classes of t-motives M' over K, for which there exists
a separable isogeny M'
→ M of degree not
divisible by ℘_{0}. For the t-motive associated to a Drinfeld module
this was proved by Taguchi. In this article we prove it for
the t-motive
associated to any direct sum of Drinfeld modules of
characteristic ℘_{0} different from 0.
On
Weil restriction of reductive groups and a theorem of
Prasad
Math. Z. 248 (2004), no. 3, 449-457.
Abstract: Let
G be a connected simple semisimple algebraic group over
a local field F of arbitrary characteristic. In a
previous article by the author the Zariski dense compact
subgroups of G(F) were classified. In the
present paper this information is used to give another proof
of a theorem of Prasad (also proved by Margulis) which asserts
that, if G is isotropic, every non-discrete closed
subgroup of finite covolume contains the image of G^{~}(F) where G^{~} denotes the
universal covering of G. This result played a central
role in Prasad's proof of strong approximation. The present
proof relies on some basic properties of Weil restrictions
over possibly inseparable field extensions, which are also
proved here.
Abstract:
A theorem of Green, Lazarsfeld and Simpson (formerly a
conjecture of Beauville and Catanese) states that certain
naturally defined subvarieties of the Picard variety of a
smooth projective complex variety are unions of translates of
abelian subvarieties by torsion points. Their proof uses
analytic methods. We refine and give a completely new proof of
their result. Our proof combines galois-theoretic methods and
algebraic geometry in positive characteristic. When the
variety has a model over a function field and its Picard
variety has no isotrivial factors, we show how to replace the
galois-theoretic results we need by results from model theory
(mathematical logic). Furthermore, we prove partial analogs of
the conjecture of Beauville and Catanese in positive
characteristic.
Abstract:
Let C be a complete non-archimedean valued
algebraically closed field of characteristic p>0 and consider the
punctured unit disc D
in C. Let q
be a power of p and
consider the arithmetic Frobenius automorphism σ: D → D, x |→ x^{q -1 }.
A sigma-bundle is a vector bundle F on D
together with an isomorphism σ^{*}F → F. The aim of this
article is to develop the basic theory of these objects and to
classify them. It is shown that every σ-bundle is isomorphic
to a direct sum of certain basic σ-bundles F_{d,r} which depend only on rational numbers d/r. This result has
close analogies with the classification of rational Dieudonne
modules and of vector bundles on the projective line or on an
elliptic curve. It has interesting consequences concerning the
uniformizability of Anderson's t-motives that will be treated in a future
paper.
Abstract:
Let A be a
semiabelian variety over an algebraically closed field of
arbitrary characteristic, endowed with a finite morphism ψ: A → A. In this paper we give
an essentially complete classification of all ψ-invariant
subvarieties of A.
For example, under some mild assumptions on (A,ψ) we prove that every
ψ-invariant subvariety is a finite union of translates of
semiabelian subvarieties. This result is then used to prove
the Manin-Mumford conjecture in arbitrary characteristic and
in full generality. Previously, it had been known only for the
group of torsion points of order prime to the characteristic
of K. The proofs
involve only algebraic geometry, though scheme theory and some
arithmetic arguments cannot be avoided.
Abstract:
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.
Abstract:
Let X be an
irreducible smooth projective curve over an algebraically
closed field of characteristic p>0. Let k be either a finite field of characteristic
p or a local field of
residue characteristic p.
Let F be a
constructible etale sheaf of k-vector spaces on X. Suppose that there exists a finite Galois
covering π: Y → X such that the generic
monodromy of π*F is
pro-p and Y is ordinary. Under
these assumptions we derive an explicit formula for the
Euler-Poincaré characteristic c(X,F) in terms of easy local and global
numerical invariants, much like the formula of
Grothendieck-Ogg-Shafarevich in the case of different
characteristic. Although the ordinariness assumption imposes
severe restrictions on the local ramification of the covering
π, it is satisfied in interesting cases such as Drinfeld
modular curves.
Abstract:
Consider a finitely generated Zariski dense subgroup Γ of a
connected simple algebraic group G over a global field F. An important aspect of
strong approximation is the question of whether the closure of
Γ in the group of points of G
with coefficients in a ring of partial adeles is open. We
prove an essentially optimal result in this direction, based
on the condition that Γ is not discrete in that ambient group.
There are no restrictions on the characteristic of F or the type of G, and simultaneous
approximation in finitely many algebraic groups is also
studied. Classification of finite simple groups is not used.
Abstract: We
develop a general theory of mixed Hodge structures over local
or global function fields which in many ways resembles the
formalism of classical Hodge structures. Our objects consist
of a finite dimensional vector space together with a weight
filtration, but instead of a Hodge filtration we require finer
information. In order to obtain a reasonable category we
impose a semistability condition in the spirit of invariant
theory and prove that the resulting category is tannakian.
This allows us to define and analyze Hodge groups and
determine them in some cases.
The analogies with classical mixed Hodge structures range from
the role of semistability to the fact that both objects arise
from the analytic behavior of motives. The precise relation of
our objects with the analytic uniformization of Anderson's t-motives will be the
subject of a separate paper. For Hodge structures arising from
Drinfeld modules we can combine the present results with a
previous one on Galois representations, obtaining a precise
analogue of the Mumford-Tate conjecture.
Motives and Hodge Structures
over Function Fields
Talk at the Arbeitstagung 1997 in Bonn, 6 p.