Tom Weston
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View article: CLASS GROUP STATISTICS FOR TORSION FIELDS GENERATED BY ELLIPTIC CURVES
CLASS GROUP STATISTICS FOR TORSION FIELDS GENERATED BY ELLIPTIC CURVES Open
For a prime p and a rational elliptic curve $E_{/\mathbb {Q}}$ , set $K=\mathbb {Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname {ker}\{E\xrightarrow {p} E\}$ . The class group $\operatorname {Cl}_K$ is a module ove…
View article: Diophantine stability for elliptic curves on average
Diophantine stability for elliptic curves on average Open
Let $K$ be a number field and $\ell \geq 5$ a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety $X_{/K}$ at a prime $\ell$. We show that there is a positive density set of elliptic curves $E_{/\math…
View article: Hilbert's tenth problem in Anticyclotomic towers of number fields
Hilbert's tenth problem in Anticyclotomic towers of number fields Open
Let $K$ be an imaginary quadratic field and $p$ be an odd prime which splits in $K$. Let $E_1$ and $E_2$ be elliptic curves over $K$ such that the $Gal(\bar{K}/K)$-modules $E_1[p]$ and $E_2[p]$ are isomorphic. We show that under certain ex…
View article: Explicit reciprocity laws and Iwasawa theory for modular forms
Explicit reciprocity laws and Iwasawa theory for modular forms Open
We prove that the Mazur-Tate elements of an eigenform $f$ sit inside the Fitting ideals of the corresponding dual Selmer groups along the cyclotomic $\mathbb Z_p$-extension (up to scaling by a single constant). Our method begins with the c…
View article: Class group statistics for torsion fields generated by elliptic curves
Class group statistics for torsion fields generated by elliptic curves Open
For a prime $p$ and a rational elliptic curve $E_{/\mathbb{Q}}$, set $K=\mathbb{Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. The class group $\operatorname{Cl}_K$ is a module over $\o…
View article: Arithmetic statistics for Galois deformation rings
Arithmetic statistics for Galois deformation rings Open
Given an elliptic curve $E$ defined over the rational numbers and a prime $p$ at which $E$ has good reduction, we consider the Galois deformation ring parametrizing lifts of the residual representation on the $p$-torsion group $E[p]$. For …
View article: Unobstructed deformation problems for GSp(4)
Unobstructed deformation problems for GSp(4) Open
We prove, for many cuspidal automorphic representations for GSp(4), that the local obstructions to the deformation theory of the associated residual Galois representations generically vanish.