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Bradley W. Brock

Publications and source records attributed to Bradley W. Brock.

5 recordsLinked to original sources
arXiv2025

Purely inseparable Richelot isogenies

We show that if $C$ is a supersingular genus-$2$ curve over an algebraically-closed field of characteristic $2$, then there are infinitely many Richelot isogenies starting from $C$. This is in contrast to what happens with non-supersingular curves in characteristic $2$, or to arbitrary curves in characteristic not $2$: In these situations, there are at most fifteen Richelot isogenies starting from a given genus-$2$ curve. More specifically, we show that if $C_1$ and $C_2$ are two arbitrary supersingular genus-$2$ curves over an algebraically-closed field of characteristic $2$, then there are exactly sixty Richelot isogenies from $C_1$ to $C_2$, unless either $C_1$ or $C_2$ is isomorphic to the curve $y^2 + y = x^5$. In that case, there are either twelve or four Richelot isogenies from $C_1$ to $C_2$, depending on whether $C_1$ is isomorphic to $C_2$. (Here we count Richelot isogenies up to isomorphism.) We give explicit constructions that produce all of the Richelot isogenies between two supersingular curves.

math.AG↗
arXiv2023

A composition law and refined notions of convergence for periodic continued fractions

We define an equivalence relation on periodic continued fractions with partial quotients in a ring $\mathcal{O} \subseteq \mathbf{C}$, a group law on these equivalence classes, and a map from these equivalence classes to matrices in $\mathrm{GL}_2(\mathcal{O})$ with determinant $\pm1$. We prove this group of equivalence classes is isomorphic to $\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ and study certain of its one- and two-dimensional representations. For a periodic continued fraction with period $k$, we give a refined description of the limits of the $k$ different $k$-decimations of its sequence of convergents. We show that for a periodic continued fraction associated to a matrix with eigenvalues of different magnitudes, all $k$ of these limits exist in $\mathbb{P}^1(\mathbf{C})$ and a strict majority of them are equal.

math.NT↗
arXiv2020

Linear independence in linear systems on elliptic curves

Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $\operatorname{char} k \nmid N$. For $P \in C$, let $s_P$ be a rational function with divisor $N \cdot P - N \cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional $(E,C)$ when $N$ is a prime by using the geometry of the universal generalized elliptic curve over $X_1(N)$. The problem can be recast in terms of sections of an arbitrary degree $N$ line bundle on $E$.

math.NT↗
arXiv2019

Periodic continued fractions over $S$-integers in number fields and Skolem's $p$-adic method

We generalize the classical theory of periodic continued fractions (PCFs) over ${\mathbf Z}$ to rings ${\mathcal O}$ of $S$-integers in a number field. Let ${\mathcal B}=\{β, {β^*}\}$ be the multi-set of roots of a quadratic polynomial in ${\mathcal O}[x]$. We show that PCFs $P=[b_1,\ldots,b_N,\bar{a_1\ldots ,a_k}]$ of type $(N,k)$ potentially converging to a limit in ${\mathcal B}$ are given by ${\mathcal O}$-points on an affine variety $V:=V({\mathcal B})_{N,k}$ generically of dimension $N+k-2$. We give the equations of $V$ in terms of the continuant polynomials of Wallis and Euler. The integral points $V({\mathcal O})$ are related to writing matrices in $\textrm{SL}_2({\mathcal O})$ as products of elementary matrices. We give an algorithm to determine if a PCF converges and, if so, to compute its limit. Our standard example generalizes the PCF $\sqrt{2}=[1,\bar{2}]$ to the ${\mathbf Z}_2$-extension of ${\mathbf Q}$: $F_n={\mathbf Q}(α_n)$, $α_{n}:=2\cos(2π/2^{n+2})$, with integers ${\mathcal O}_n={\mathbf Z}[α_n]$. We want to find the PCFs of $α_{n+1}$ over ${\mathcal O}_{n}$ of type $(N,k)$ by finding the ${\mathcal O}_{n}$-points on $V({\mathcal B}_{n+1})_{N,k}$ for ${\mathcal B}_{n+1}:=\{α_{n+1}, -α_{n+1}\}$. There are three types $(N,k)=(0,3), (1,2), (2,1)$ such that the associated PCF variety $V({\mathcal B})_{N,k}$ is a curve; we analyze these curves. For generic ${\mathcal B}$, Siegel's theorem implies that each of these three $V({\mathcal B})_{N,k}({\mathcal O})$ is finite. We find all the ${\mathcal O}_n$-points on these PCF curves $V({\mathcal B}_{n+1})_{N,k}$ for $n=0,1$. When $n=1$ we make extensive use of Skolem's $p$-adic method for $p=2$, including its application to Ljunggren's equation $x^2 + 1 =2y^4$.

math.NT↗