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

Abstract

In the first part we construct algorithms (over Q) 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. The first type of our algorithms uses modular symbols, and the second type combines explicit height bounds with efficient sieves. In particular we construct a refined sieve for S-unit equations which combines Diophantine approximation techniques of de Weger with new geometric ideas. 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-Sprindžuk. In fact we get refined versions of their theorems, which improve the actual best results in many fundamental cases. We also conduct some effort to work out optimized height bounds for S-unit and Mordell equations which are used in our algorithms of the first part. Our results and algorithms all ultimately rely on the method of Faltings (Arakelov, Paršin, 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). Here we combine a geometric interpretation of the known elliptic logarithm reduction (initiated by Zagier) with several conceptually new ideas. The resulting "elliptic logarithm sieve" is crucial for some of our algorithms of the first part. Moreover, it considerably extends the class of elliptic Diophantine equations which can be solved in practice: To demonstrate this we solved many notoriously difficult equations by combining our sieve with known height bounds based on the theory of logarithmic forms.

Integral points on moduli schemes of elliptic curves

“…Here h φ : Y ( Q) → R is a height, defined in (3.4), which depends on the choice of P and which has the Northcott property if |P| C < ∞, see Section 3.2. However the bound for h φ in (i) can also be useful when |P| C = ∞: For example cubic Thue-Mahler equations define moduli schemes Y = M P with |P| C = ∞, see [vKM16,p.60]. We remark that the case g = 1 of Theorem A was established in [vK14,Thm 7.1].…”

“…For example, it was shown in [vK14] that the case Thm 1.1]; see also Frey [Fre97,p.544]). Furthermore, it was demonstrated in [vKM16] that these explicit Weil height bounds combined with efficient sieves constructed in [vKM16] allow to solve these classical equations in practice. We are currently trying to work out similar explicit applications of Theorem A for equations defining Hilbert moduli schemes with g ≥ 2.…”

Potential good reduction of hyperelliptic curves

“…By Corollary 7, we proceed by solving the S-unit equation x + y = 1, where x, y ∈ O × S . These solutions can be computed using existing algorithms, such as those described by von Känel and Matschke [10]. Using their Sage [8] implementation, we obtained 21 solutions, and can conclude that any such curve must be isomorphic to one of the following curves:…”

A robust implementation for solving the $S$-unit equation and several applications

“…Item (3) paraphrases an interesting idea due to Wildanger [43], which was generalized by Smart [38]. This is an extremely promising and potentially effective method of reducing the search space, and has been implemented recently in special cases by several people, including Kousianas [22], Bennett, Gherga, and Rechnitzer [5], von Känel and Matschke [29], and others. Future work will certainly focus on including this sieving technique for our functions.…”

“…It is worth mentioning the recent results of von Känel and Matschke[29], who solve S-unit equations using modularity.…”