We give conditions on the rational numbers a, b, c which imply that there are infinitely many triples (x, y, z) of rational numbers such that x+ y + z = a+ b+ c and xyz = abc. We do the same for the equations x + y + z = a + b + c and x + y + z = a + b + c. These results rely on exhibiting families of positive-rank elliptic curves.

Graph curves are a useful construction for considering various problems in algebraic geometry. In this paper, we explain some of the results about graph curves from the seminal paper by Bayer and Eisenbud [3]. We provide background in simplicial cohomology and the canonical series of a curve for the beginning reader. In our study of algebraic curves, it is helpful to consider degenerations – ob...

We study closed smooth convex plane curves Γ enjoying the following property: a pair of points x, y can traverse Γ so that the distances between x and y along the curve and in the ambient plane do not change; such curves are called bicycle curves. Motivation for this study comes from the problem how to determine the direction of the bicycle motion by the tire tracks of the bicycle wheels; bicyc...

An amicable pair for an elliptic curve E/Q is a pair of primes (p, q) of good reduction for E satisfying #Ẽp(Fp) = q and #Ẽq(Fq) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliqout cycles, but that CM elliptic curves (with j 6= 0) have no aliqout cycles of length greate...

Some of these curves are considered in a rather incidental way in the writings of Poincare.** However, the full concept of pseudo closed trajectories does not seem to have been discussed explicitly heretofore. We assume that all of the variables in the equations (1) are real, that the functions X and Y are continuous in an open connected region R in the xy-plane, that these functions satisfy Li...

We describe degenerations and smoothings of linear series on some reducible algebraic curves. Applications include a proof that the moduli space of curves of genus g has general type for all g > 24, a proof that the monodromy action is transitive on the set of linear series of dimension r and degree d on a general curve of genus g when p := g — (r + l)(g — d+r) = 0, a proof that there exist Wei...

We present an algorithm for counting points on superelliptic curves y = f(x) over a finite field Fq of small characteristic different from r. This is an extension of an algorithm for hyperelliptic curves due to Kedlaya. In this extension, the complexity, assuming r and the genus are fixed, is O(log q) in time and space, just like for hyperelliptic curves. We give some numerical examples obtaine...

Given a symmetric polynomial 8(x, y) over a perfect field k of characteristic zero, the Galois graph G(8) is defined by taking the algebraic closure k̄ as the vertex set and adjacencies corresponding to the zeroes of 8(x, y). Some graph properties of G(8), such as lengths of walks, distances and cycles are described in terms of 8. Symmetry is also considered, relating the Galois group Gal(k̄/k) t...

The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences (1/2, 1/4, 1/4), (1/2, 1/3, 1/6) and (1/3, 1/3, 1/3). These form a special class of algebraic transformations of Gauss hypergeometric functions, of arbitrary high degree. The Gauss hypergeometric functions can be identified as elliptic integrals on the genus 1 curves y = x − x or...

We classify primes p for which there exist elliptic curves E/Q with conductor NE ∈ {18p, 36p, 72p} and nontrivial rational 2-torsion, and, in consequence, show that, for “almost all” primes p, the Diophantine equation x + y = pz has at most finitely many solutions in coprime nonzero integers x, y and z and positive integers α and n ≥ 4. To prove this result, we appeal to such disparate techniqu...