Related papers: Using Fricke modular polynomials to compute isogen…

We give an algorithm to compute $(\ell,\ell,\ell)$-isogenies from the Jacobians of genus three hyperelliptic curves to the Jacobians of non-hyperelliptic curves. An important application is to reduce the discrete logarithm problem in the…

We investigate the behaviour of elliptic Feynman integrals under modular transformations. This has a practical motivation: Through a suitable modular transformation we can achieve that the nome squared is a small quantity, leading to fast…

Given an elliptic curve $E$ and a finite Abelian group $G$, we consider the problem of counting the number of primes $p$ for which the group of points modulo $p$ is isomorphic to $G$. Under a certain conjecture concerning the distribution…

Let E be an elliptic curve of conductor N and rank one over Q. So there is a non-constant morphism X+0(N) --> E defined over Q, where X+0(N) = X0(N)/wN and wN is the Fricke involution of the modular curve X+0(N). Under this morphism the…

We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…

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…

A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of…

We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a "determinant" map from this moduli surface to (Z/NZ)*; its fibers are the components of the…

We consider the $q^\text{Volume}$ lozenge tiling model on a large, finite hexagon. It is well-known that random lozenge tilings of the hexagon correspond to a two-dimensional determinantal point process via a bijection with ensembles of…

We show that a character sum attached to a family of 3-isogenies defined on the fibers of a certain elliptic surface over $\mathbb{F}_p$ relates to the class number of the quadratic imaginary number field $\Q(\sqrt{-p})$. In this sense,…

We investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple proof of an elliptic…

We give a new approach to the elliptic curve discrete logarithm problem over cubic extension fields $\mathbb{F}_{q^3}$. It is based on a transfer: First an $\mathbb{F}_q$-rational $(\ell,\ell,\ell)$-isogeny from the Weil restriction of the…

We compute the $F$-pure threshold of a degree three homogeneous polynomial in three variables with an isolated singularity. The computation uses elementary methods to prove a known result of Bhatt and Singh.

In this paper, we propose the use of Ramanujan class of polynomials for the construction of prime order elliptic curves using the CM-method. We compare (theoretically and experimentally) the efficiency of using this new class against the…

For an elliptic curve $E$ over a finite field $\F_q$, where $q$ is a prime power, we propose new algorithms for testing the supersingularity of $E$. Our algorithms are based on the Polynomial Identity Testing (PIT) problem for the $p$-th…

In this paper, we provide a classification of certain points on Hilbert modular varieties over finite fields under a mild assumption on Newton polygon. Furthermore, we use this characterization to prove estimates for the size of isogeny…

We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still…

We describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface $A$ over $\mathbb{Q}$ with geometric endomorphism ring equal to $\mathbb{Z}$, computes all the other p.p. abelian surfaces over $\mathbb{Q}$ that…

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…

The Deligne-Ogus-Shioda theorem guarantees the existence of isomorphisms between products of supersingular elliptic curves over finite fields. In this paper, we present methods for explicitly computing these isomorphisms in polynomial time,…