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Let q be a prime power and E a non-isotrivial elliptic curve over Fq(T) given by a Weierstrass model. We survey the construction, with an explicit point of view, of the modular parametrization of E by the associated Drinfeld modular curve.…

This work builds on earlier results. We define universal elliptic Gau{\ss} sums for Atkin primes in Schoof's algorithm for counting points on elliptic curves. Subsequently, we show these quantities admit an efficiently computable…

We compute the $L$-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local $L$-factor and the…

We outline a general algorithm for computing an explicit model over a number field of any curve of genus 2 whose (unpolarized) Jacobian is isomorphic to the product of two elliptic curves with CM by the same order in an imaginary quadratic…

Let $E$ be an elliptic curve over $\Q$ without complex multiplication, and which is not isogenous to a curve with non-trivial rational torsion. For each prime $p$ of good reduction, let $|E(\F_p)|$ be the order of the group of points of the…

In Physics, we are generally interested in real solutions involving natural phenomena, where knowledge of real functions of real variables is sufficient to obtain physically relevant results. However, the complexity of phenomena associated…

We describe first-degree prime ideals of biquadratic extensions in terms of first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms.…

We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic as a special case of…

We obtain new uniform bounds for the symmetric tensor rank of multiplication in finite extensions of any finite field Fp or Fp2 where p denotes a prime number greater or equal than 5. In this aim, we use the symmetric Chudnovsky-type…

Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which…

We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the…

This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence…

This paper contains an account of arbitrary cubic function fields of characteristic three. We define a standard form for an arbitrary cubic curve and consider its function field. By considering an integral basis for the maximal order of…

For any prime p we consider the density of elements in the multiplicative group of the finite field F_p having order, respectively index, congruent to a(mod d). We compute these densities on average, where the average is taken over all…

Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet $L$-functions is true, we then establish…

Let $E/\mathbb{Q}$ be an elliptic curve and $p > 2$ be a prime of good ordinary reduction for $E$. Assume that the residue representation associated with $(E, p)$ is irreducible. In this paper, we prove more cases on several Iwasawa main…

Polynomial series approximations are a central theme in approximation theory due to their utility in an abundance of numerical applications. The two types of series, which are featured most prominently, are Taylor series expansions and…

In this article we show the rough outline of a computer algorithm to generate lower bounds on the exponential function of (in principle) arbitrary precision. We implemented this to generate all necessary analytic terms for the Boltzmann…

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints,…

We consider the construction of the fundamental function and Abelian differentials of the third kind on a plane algebraic curve over the field of complex numbers that has no singular points. The algorithm for constructing differentials of…