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Charlotte Ure

Publications and source records attributed to Charlotte Ure.

10 recordsLinked to original sources
arXiv2026

A septic covariant and the Hermite--Joubert problem in degree seven

We prove the degree-seven case of the Hermite--Joubert problem in characteristic zero: if $F$ is a field of characteristic zero and $E/F$ is a field extension of degree seven, then $E$ is generated by an element $a$ with $\text{tr}_{E/F}(a)=\text{tr}_{E/F}(a^{3})=0$, that is, with minimal polynomial of the form $λ^{7}+c_{2}λ^{5}+c_{4}λ^{3}+c_{5}λ^{2}+c_{6}λ+c_{7}$, where the $c_i$'s belong to $F$. This is given by an explicit formula: a covariant of the binary septic of coefficient degree seven and order five, evaluated at a generator $θ$ and divided by the derivative of its minimal polynomial evaluated at $θ$. We also announce the general theorem, which will be proved in a companion paper in preparation: over every infinite field, in every characteristic, every étale algebra of degree seven contains a primitive element $a$ with $c_{1}(a)=c_{3}(a)=0$. Moreover, every field extension of degree seven of an arbitrary field has a generator $a$ with $c_{1}(a)=c_{3}(a)=0$.

math.NT↗
arXiv2026

The Bloch--Kato conjecture, decomposing fields, and generating cohomology in degree one

The famous Bloch--Kato conjecture implies that for a field $F$ containing a primitive $p$th root of unity, the cohomology ring of the absolute Galois group $G_F$ of $F$ with $\mathbb{F}_p$ coefficients is generated by degree one elements. We investigate other groups with this property and characterize all such groups that are finite. Restricting to the case of $p$-groups, our work answers a question of Quadrelli, Snopce and Vanacci posed in 2022. As a further step in this program, we study implications of the Bloch--Kato conjecture to cohomological invariants of finite field extensions. Conversely, these cohomological invariants have implications for refining the Bloch--Kato conjecture. In service of such a refinement, we define the notion of a decomposing field for a cohomology class of a finite field extension and study minimal decomposing fields of degree two cohomology classes arising from degree $p$ extensions. We illustrate this refinement by explicitly computing the cohomology rings of superpythagorean fields and $p$-rigid fields. Finally, we construct nontrivial examples of cohomology classes and their decomposing fields, which rely on computations by David Benson in the appendix.

math.NT↗
arXiv2024

Twisting Manin's universal quantum groups and comodule algebras

We introduce the notion of quantum-symmetric equivalence of two connected graded algebras, based on Morita-Takeuchi equivalences of their universal quantum groups, in the sense of Manin. We study homological and algebraic invariants of quantum-symmetric equivalence classes, and prove that numerical $\mathrm{Tor}$-regularity, Castelnuovo-Mumford regularity, Artin-Schelter regularity, and the Frobenius property are invariant under any Morita-Takeuchi equivalence. In particular, by combining our results with the work of Raedschelders and Van den Bergh, we prove that Koszul Artin-Schelter regular algebras of a fixed global dimension form a single quantum-symmetric equivalence class. Moreover, we characterize 2-cocycle twists (which arise as a special case of quantum-symmetric equivalence) of Koszul duals, of superpotentials, of superpotential algebras, of Nakayama automorphisms of twisted Frobenius algebras, and of Artin-Schelter regular algebras. We also show that finite generation of Hochschild cohomology rings is preserved under certain 2-cocycle twists.

math.QA↗
arXiv2023

Period-index in top cohomology over semiglobal fields

We prove a common slot lemma for symbols in top cohomology classes over semiglobal fields. Furthermore, we prove that period and index agree for general top cohomology classes over such fields. We discuss applications to quadratic forms and related open problems.

math.NT↗
arXiv2022

A cogroupoid associated to preregular forms

We construct a family of cogroupoids associated to preregular forms and recover the Morita-Takeuchi equivalence for Artin-Schelter regular algebras of dimension two, observed by Raedschelders and Van den Bergh. Moreover, we study the 2-cocycle twists of pivotal analogues of these cogroupoids, by developing a categorical description of preregularity in any tensor category that has a pivotal structure.

math.RA↗
arXiv2022

Prime Torsion in the Brauer Group of an Elliptic Curve

We give an algorithm to explicitly determine all elements of the $q$-torsion (for $q$ an odd prime) of the Brauer group of an elliptic curve over any base field of characteristic different from $q$, containing a primitive $q$-th root of unity. These elements of the Brauer group are given as tensor products of symbol algebras over the function field of the elliptic curve. We give sufficient conditions to determine if the Brauer classes that arise are trivial. Using our algorithm, we derive an upper bound on the symbol length of the prime torsion of $\mathrm{Br}(E)/\mathrm{Br}(k)$.

math.AG↗
arXiv2022

Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation

Let $H$ be a Hopf algebra that is $\mathbb Z$-graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of $H$ to be a Zhang twist of $H$. In particular, we introduce the notion of a twisting pair for $H$ such that the Zhang twist of $H$ by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated to quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction.

math.RA↗
arXiv2022

Symbol Length in Brauer Groups of Elliptic Curves

Let $\ell$ be an odd prime, and let $K$ be a field of characteristic not $2,3,$ or $\ell$ containing a primitive $\ell$-th root of unity. For an elliptic curve $E$ over $K$, we consider the standard Galois representation $$ρ_{E,\ell}: \text{Gal}(\overline{K}/K) \rightarrow \text{GL}_2(\mathbb{F}_{\ell}),$$ and denote the fixed field of its kernel by $L$. Recently, the last author gave an algorithm to compute elements in the Brauer group explicitly, deducing an upper bound of $2(\ell+1)(\ell-1)$ on the symbol length in $\mathbin{_{\ell}\text{Br}(E)} / \mathbin{_{\ell}\text{Br}(K)}$. More precisely, the symbol length is bounded above by $2[L:K]$. We improve this bound to $[L:K]-1$ if $\ell \nmid [L:K]$. Under the additional assumption that $\text{Gal}(L/K)$ contains an element of order $d > 1$, we further reduce it to $(1-\frac{1}{d})[L:K]$. In particular, these bounds hold for all CM elliptic curves, in which case we deduce a general upper bound of $\ell + 1$. We provide an algorithm implemented in SageMath to compute these symbols explicitly over number fields.

math.NT↗
arXiv2021

A moduli interpretation of untwisted binary cubic forms

We give a moduli interpretation to the quotient of (nondegenerate) binary cubic forms with respect to the natural $\text{GL}_2$-action on the variables. In particular, we show that these $\text{GL}_2$ orbits are in bijection with pairs of $j$-invariant $0$ elliptic curves together with $3$-torsion Brauer classes that are invariant under complex multiplication. The binary cubic generic Clifford algebra plays a key role in the construction of this correspondence.

math.AG↗