Related papers: On Lehmer's Conjecture for Polynomials and for Ell…

We prove the conjectural relations between Mahler measures and $L$-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for $L$-values of CM elliptic curves of conductors 27 and 36. Furthermore,…

Let $N>1$ and let $\Phi_N(X,Y)\in\mathbb{Z}[X,Y]$ be the modular polynomial which vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of degree $N$. In this note we study the divisibility of the…

We consider the issue of when the L-polynomial of one curve over $\F_q$ divides the L-polynomial of another curve. We prove a theorem which shows that divisibility follows from a hypothesis that two curves have the same number of points…

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…

We obtain an explicit upper bound on the size of the coefficients of the elliptic modular polynomials $\Phi_N$ for any $N\geq1$. These polynomials vanish at pairs of $j$-invariants of elliptic curves linked by cyclic isogenies of degree…

We prove that under certain explicit conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of elliptic curve $L$-values and Bloch-Wigner dilogarithmmic values, conditionally on Beilinson's conjecture. In…

The classical modular polynomial for $j$-invariants describes the relation between two elliptic curves connected by isogenies. This polynomial has been applied to various algorithms in computational number theory, such as point counting on…

We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials $\Phi_N$ for the elliptic $j$-function. These bounds make explicit the best previously known asymptotic bounds. We then give an explicit…

We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $p$ by revisiting the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on…

We prove a conjecture of Boyd and Rodriguez Villegas relating the Mahler measure of the polynomial $(1+x)(1+y)+z$ and the value at $s=3$ of the $L$-function of an elliptic curve of conductor $15$. The proof makes use of the computation by…

We study congruences of the form F(j(z)) | U(p) = G(j(z)) mod p, where U(p) is the p-th Hecke operator, j is the basic modular invariant 1/q+744+196884q+... for SL2(Z), and F,G are polynomials with integer coefficients. Using the interplay…

In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1} +\alpha X^{pq}+X^{q+p-1}$, on finite fields $\mathbb{F}_{q^2}, q=p^k$,…

Let $g(x)$ be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if $g(h(x))$ has boundedly many terms, then $h(x)\in \C[x]$ must also have boundedly many terms. Solving an older conjecture raised by R\'enyi and by…

We present a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography. Our…

Let $F_q$ be a finite field of characteristic $p=2,3$. We give the number of irreducible polynomials $x^m+a_{m-1}x^{m-1}+...+a_0\in\F_q[x]$ with $a_{m-1}$ and $a_{m-3}$ prescribed for any given $m$ if $p=2$, and with $a_{m-1}$ and $a_1$…

We generalize the polynomial Szemer\'{e}di theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new…

Given an integer $q$ and a polynomial $f\in \mathbb Z_{q}[X]$ of degree $d$ with coefficients in the residue ring $\mathbb Z_q=\mathbb Z/q\mathbb Z,$ we obtain new results concerning the number of solutions to congruences of the form…

The paper [GLZ] "L-functions of Carlitz modules, resultantal varieties and rooted binary trees" is devoted to a description of some resultantal varieties related to L-functions of Carlitz modules. It contains a conjecture that some of these…

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 prove a conjectured formula relating the Mahler measure of the Laurent polynomial $1+X+X^{-1}+Y+Y^{-1}$, to the $L$-series of a conductor 15 elliptic curve.