arXiv subjects

Frauke M. Bleher

Publications and source records attributed to Frauke M. Bleher.

At least 19 recordsLinked to original sources

arXiv2026

On the radii of Voronoi cells of rings of integers

Since the time of Minkowski a basic problem in number theory has been to find lower bounds for the absolute value $Δ(K)$ of the discriminant of a number field $K$ in terms of the degree $n(K)$ of $K$. In this paper we study another measure of the size of $K$ given by the covering radius $μ(K)$ of the ring of integers $O_K$ of $K$. Here $μ(K)$ is the $L^2$ radius $||V_2(K)||_2$ of the $L^2$ Voronoi cell $V_2(K)$ of $O_K$, where $V_2(K)$ is the set of points in $\mathbb{R} \otimes_{\mathbb{Q}} K$ that are at least as close to the origin as they are to any non-zero element of $O_K$. To put a limit on what lower bounds one can prove for $μ(K)$ in terms of $n(K)$, we study infinite families of $K$ of increasing degree for which $μ(K)$ can be bounded above by an explicit power of $n(K)$. We also study analogous questions when the $L^2$ norm is replaced by the $L^\infty$ norm.

math.NT↗

arXiv2026

Explicit Approximations to Class Field Towers

We answer a question of Peikert and Rosen by giving for each $ε> 0$ an efficient construction of infinite families of number fields $N$ such that the root discriminant $D_N^{1/[N:\mathbb{Q}]}$ is bounded above by a constant times $[N:\mathbb{Q}]^ε$.

math.NT↗

arXiv2025

Holomorphic differentials of alternating four covers

Suppose $k$ is an algebraically closed field of characteristic two, let $A_4$ be an alternating group on four letters, and let $H$ be the unique Sylow two-subgroup of $A_4$. Let $X$ be a smooth projective irreducible curve over $k$ with a faithful $A_4$-action such that the quotient curve $X/H$ is a projective line and the $H$-cover $X\to X/H$ is totally ramified, in the sense that it is ramified and every branch point is totally ramified. Under these assumptions, we determine the precise $kA_4$-module structure of the space of holomorphic differentials of $X$ over $k$. We show that there are infinitely many different isomorphism classes of indecomposable $kA_4$-modules that can occur as direct summands, and we give precise formulas for the multiplicities with which they occur.

math.AG↗

arXiv2025

Triple Massey products for higher genus curves

We study the vanishing of triple Massey products for absolutely irreducible smooth projective curves over a number field. For each genus $g > 1$ and each prime $\ell > 3$, we construct examples of hyperelliptic curves of genus $g$ for which there are non-empty triple Massey products with coefficients in $\mathbb{Z}/\ell$ that do not contain $0$.

math.AG↗

arXiv2024

Cup products on curves over finite fields

Suppose $k$ is a finite field, that $C$ is a smooth projective geometrically irreducible curve over $k$, and that $n$ is a positive integer not divisible by the characteristic of $k$. In this paper we compute cup products of elements of the étale cohomology groups $\mathrm{H}^1(C,\mathbb{Z}/n)$ and $\mathrm{H}^1(C,μ_n)$. Over the algebraic closure $\overline{k}$ of $k$, such cup products are connected to values of the Weil pairing on the $n$-torsion of the Jacobian of $\overline{C} = \overline{k} \otimes_k C$ by using a fixed isomorphism between $\mathbb{Z}/n$ and $μ_n$ over $\overline{C}$. Over $k$, such cup products are more subtle due to the fact that they take values in the group $\mathrm{H}^2(C,μ_n)=\mathrm{Pic}(C)/n\cdot \mathrm{Pic}(C)$ rather than in the group $\mathrm{H}^2(\overline{C},μ_n) = \mathbb{Z}/n$.

math.AG↗

arXiv2023

Massey products and elliptic curves

We study the vanishing of Massey products of order at least $3$ for absolutely irreducible smooth projective curves over a perfect field with coefficients in $\mathbb{Z}/\ell$. We mainly focus on elliptic curves, for which we obtain a complete characterization of when triple Massey products do not vanish.

math.AG↗

arXiv2023

The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces

Suppose $X$ is a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Let $G$ be a finite group acting faithfully on $X$ over $k$ such that $G$ has non-trivial, cyclic Sylow $p$-subgroups. If $E$ is a $G$-invariant Weil divisor on $X$ with $\mathrm{deg}(E)> 2g(X)-2$, we prove that the decomposition of $\mathrm{H}^0(X,\mathcal{O}_X(E))$ into a direct sum of indecomposable $kG$-modules is uniquely determined by the class of $E$ modulo $G$-invariant principal divisors, together with the ramification data of the cover $X\to X/G$. The latter is given by the lower ramification groups and the fundamental characters of the closed points of $X$ that are ramified in the cover. As a consequence, we obtain that if $m>1$ and $g(X)\ge 2$, then the $kG$-module structure of $\mathrm{H}^0(X,Ω_X^{\otimes m})$ is uniquely determined by the class of a canonical divisor on $X/G$ modulo principal divisors, together with the ramification data of $X\to X/G$. This extends to arbitrary $m > 1$ the $m = 1$ case treated by the first author with T. Chinburg and A. Kontogeorgis. We discuss applications to the tangent space of the global deformation functor associated to $(X,G)$ and to congruences between prime level cusp forms in characteristic $0$. In particular, we complete the description of the precise $k\mathrm{PSL}(2,\mathbb{F}_\ell)$-module structure of all prime level $\ell$ cusp forms of even weight in characteristic $p=3$.

math.AG↗

arXiv2023

Holomorphic differentials of Klein four covers

Let $k$ be an algebraically closed field of characteristic two, and let $G$ be isomorphic to $\mathbb{Z}/2\times\mathbb{Z}/2$. Suppose $X$ is a smooth projective irreducible curve over $k$ with a faithful $G$-action, and assume that the cover $X\to X/G$ is totally ramified, in the sense that it is ramified and every branch point is totally ramified. We study to what extent the lower ramification groups of the closed points of $X$ determine the isomorphism types of the indecomposable $kG$-modules and the multiplicities with which they occur as direct summands of the space $\mathrm{H}^0(X,Ω_{X/k})$ of holomorphic differentials of $X$ over $k$. In the case when $X/G=\mathbb{P}^1_k$, we completely determine the decomposition of $\mathrm{H}^0(X,Ω_{X/k})$ into a direct sum of indecomposable $kG$-modules. Moreover, we show that the isomorphism classes of indecomposable $kG$-modules that actually occur as direct summands belong to an infinite list of non-isomorphic indecomposable $kG$-modules that contain modules of arbitrarily large $k$-dimension. In particular, our results show that [14, Theorem 6.4] is incorrect.

math.AG↗

arXiv2021

On representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, $\widehat{GT}$ and $\mathrm{Aut}(\hat{F}_2)$

By work of Belyi, the absolute Galois group $G_{\mathbb{Q}}=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be embedded into $A=\mathrm{Aut}(\widehat{F_2})$, the automorphism group of the free profinite group $\widehat{F_2}$ on two generators. The image of $G_{\mathbb{Q}}$ lies inside $\widehat{GT}$, the Grothendieck-Teichmüller group. While it is known that every abelian representation of $G_{\mathbb{Q}}$ can be extended to $\widehat{GT}$, Lochak and Schneps put forward the challenge of constructing irreducible non-abelian representations of $\widehat{GT}$. We do this virtually, namely by showing that a rich class of arithmetically defined representations of $G_{\mathbb{Q}}$ can be extended to finite index subgroups of $\widehat{GT}$. This is achieved, in fact, by extending these representations all the way to finite index subgroups of $A=\mathrm{Aut}(\widehat{F_2})$. We do this by developing a profinite version of the work of Grunewald and Lubotzky, which provided a rich collection of representations for the discrete group $\mathrm{Aut}(F_d)$.

math.NT↗

arXiv2020

Galois structure of the holomorphic differentials of curves

Let $X$ be a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Suppose $G$ is a finite group acting faithfully on $X$ such that $G$ has non-trivial cyclic Sylow $p$-subgroups. We show that the decomposition of the space of holomorphic differentials of $X$ into a direct sum of indecomposable $k[G]$-modules is uniquely determined by the lower ramification groups and the fundamental characters of closed points of $X$ that are ramified in the cover $X\to X/G$. We apply our method to determine the $\mathrm{PSL}(2,\mathbb{F}_\ell)$-module structure of the space of holomorphic differentials of the reduction of the modular curve $\mathcal{X}(\ell)$ modulo $p$ when $p$ and $\ell$ are distinct odd primes and the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on this reduction is not tamely ramified. This provides some non-trivial congruences modulo appropriate maximal ideals containing $p$ between modular forms arising from isotypic components with respect to the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on $\mathcal{X}(\ell)$.

math.AG↗

arXiv2018

Cup products in the etale cohomology of number fields

This paper concerns cup product pairings in étale cohomology related to work of M. Kim and of W. McCallum and R. Sharifi. We will show that by considering Ext groups rather than cohomology groups, one arrives at a pairing which combines invariants defined by Kim with a pairing defined by McCallum and Sharifi. We also prove a formula for Kim's invariant in terms of Artin maps in the case of cyclic unramified Kummer extensions. One consequence is that for all $n > 1$, there are infinitely many number fields $F$ over which there are both trivial and non-trivial Kim invariants associated to cyclic groups of order $n$.

math.NT↗

arXiv2018

Universal deformation rings and self-injective Nakayama algebras

Let $k$ be a field and let $Λ$ be an indecomposable finite dimensional $k$-algebra such that there is a stable equivalence of Morita type between $Λ$ and a self-injective split basic Nakayama algebra over $k$. We show that every indecomposable finitely generated $Λ$-module $V$ has a universal deformation ring $R(Λ,V)$ and we describe $R(Λ,V)$ explicitly as a quotient ring of a power series ring over $k$ in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to $p$-modular blocks of finite groups with cyclic defect groups.

math.GR↗

arXiv2017

Deformations of complexes for finite dimensional algebras

Let $k$ be a field and let $Λ$ be a finite dimensional $k$-algebra. We prove that every bounded complex $V^\bullet$ of finitely generated $Λ$-modules has a well-defined versal deformation ring $R(Λ,V^\bullet)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. We also prove that nice two-sided tilting complexes between $Λ$ and another finite dimensional $k$-algebra $Γ$ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra.

math.RT↗

arXiv2017

Universal deformation rings, endo-trivial modules, and semidihedral and generalized quaternion 2-groups

Let $k$ be a field of characteristic $p>0$, and let $W$ be a complete discrete valuation ring of characteristic $0$ that has $k$ as its residue field. Suppose $G$ is a finite group and $G^{\mathrm{ab},p}$ is its maximal abelian $p$-quotient group. We prove that every endo-trivial $kG$-module $V$ has a universal deformation ring that is isomorphic to the group ring $WG^{\mathrm{ab},p}$. In particular, this gives a positive answer to a question raised by Bleher and Chinburg for all endo-trivial modules. Moreover, we show that the universal deformation of $V$ over $WG^{\mathrm{ab},p}$ is uniquely determined by any lift of $V$ over $W$. In the case when $p=2$ and $G=\mathrm{D}$ is a $2$-group that is either semidihedral or generalized quaternion, we give an explicit description of the universal deformation of every indecomposable endo-trivial $k\mathrm{D}$-module $V$.

math.GR↗

arXiv2016

Automorphisms of Harbater-Katz-Gabber curves

Let k be a perfect field of characteristic p > 0, and let G be a finite group. We consider the pointed G-curves over k associated by Harbater, Katz, and Gabber to faithful actions of G on k[[t]] over k. We use such "HKG G-curves" to classify the automorphisms of k[[t]] of p-power order that can be expressed by particularly explicit formulas, namely those mapping t to a power series lying in a Z/pZ Artin-Schreier extension of k(t). In addition, we give necessary and sufficient criteria to decide when an HKG G-curve with an action of a larger finite group J is also an HKG J-curve.

math.AG↗

arXiv2014

The geometry of finite dimensional algebras with vanishing radical square

Let $Λ$ be a basic finite dimensional algebra over an algebraically closed field, with the property that the square of the Jacobson radical $J$ vanishes. We determine the irreducible components of the module variety $\text{Mod}_{\bf d}(Λ)$ for any dimension vector $\bf d$. Our description leads to a count of the components in terms of the underlying Gabriel quiver. A closed formula for the number of components when $Λ$ is local extends existing counts for the two-loop quiver to quivers with arbitrary finite sets of loops. For any algebra $Λ$ with $J^2 = 0$, our criteria for identifying the components of $\text{Mod}_{\bf d}(Λ)$ permit us to characterize the modules parametrized by the individual irreducible components. Focusing on such a component, we explore generic properties of the corresponding modules by establishing a geometric bridge between the algebras with zero radical square on one hand and their stably equivalent hereditary counterparts on the other. The bridge links certain closed subvarieties of Grassmannians parametrizing the modules with fixed top over the two types of algebras. By way of this connection, we transfer results of Kac and Schofield from the hereditary case to algebras of Loewy length $2$. Finally, we use the transit of information to show that any algebra of Loewy length $2$ which enjoys the dense orbit property in the sense of Chindris, Kinser and Weyman has finite representation type.

math.RT↗

arXiv2014

Orbit closures and rational surfaces

In this paper we study the Grassmannian of submodules of a given dimension inside a finitely generated projective module $P$ for a finite dimensional algebra $Λ$ over an algebraically closed field. The orbit of such a submodule $C$ under the action of $\mathrm{Aut}_Λ( P )$ on the Grassmannian encodes information on the degenerations of $P/C$ and has been considered by a number of authors. The goal of this article is to bound the geometry of two-dimensional orbit closures in terms of representation-theoretic data. Several examples are given to illustrate the interplay between the geometry of the projective surfaces which arise and the corresponding posets of degenerations.

math.RT↗

arXiv2013

Universal deformation rings and tame blocks

Let k be an algebraically closed field of positive characteristic, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group and B is a block of kG of infinite tame representation type. We find all finitely generated kG-modules V that belong to B and whose endomorphism ring is isomorphic to k and determine the universal deformation ring R(G,V) for each of these modules.

math.RT↗