Jeffrey Yelton

Jeffrey Yelton
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Jeffrey Yelton
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Mathematics - Number Theory (4)
 
Mathematics - Algebraic Geometry (1)

Publications Authored By Jeffrey Yelton

Let $K$ be a field of characteristic different from $2$ and let $E$ be an elliptic curve over $K$, defined either by an equation of the form $y^{2} = f(x)$ with degree $3$ or as the Jacobian of a curve defined by an equation of the form $y^{2} = f(x)$ with degree $4$. We obtain generators over $K$ of the $8$-division field $K(E[8])$ of $E$ given as formulas in terms of the roots of the polynomial $f$, and we explicitly describe the action of a particular automorphism in $\mathrm{Gal}(K(E[8]) / K)$. Read More

Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}$ of discriminant $\Delta$ and some element $\lambda \in \mathcal{O}_{K}$. Our main result says that if there is a prime $\mathfrak{p}$ of $K$ dividing $(f(\lambda))$ but not $(2\Delta)$, then the image of the natural $2$-adic Galois representation is open in $\mathrm{GSp}(T_{2}(J))$ and contains a certain congruence subgroup of $\mathrm{Sp}(T_{2}(J))$ depending on the maximal power of $\mathfrak{p}$ dividing $(f(\lambda))$. Read More

Let $k$ be a field of characteristic $0$, and let $\alpha_{1}$, $\alpha_{2}$, ... Read More

Let $k$ be a subfield of $\mathbb{C}$ which contains all $2$-power roots of unity, and let $K = k(\alpha_{1}, \alpha_{2}, ... Read More