The paper is largely expository. The first part is devoted to studying integer solutions to Pell’s Equation: u2 − dv2 = 1. The authors present the classic construction of a fundamental solution via continued fractions, from which all solutions can be derived. The primary focus of the second part is on rational solutions to the Thue’s equation, u3−dv3 = 1. The authors explain why these rational ...

We study the arithmetic structure of elliptic curves over k(t), where k is an algebraically closed field. In [Shi86] Shioda shows how one may determine rank of the Néron-Severi group of a Delsarte surface–a surface that may be defined by four monomial terms. To this end, he describes an explicit method of computing the Lefschetz number of a Delsarte surface. He proves the universal bound of 56 ...

To get a feeling for our level of ignorance in the face of such questions, consider that, before Faltings, there was not a single curve X (of genus > 1) for which we knew this statement to be true for all number fields K over which X is defined! Already in the twenties, Weil and Siegel made serious attempts to attack the problem. Siegel, influenced by Weil's thesis, used methods of diophantine ...

Matsumoto et al. define the Mordell-Tornheim L-functions of depth k by LMT(s1, . . . , sk+1;χ1, . . . , χk+1) := ∞ ∑

Let E be an elliptic curve over a number field K. Descent calculations on E can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this group. The general method of 4-descent, developed in the PhD theses of Siksek, Womack and Stamminger, has been implemented in Magma (when K = Q) and works well for elli...

Suppose E is an elliptic curve defined over Q. At the 1983 ICM the first author formulated some conjectures that propose a close relationship between the explicit class field theory construction of certain abelian extensions of imaginary quadratic fields and an explicit construction that (conjecturally) produces almost all of the rational points on E over those fields. Those conjectures are to ...

Let K be an algebraically closed field of prime characteristic p, let N ∈ N, let Φ : Gm −→ Gm be a self-map defined over K, let V ⊂ Gm be a curve defined over K, and let α ∈ Gm(K). We show that the set S = {n ∈ N : Φn(α) ∈ V } is a union of finitely many arithmetic progressions, along with a finite set and finitely many p-arithmetic sequences, which are sets of the form {a+ bpkn : n ∈ N} for so...

Based on ideas from recent joint work with Bjorn Poonen, we describe an algorithm that can in certain cases determine the set of rational points on a curve C, given only the p-Selmer group S of its Jacobian (or some other abelian variety C maps to) and the image of the p-Selmer set of C in S. The method is more likely to succeed when the genus is large, which is when it is usually rather diffic...

Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K of degree d, and denote its Jacobian by J . Denote the Mordell–Weil rank of J(K) by r. We give an explicit and practical Chabauty-style criterion for showing that a given subset K ⊆ C(K) is in fact equal to C(K). This criterion is likely to be successful if r ≤ d(g − 1). We also show that the only sol...