Related papers: Refinements on vertical Sato-Tate

It is well-known that if $E$ is an elliptic curve over the finite field $\mathbb{F}_p$, then $E(\mathbb{F}_p)\simeq\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/mk\mathbb{Z}$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs…

We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these…

Let $p \in \{3, 5\}$ and consider a cyclic $p$-extension $L/\mathbb{Q}$. We show that there exists an effective positive density of elliptic curves $ E $ defined over $ \mathbb{Q} $, ordered by height, that are diophantine stable in $ L $.

We assemble a statistical set of global mass models for ~2,000 nearly spherical SDSS galaxies at a mean redshift of 0.12 based on their aperture velocity dispersions and newly derived luminosity profiles in conjunction with published…

Let E be a non-CM elliptic curve defined over Q. For each prime p of good reduction, E reduces to a curve E_p over the finite field F_p. For a given squarefree polynomial f(x,y), we examine the sequences f_p(E) := f(a_p(E), p), whose values…

Let $A$ be an abelian variety over $\mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $\rho_A$ is open in $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. We investigate the arithmetic of the…

This is an updated version of ANT-0166. Generalizing results of Stroeker and Top we show that the 2-ranks of the Tate-Shafarevich groups of the elliptic curves $y^2 = (x+k)(x^2+k^2)$ can become arbitrarily large. We also present a…

Given an elliptic curve $E$ over a finite field $\mathbb{F}_q$ we study the finite extensions $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ such that the number of $\mathbb{F}_{q^n}$-rational points on $E$ attains the Hasse upper bound. We obtain an…

The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb…

Suppose that $\pi \: Y \to X$ is a finite map of normal varieties over a perfect field of characteristic $p > 0$. Previous work of the authors gave a criterion for when Frobenius splittings on $X$ (or more generally any $p^{-e}$-linear map)…

We study the kernel and cokernel of the Frobenius map on the $p$-typical Witt vectors of a commutative ring, not necessarily of characteristic $p$. We give some equivalent conditions to surjectivity of the Frobenus map on both finite and…

For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of…

U(1) symmetries play a central role in constructing phenomenologically viable F-theory compactifications that realize Grand Unified Theories (GUTs). In F-theory, gauge symmetries with abelian gauge factors are modeled by singular elliptic…

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite…

Cremona, Mazur, and others have studied what they call visibility of elements of Shafarevich-Tate groups of elliptic curves. The analogue for an abelian number field $K$ is capitulation of ideal classes of $K$ in the minimal cyclotomic…

Distributions, i.e., subsets of tangent bundles formed by piecing together subspaces of tangent spaces, are commonly encountered in the theory and application of differential geometry. Indeed, the theory of distributions is a fundamental…

We consider the extreme value statistics of $N$ independent and identically distributed random variables, which is a classic problem in probability theory. When $N\to\infty$, fluctuations around the maximum of the variables are described by…

We study the limiting distributions of expanding translates of a compact segment of a smooth curve under a diagonal subgroup of $G=\mathrm{SO}(n_1,1)\times\cdots\times\mathrm{SO}(n_k,1)$, where $G$ acts on a finite volume homogeneous space…

We introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves and sits nicely in…

We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of…