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Richard Pink

Publications and source records attributed to Richard Pink.

At least 19 recordsLinked to original sources
arXiv2026

Schur $\sigma$-groups of type $(3,3)$ for $p=3$

For any imaginary quadratic field $K$, the Galois group $G_K$ of its maximal unramified pro-$3$-extension is a Schur $\sigma$-group. If this has Zassenhaus type $(3,3)$, there are 13 possibilities for the isomorphism class of the finite quotient $G_K/D_4(G_K)$. We prove that for 10 of these 13 cases $G_K$ is either finite or isomorphic to an open subgroup of a form of $\mathop{\rm PGL}_2$ over $\mathbb{Q}_3$. Combined with the Fontaine-Mazur conjecture, or with earlier work on an analogue of the Cohen--Lenstra heuristic for Schur $\sigma$-groups, this lends credence to the "if" part of a conjecture of McLeman. Using explicit computations of triple Massey products, we also test the heuristic for all imaginary quadratic fields $K$ with $d(G_K)=2$ and discriminant $-10^8 < d_K < 0$ and find a reasonably good agreement.

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arXiv2025

Weil representations associated to isocrystals over function fields

Every Anderson $A$-motive $M$ over a field determines a compatible system of Galois representations on its Tate modules at almost all primes of $A$. This adapts easily to $F$-isocrystals, which are rational analogues of $A$-motives for the global function field $F:=\operatorname{Quot}(A)$. We extend this compatible system by constructing a Weil group representation associated to $M$ for every place of $F$. To this end we generalize the Tate module construction to a tensor functor on $F_{\mathfrak{p}}$-isocrystals that are not necessarily pure. To prove that this yields a compatible system, we work out how that construction behaves under reduction of $M$. As an offshoot we obtain a new kind of $\wp$-adic Weil representations associated to Drinfeld modules of special characteristic $\wp$.

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arXiv2025

A Cohen-Lenstra Heuristic for Schur $\sigma$-Groups

For any odd prime $p$ and any imaginary quadratic field $K$, the $p$-tower group $G_K$ associated to $K$ is the Galois group over $K$ of the maximal unramified pro-$p$-extension of $K$. This group comes with an action of a finite group $\{1,\sigma\}$ of order $2$ induced by complex conjugation and is known to possess a number of other properties, making it a so-called Schur $\sigma$-group. Its maximal abelian quotient is naturally isomorphic to the $p$-primary part of the narrow ideal class group of ${\mathcal O}_K$, and the Cohen-Lenstra heuristic gives a probabilistic explanation for how often this group is isomorphic to a given finite abelian $p$-group. The present paper develops an analogue of this heuristic for the full group $G_K$. It is based on a detailed analysis of general pro-$p$-groups with an action of $\{1,\sigma\}$, which we call $\sigma$-pro-$p$-groups. We construct a probability space whose underlying set consists of $\sigma$-isomorphism classes of weak Schur $\sigma$-groups and whose measure is constructed from the principle that the relations defining $G_K$ should be randomly distributed according to the Haar measure. We also compute the measures of certain basic subsets, the result being inversely proportional to the order of the $\sigma$-automorphism group of a certain finite $\sigma$-$p$-group, as has often been observed before. Finally, we show that the $\sigma$-isomorphism classes of weak Schur $\sigma$-groups for which each open subgroup has finite abelianization form a subset of measure $1$.

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arXiv2025

Schur $\sigma$-groups of type (3,3)

For any odd prime $p$, the Galois group of the maximal unramified pro-$p$-extension of an imaginary quadratic field is a Schur $\sigma$-group. But Schur $\sigma$-groups can also be constructed and studied abstractly. We prove that if $p>3$, any Schur $\sigma$-group of Zassenhaus type $(3,3)$, for which every open subgroup has finite abelianization, is isomorphic to an open subgroup of a form of ${\rm PGL}_2$ over ${\mathbb Q}_p$. Combined with earlier work on an analogue of the Cohen-Lenstra heuristic for Schur $\sigma$-groups, or with the Fontaine-Mazur conjecture, this lends credence to the ``if'' part of a conjecture of McLeman.

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arXiv2024

Reduction of Hyperelliptic Curves in Residue Characteristic 2

Consider a hyperelliptic curve of genus $g$ over a field $K$ of characteristic zero. After extending $K$ we can view it as a marked curve with its $2g+2$ Weierstrass points. We present an explicit algorithm to compute the stable reduction of this marked curve for a valuation of residue characteristic $2$ over a finite extension of $K$. In the cases $g\le2$ we work out relatively simple conditions for the structure of this reduction.

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arXiv2024

Local Kummer theory for Drinfeld modules

Let $\phi$ be a Drinfeld $A$-module of finite residual characteristic $\bar{\mathfrak{p}}$ over a local field $K$. We study the action of the inertia group of $K$ on a modified adelic Tate module $\smash{T^\circ_{\text{ad}}}(\phi)$ which differs from the usual adelic Tate module only at the $\bar{\mathfrak{p}}$-primary component. After replacing $K$ by a finite extension we can assume that $\phi$ is the analytic quotient of a Drinfeld module $\psi$ of good reduction by a lattice $M\subset K$. The image of inertia acting on $T^\circ_{\text{ad}}(\phi)$ is then naturally a subgroup of $\operatorname{Hom}_A(M,T^\circ_\text{ad}(\psi))$. This subgroup is described by a canonical local Kummer pairing that we study extensively in this article. In particular we give an effective formula for the image of inertia up to finite index, and obtain a necessary and sufficient condition for this image to be open. We also determine the image of the ramification filtration

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arXiv2021

Reduction of Hyperelliptic Curves in Characteristic $\not=2$

Let $K$ be the quotient field of a discrete valuation ring $R$ with residue characteristic $\not=2$, and let $C$ be a hyperelliptic curve over $K$. We assume that all geometric branch points of the double covering $C\twoheadrightarrow{\mathbb P}^1_K$ are rational and mark both $C$ and ${\mathbb P}^1_K$ with these branch points. After possibly replacing $R$ by a ramified extension of degree $2$, we give a direct construction for the stable model of $C$ as a marked curve over $R$. We deduce that the closed fiber of this stable model is determined completely by the closed fiber of the stable model of the marked ${\mathbb P}^1_K$. In particular, the dual graph and other information for the former can be read off directly from the corresponding information for the latter.

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arXiv2019

Compactification of Drinfeld Moduli Spaces as Moduli Spaces of $A$-Reciprocal Maps and Consequences for Drinfeld Modular Forms

We construct a compactification of the moduli space of Drinfeld modules of rank $r$ and level $N$ as a moduli space of $A$-reciprocal maps. This is closely related to the Satake compactification, but not exactly the same. The construction involves some technical assumptions on $N$ that are satisfied for a cofinal set of ideals $N$. In the special case $A={\mathbb F}_q[t]$ and $N=(t^n)$ we obtain a presentation for the graded ideal of Drinfeld cusp forms of level $N$ and all weights and can deduce a dimension formula for the space of cusp forms of any weight. We expect the same results in general, but the proof will require more ideas.

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arXiv2018

Drinfeld modular forms of arbitrary rank, Part I: Analytic Theory

This is the first of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank r. In the present part, we develop the analytic theory. Most of the work goes into defining and studying the u-expansion of a weak Drinfeld modular form, whose coefficients are weak Drinfeld modular forms of rank r-1. Based on that we give a precise definition of when a weak Drinfeld modular form is holomorphic at infinity and thus a Drinfeld modular form in the proper sense.

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arXiv2018

Drinfeld modular forms of arbitrary rank, Part II: Comparison with Algebraic Theory

This is the second of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present part, we compare the analytic theory with the algebraic one that was begun in a paper of the third author. For any arithmetic congruence subgroup and any integral weight we establish an isomorphism between the space of analytic modular forms with the space of algebraic modular forms defined in terms of the Satake compactification. From this we deduce the important result that this space is finite dimensional.

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arXiv2018

Drinfeld modular forms of arbitrary rank, Part III: Examples

This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define Eisenstein series, as well as the action of Hecke operators upon them, coefficient forms and discriminant forms. In the special case A=F_q[t] we show that all modular forms for GL_r(Γ(t)) are generated by certain weight one Eisenstein series, and all modular forms for GL_r(A) and SL_r(A) are generated by certain coefficient forms and discriminant forms. We also compute the dimensions of the spaces of such modular forms.

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arXiv2016

Finding Endomorphisms of Drinfeld modules

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.

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arXiv2016

The Strong Nullstellensatz for Certain Normed Algebras

Consider the polynomial ring in any finite number of variables over the complex numbers, endowed with the $\ell_1$-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.

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arXiv2014

$F$-zips with additional structure

An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module 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 $G$. These objects come in two incarnations. One incarnation is an exact linear tensor functor from the category of finite dimensional representations of $G$ to the category of $F$-zips over $S$. Locally any such functor has a type $χ$, which is a cocharacter of $G$. 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 $[E_{G,χ}\backslash G_k]$ that was studied in an earlier paper. By the results obtained there they are therefore smooth algebraic stacks of dimension 0 over $k$. Using our earlier results 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. The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof such as smooth proper Deligne-Mumford stacks) 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.

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arXiv2014

Orbit length generating functions of automorphisms of a rooted regular binary tree

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 Γassociated to a postcritically finite quadratic polynomial over C. For such Γwe 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.

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arXiv2013

Profinite iterated monodromy groups arising from quadratic polynomials

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.

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arXiv2013

Profinite iterated monodromy groups arising from quadratic morphisms with infinite postcritical orbits

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.

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