Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture

In the first part we construct algorithms which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we used the resulting data to motivate various conjectures and questions, including Baker's explicit abc-conjecture and a new conjecture on S-integral points of any hyperbolic genus one curve over Q. In the second part we establish new results for certain old Diophantine problems (e.g. the difference of squares and cubes) related to Mordell equations, and we prove explicit height bounds for cubic Thue, cubic Thue--Mahler and generalized Ramanujan--Nagell equations. As a byproduct, we obtain here an alternative proof of classical theorems of Baker, Coates and Vinogradov-Sprindzuk. In fact we get refined versions of their theorems, which improve the actual best results in many fundamental cases. Our results and algorithms all ultimately rely on the method of Faltings (Arakelov, Parshin, Szpiro) combined with the Shimura-Taniyama conjecture, and they all do not use lower bounds for linear forms in (elliptic) logarithms. In the third part we solve the problem of constructing an efficient sieve for the S-integral points of bounded height on any elliptic curve E over Q with given Mordell-Weil basis of E(Q).Comment: Comments are always very welcom

Descent for the punctured universal elliptic curve, and the average number of integral points on elliptic curves

We show that the average number of integral points on elliptic curves, counted modulo the natural involution on a punctured elliptic curve, is bounded from above by $2.1 \times 10^8$. To prove it, we design a descent map, whose prototype goes at least back to Mordell, which associates a pair of binary forms to an integral point on an elliptic curve. Other ingredients of the proof include the upper bounds for the number of solutions of a Thue equation by Evertse and Akhtari-Okazaki, and the estimation of the number of binary quartic forms by Bhargava-Shankar. Our method applies to $S$-integral points to some extent, although our present knowledge is insufficient to deduce an upper bound for the average number of them. We work out the numerical example with $S=\{2\}$.Comment: 22 pages, 2 tables. Major changes include: The statement of the main theorem was incorrect in the previous version. Section 7 is removed. Introduction is rewritte

Elliptic curves with good reduction outside of the first six primes

We present a database of rational elliptic curves, up to Q-isomorphism, with good reduction outside {2,3,5,7,11,13}. We provide a heuristic involving the abc and BSD conjectures that the database is likely to be the complete set of such curves. Moreover, proving completeness likely needs only more computation time to conclude. We present data on the distribution of various quantities associated to curves in the set. We also discuss the connection to S-unit equations and the existence of rational elliptic curves with maximal conductor.Comment: 19 pages, 4 figures; the data is available at https://github.com/elliptic-curve-data/ec-data-S

Open Diophantine Problems

We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.Comment: 58 pages. to appear in the Moscow Mathematical Journal vo. 4 N.1 (2004) dedicated to Pierre Cartie

Words and Transcendence

Is it possible to distinguish algebraic from transcendental real numbers by considering the $b$-ary expansion in some base $b\ge2$? In 1950, \'E. Borel suggested that the answer is no and that for any real irrational algebraic number $x$ and for any base $g\ge2$, the $g$-ary expansion of $x$ should satisfy some of the laws that are shared by almost all numbers. There is no explicitly known example of a triple $(g,a,x)$, where $g\ge3$ is an integer, $a$ a digit in $\{0,...,g-1\}$ and $x$ a real irrational algebraic number, for which one can claim that the digit $a$ occurs infinitely often in the $g$-ary expansion of $x$. However, some progress has been made recently, thanks mainly to clever use of Schmidt's subspace theorem. We review some of these results

Integral points on Hilbert moduli schemes

We use the method of Faltings (Arakelov, Par\v{s}in, Szpiro) in order to explicitly study integral points on a class of varieties over $\mathbb Z$ called Hilbert moduli schemes. For instance, integral models of Hilbert modular varieties are classical examples of Hilbert moduli schemes. Our main result gives explicit upper bounds for the height and the number of integral points on Hilbert moduli schemes.Comment: Comments are always very welcom

Perfect powers generated by the twisted Fermat cubic

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. It is shown that there are finitely many perfect powers in such a sequence whose first term is greater than 1. Moreover, if the first term is divisible by 6 and the generating point is triple another rational point then there are no perfect powers in the sequence except possibly an lth power for some l dividing the order of 2 in the first term.Comment: 10 page

The density of rational points in curves and surfaces

Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes from the counting function those points that lie on lines in the surface. The bounds are uniform for all $X$ of a given degree. They are best possible in the case of curves. As an application it is shown that if $F$ is an irreducible binary form of degree 3 or more then almost all integers represented by $F$ have essentially one such representation.Comment: 46 pages, published version; appendix by J.-L. Colliot-Th\'el\`en

Diophantische Approximationen

This number theoretic conference was focused on a broad variety of subjects in (or closely related to) Diophantine approximation, including the following: metric Diophantine approximation, Mahler’s method in transcendence, geometry of numbers, theory of heights, arithmetic dynamics, function fields arithmetic

Computing all S-integral points on elliptic curves

In this note we combine the advantages of the methods of Siegel-Baker-Coates and of Lang-Zagier for the computation of S-integral points on elliptic curves in Weierstrass normal form over the rationals. In this way we are able to overcome the absence of an explicit lower bound for linear forms in q-adic elliptic logarithms. We present an efficient algorithm for determining all S-integral points on such curves