Let E/Q be an elliptic curve. Silverman and Stange define primes p and q to be an elliptic amicable pair if #E(Fp) = q and #E(Fq) = p. More generally, they define the notion of aliquot cycles for elliptic curves. Here we study the same notion in the case that the elliptic curve is defined over a number field K. We focus on proving the existence of an elliptic curve E/K with aliquot cycle (p1, ....

Efficient computation of elliptic curve scalar multiplication has been a significant problem since Koblitz [13] and Miller [14] independently proposed elliptic curve cryptography, and several efficient methods of scalar multiplication have been proposed (e.g., [8], [9], [12]). A standard approach for computing scalar multiplication is to use the Frobenius endomorphism. If we compute the s-multi...

A genus one curve defined over Q which has points over Qp for all primes p may not have a rational point. It is natural to study the classes of Q-extensions over which all such curves obtain a global point. In this article, we show that every such genus one curve with semistable Jacobian has a point defined over a solvable extension of Q.

Let Y be a smooth projective irreducible curve defined over a finite field F q. We assume that G is a finite subgroup of the automorphism group of Y whose order divides q − 1. We show that relations among idempotents corresponding to the irreducible characters of subgroups H of G imply similar relations among generalized pranks of Y. We also relate this result to the realization of finite group...

A Q-curve is an elliptic curve over a number field K which is geometrically isogenous to each of its Galois conjugates. K. Ribet [17] asked whether every Q-curve is modular, and he showed that a positive answer would follow from J.-P. Serre’s conjecture on mod p Galois representations. We answer Ribet’s question in the affirmative, subject to certain local conditions at 3.

We discuss Chapter 6 from Silverman & Tate’s Rational Points on Elliptic Curves, in which the authors outline a means of proving Kronecker’s Jugendtraum for Q and Q(i). One considers C[n], the kernel of the multiplication-by-n map on an elliptic curve C. After defining the number field Q(C[n]), we examine instances in which its Galois group over a field K is abelian. For the elliptic curve

We prove that every algebraic curve X/Q is birational over C to a Teichmüller curve. keywords: algebraic curve, mapping class group, Teichmüller curve, Veech group. MSC code: 32G15, 37D40.

in x and y. The degree of the curve is the degree of the polynomial f(x, y). A parametric curve α(t) = (p(t), q(t)), where p(t) and q(t) are polynomials in t, is an algebraic curve. This is because we can eliminate t from the two polynomial equations x = p(t) and y = q(t) using the resultant technique, and obtain a polynomial equation, called their resultant, in x and y. The resultant has a deg...