Related papers: Refinements on vertical Sato-Tate

Let $K$ be a number field and $r$ an integer. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree two prime ideals of $K$ with trace of Frobenius equal to $r$. Under certain restrictions…

We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form $$ y^2=x^p-1 \text{ and } y^2=x^{2p}-1,$$ where $p$ is an odd prime. Our results rely on the fact the Jacobians of…

Fix a finite field. A hyperelliptic curve determines a measure on the discrete space of rank two bundles on the projective line: the mass of a given vector bundle is the number of line bundles whose pushforward it is. In a sequence of…

We characterize quadratic twists of $y^2=x(x-a^2)(x+b^2)$ with Mordell-Weil groups and $2$-primary part of Shafarevich-Tate groups being isomorphic to $(\mathb Z/2\mathbb Z)^2$ under certain conditions. We also obtain the distribution…

We propose a refined version of the Sato-Tate conjecture about the spacing distribution of the angle determined for each prime number. We also discuss its implications on $L$-function associated with elliptic curves in the relation to…

We prove the generalised Tate conjecture for H^3 of products of elliptic curves over finite fields, by slightly modifying an argument of M. Spiess concerning the Tate conjecture. We prove it fully if the elliptic curves run among at most 3…

We study fluctuations in the distribution of families of $p$-th Fourier coefficients $a_f(p)$ of normalised holomorphic Hecke eigenforms $f$ of weight $k$ with respect to $SL_2(\mathbb{Z})$ as $k \to \infty$ and primes $p \to \infty.$ These…

Let $p\ge5$ be a prime and $T$ a Kodaira type of the special fiber of an elliptic curve. We estimate the number of elliptic curves over $\mathbb Q$ up to height $X$ with Kodaira type $T$ at $p$. This enables us find the proportion of…

Let $S_k(N)$ denote the space of cusp forms of even integer weight $k$ and level $N$. We prove an asymptotic for the Petersson trace formula for $S_k(N)$ under an appropriate condition. Using the non-vanishing of a Kloosterman sum involved…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with Mordell--Weil rank $2$ and $p$ be an odd prime of good ordinary reduction. For every imaginary quadratic field $K$ satisfying the Heegner hypothesis, there is (subject to the…

In a previous article, we obtained data on the distribution of traces of Frobenius of non-hyperelliptic genus $3$ curves over small finite fields. In the present one, we give a heuristic explanation of these data, by extrapolating from…

In 1988, Koblitz conjectured the infinitude of primes p for which |E(F_p)| is prime for elliptic curves E over Q, drawing an analogy with the twin prime conjecture. He also proposed studying the primality of |E(F_{p^l})| / |E(F_p)|, in…

We show that the reductions modulo primes $p\le x$ of the elliptic curve $$ Y^2 = X^3 + f(a)X + g(b), $$ behave as predicted by the Lang-Trotter and Sato-Tate conjectures, on average over integers $a \in [-A,A]$ and $b \in [-B,B]$ for $A$…

We investigate the average value of the Frobenius trace at p over elliptic curves in a fixed conductor range with given rank. Plotting this average as p varies over the primes yields a striking oscillating pattern, the details of which vary…

Given an elliptic curve $E$ and a point $P$ in $E(\mathbb{R})$, we investigate the distribution of the points $nP$ as $n$ varies over the integers, giving bounds on the $x$ and $y$ coordinates of $nP$ and determining the natural density of…

We define a function in terms of quotients of the $p$-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the $p$-adic setting. We prove, for primes $p > 3$, that the…

For a non-CM elliptic curve $E$ defined over the rationals, Lang and Trotter made very deep conjectures concerning the number of primes $p\leq x$ for which $a_p(E)$ is a fixed integer (and for which the Frobenius field at $p$ is a fixed…

Given an elliptic curve $E$ and a positive integer $N$, we consider the problem of counting the number of primes $p$ for which the reduction of $E$ modulo $p$ possesses exactly $N$ points over $\mathbb F_p$. On average (over a family of…

We show that the number of copies of ${\Bbb Q}_p/{\Bbb Z}_p$ in the Tate-Shafarevich group of an elliptic curve $E$ over ${\Bbb Q}$ with complex multipication, is at most $2p - g$, where $g$ is the rank of $E({\Bbb Q})$, and for all…

In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining…