Related papers: Algorithms for Modular Parametrizations of Ellipti…

From the Modularity Theorem proven by Wiles, Taylor, et al, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented…

Suppose $p$ is a prime of the form $u^2+64$ for some integer $u$, which we take to be 3 mod 4. Then there are two Neumann--Setzer elliptic curves $E_0$ and $E_1$ of prime conductor $p$, and both have Mordell--Weil group $\Z/2\Z$. There is a…

A $\textit{portrait}$ $\mathcal{P}$ on $\mathbb{P}^N$ is a pair of finite point sets $Y\subseteq{X}\subset\mathbb{P}^N$, a map $Y\to X$, and an assignment of weights to the points in $Y$. We construct a parameter space…

An efficient integer factorization algorithm would reduce the security of all variants of the RSA cryptographic scheme to zero. Despite the passage of years, no method for efficiently factoring large semiprime numbers in a classical…

In this paper we classify the complex elliptic curves $E$ for which there exist cyclic subgroups $C\leq (E,+)$ of order $n$ such that the elliptic curves $E$ and $E/C$ are isomorphic, where $n$ is a positive integer. Important examples are…

For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials of rational elliptic curves $E$ of conductor $p$. Ono made the observation that these zeros of are often $j$-invariants of supersingular elliptic curves over…

Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on…

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.

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…

The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary…

Let X be a product of Drinfeld modular curves over a general base ring A of odd characteristic. We classify those subvarieties of X which contain a Zariski-dense set of CM points. This is an analogue of the Andr\'e-Oort conjecture. As an…

Let $C$ be a curve of genus 2 and $\psi_1:C \lar E_1$ a map of degree $n$, from $C$ to an elliptic curve $E_1$, both curves defined over $\bC$. This map induces a degree $n$ map $\phi_1:\bP^1 \lar \bP^1$ which we call a Frey-Kani covering.…

In this paper we study quadratic points on the non-split Cartan modular curves $X_{ns}(p)$, for $p = 7, 11,$ and $13$. Recently, Siksek proved that all quadratic points on $X_{ns}(7)$ arise as pullbacks of rational points on $X_{ns}^+(7)$.…

Let E be an elliptic curve over a real quadratic field K and F/K a totally real finite Galois extension. We prove that E/F is modular.

Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in E(K)$ be a point of infinite order, and let $N^{-1}\alpha$ be the set of $N$-division points of $\alpha$ in $E(\bar{K})$. We prove strong effective and uniform…

We consider a map $F$ of class $C^r$ with a fixed point of parabolic type whose differential is not diagonalizable and we study the existence and regularity of the invariant manifolds associated with the fixed point using the…

Elliptic curves with a known number of points over a given prime field with n elements are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication…

We extend the finite element interpolated neural network (FEINN) framework from partial differential equations (PDEs) with weak solutions in $H^1$ to PDEs with weak solutions in $H(\textbf{curl})$ or $H(\textbf{div})$. To this end, we…

As a homomorphic image of the hyperalgebra $U_{q,R}(m|n)$ associated with the quantum linear supergroup $U_\upsilon(\mathfrak{gl}_{m|n})$, we first give a presentation for the $q$-Schur superalgebra $S_{q,R}(m|n,r)$ over a commutative ring…

We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne-Mumford stability. For every pair of integers 0<m<n, we…