Charlotte Ure

Research Interest

My research interest is in noncommutative algebra and algebraic geometry. In particular, I am interested in the study of

For more information on my research interests, please see my Research Statement.

Papers

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The Brauer group of a field consists of equivalence classes of central simple algebras and can be used to understand the arithmetic complexity of the underlying field. For example, it is trivial if the field is algebraically closed, finite, or the function field of an algebraic curve over an algebraically closed field. The Brauer group of the real numbers is the cyclic group of order two and is generated by the equivalence class of the quaternion algebra. Some well-understood objects in the Brauer group of a field $F$ are symbol (or cyclic) algebras. These are algebras with a presentation $$(a,b)_{F,d,\rho} = \left< x,y : x^d = a, y^d = b, xy = \rho yx \right>,$$ where $a,b \in F^\times, d \in \mathbb{N}, d\geq 2$ i, and $\rho$ is a primitive $d$-th root of unity. Further, the Merkurjev-Suslin theorem implies that every element in the Brauer group of $F$ of period (or exponent) $d$ is Brauer equivalent to tensor product of symbol algebras. The symbol length problem asks for the minimal number of symbol algebras required to write every element in the Brauer group of $F$. The starting point for our investigation of the symbol length is Kummer spaces. Let $A$ be the tensor product $\bigotimes_{j=1}^n\left(a_j,b_j\right)_{d,F,\omega}$. An element $v \in A$ is Kummer if $v^d \in F$. A Kummer space is a subspace of $A$ consisting of Kummer elements. Matzri bound the symbol length of $C_r$-fields by using the existence of certain Kummer spaces of maximal dimension in a tensor product of cyclic algebras. Motivated by this result we proved the following theorem.

Theorem. Let $F = k\left( a_j, b_j: 1 \leq j \leq n \right)$ be the function field in $2n$ variables over a field $K$ of characteristic $0$ containing the primitive $4$th root of unity $i$, and let $A = \bigotimes \left( a_j, b_j\right)_{j,F,i}$ be the generic tensor product of $n$ cyclic algebras of degree $4$. The maximal dimension of a Kummer space in $A$ is $4n + 1$.

Preprints

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Let $k$ be a field of characteristic prime to $2$ and $3$. Let $q$ be a positive prime. Assume additionally that $q$ is prime to the characteristic of $k$ and that $k$ contains a primitive $d$-th root of unity. By the purity theorem we can identify $Br(E)$ with the unramified Brauer group $Br_{\text{ur}} k(E)$ inside the Brauer group of the function field $k(E)$ of $E$. The Merkurjev-Suslin theorem implies that symbol algebras span $Br k(E)$. Thus every $q$-torsion element in the Brauer group is Brauer equivalent to a tensor product of algebras of the form $$(f,g)_{q,k(E)} = k(E)\left< x,y: x^q = f, y^q = g, xy = \rho yx \right>$$ for some $f,g \in k(E)^\times$. The main objective of this paper is to describe generators and relations of $Br(E)$ in terms of these tensor products. We prove the following theorem, which is a special case of a description of $Br_{\text{ur}} k(C)$ given by Chernousov, Rapinchuk, and Rapinchuk, where $C$ is any projective geometrically irreducible curve whose Jacobian has $k$-rational $q$-torsion.

Theorem. Fix two linearly independent $q$-torsion points $P$ and $Q$ of $E$. Assume that $P,Q$ are $k$-rational and there are elements $t_P,t_Q \in k(E)$ with $div(t_P) = qP - qO$ and $div(t_Q) = qQ- qO$, where $O$ is the point at infinity of $E$. Then the $q$-torsion of $Br(E)$ decomposes as $${}_{q}{Br(E)} = {}_qBr(k) \oplus I$$ and every element in $I$ can be represented as a tensor product $$\left( a, t_P \right)_{q,k(E)} \otimes \left( b, t_Q \right)_{q,k(E)}$$ with $a,b \in k^\times$.

This theorem also holds true for $q$ not prime, but coprime to the characteristic of $k$. The description of the Brauer groups becomes significantly more complicated when we drop the assumption that the $q$-torsion is $k$-rational. Chernousov and Guletskii describe elements in the two-torsion ${}_2Br(E)$ as tensor products of quaternion algebras. Inspired by their work we give an algorithm to determine generators and relations of ${}_qBr(E)$ over any field of characteristic different from $2,3$ and $q$ containing a primitive $q$-th root of unity. An easy consequence of the algorithm is an upper bound on the symbol length in ${}_qBr(E)$. This is the minimal number of symbol algebras required to write every element in the group.

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The classical definition of Clifford algebras associated to quadratic forms can be generalized to higher degree forms in a natural way. For example for a cubic form $f$ on a $2$-dimensional $k$-vector space $V$ with basis $\{u,v\}$, the Clifford algebra $C_f$ associated to $f$ is the quotient of the tensor algebra $TV$ by the ideal generated by elements of the form $$f(u,v) - (\lambda u + \nu v)^3$$ with $\lambda,\nu \in k$. The center of $C_f$ is the affine coordinate ring of an elliptic curve $E$ and it extends to an Azumaya algebra over $E$. Furthermore its Brauer class is trivial if and only if the curve $C$ given by the affine equation $w^3 - f(u,v)$ has a $k$-rational point. We study this phenomenon in families. Identify the affine four dimensional space with the set of binary cubic forms by identifying a point $(a, b, c,d)$ with the form $au^3 + b u^2v + c uv^2 + d v^3$. Let $U$ be the Zariski-open subset that corresponds to the non-degenerate forms. The generic binary cubic Clifford algebra $A$ is the quotient of the tensor algebra $TV$ by elements of the form $$\left[ \lambda u + \nu v, ( \lambda u + \nu v )^3 \right],$$ where $[-,-]$ denotes the commutator and $\lambda, \nu \in k$. This defines a sheaf of algebras over $U$ and at every point $f \in U$, it coincides with the Clifford algebra associated to $f$ as before. We prove that $A$ has a geometric construction as follows: Let $C$ be the relative curve given by $w^3 - (au^3 + b u^2v + c uv^2 + d v^3)$ over $U$. Tere is a finite Galois cover $U'$ of $U$ such that $C' = C \times_U U'$ admits a section. Denote the relative Jacobian of $C'$ by $J'$, let $\mathcal{P}$ be the Poincare-bundle on $C' \times J'$, and denote by $\pi: C' \times J' \rightarrow J'$ the projection. Then $End\left(\pi_*\mathcal{P}\right)$ is an Azumaya algebra and defines therefore an element in $Br(J')$.

Theorem. $End\left(\pi_*\mathcal{P}\right)$ descents to the binary cubic generic Clifford algebra. Furthermore, if there is a section $s: U \rightarrow C$, then the Brauer class of the generic Clifford algebra is trivial.

Bachelor Thesis at the University of Heidelberg (in German)

In my undergraduate thesis, I focused on the representation theory of affine linear groups. In particular, I explored representations of $GL_n$ and $SL_n$ in positive characterisitic. My thesis first examines irreducible representation of $SL_2$ and $GL_2$. Using Steinberg's tensor product theorem they can be described as tensor products and Frobenius twists of the symmetric tensor algebra of degree $r$ and the exterior tensor algebra of degree two. Further, we describe the Jordan-Hölder factors of the symmetric tensor algebra of degree $r$ over $GL_2$ and $GL_n$. For the computation of these factors I used the software Sage.

The thesis can be found here.

© Charlotte Ure 2018