Related papers: Explicit Bound for the Prime Ideal Theorem in Resi…

Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime p and positive integer m=o(sqrt(p)/(log p)^4), outputs an elliptic curve E over the finite field F_p for which the cardinality of E(F_p) is…

We give an algorithm to explicitly determine all elements of the $q$-torsion (for $q$ an odd prime) of the Brauer group of an elliptic curve over any base field of characteristic different from $q$, containing a primitive $q$-th root of…

The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use arbitrary elliptic curves and rational points of infinite order to generate sequences that are…

For a fixed number field and an elliptic curve defined and semi-stable over this number field, we consider the set of prime numbers p such that the Galois representation attached to the p-torsion points of the elliptic curve is reducible.…

Fix m >= 1 and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these…

Chebyshev interpolation is a highly effective, intensively studied method and enjoys excellent numerical properties. The interpolation nodes are known beforehand, implementation is straightforward and the method is numerically stable. For…

The complexity of the elliptic curve method of factorization (ECM) is proven under the celebrated conjecture of existence of smooth numbers in short intervals. In this work we tackle a different version of ECM which is actually much more…

Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.

A Chebyshev expansion is a series in the basis of Chebyshev polynomials of the first kind. When such a series solves a linear differential equation, its coefficients satisfy a linear recurrence equation. We interpret this equation as the…

We give practical algorithms for computing the divisor class group and the gonality of a curve over a finite field, achieving several orders of magnitude speedup over existing methods for sufficiently large genus or residue field. The…

Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…

In this paper, we propose algorithms to compute differential Chow forms for prime differential ideals which are given by their characteristic sets. The main algorithm is based on an optimal bound for the order of a prime differential ideal…

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…

We give explicit formulas for the number of distinct elliptic curves over a finite field, up to isomorphism, in the families of Legendre, Jacobi, Hessian and generalized Hessian curves.

For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with…

Let $\psi_{\mathbb K}$ be the Chebyshev function of a number field $\mathbb K$. Let $\psi^{(1)}_{\mathbb K}(x):=\int_{0}^{x}\psi_{\mathbb K}(t)\,d t$ and $\psi^{(2)}_{\mathbb K}(x):=2\int_{0}^{x}\psi^{(1)}_{\mathbb K}(t)\,d t$. We prove…

We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve $E/\Fp_q(C)$ over a function field over a finite field that have rank $\geq 2$, and for their average rank. The main tools are constructions and…

Baba and Granath generalize Elkies' theorem on infinitude of supersingular primes for elliptic curves to abelian surfaces with quaternionic multiplication of discriminant $6$, whose field of moduli is $\mathbb{Q}$ and which is a Jacobian in…

We employ the generalized Remez algorithm, initially suggested by P. T. P. Tang, to perform an experimental study of Chebyshev polynomials in the complex plane. Our focus lies particularly on the examination of their norms and zeros. What…

Let $k$ be a number field, $f(x)\in k[x]$ a polynomial over $k$ with $f(0)\neq 0$, and $\O_{k,S}^*$ the group of $S$-units of $k$, where $S$ is an appropriate finite set of places of $k$. In this note, we prove that outside of some natural…