Computing elliptic curves over $\mathbb {Q}$

Bennett, Michael A.; Gherga, Adela; Rechnitzer, Andrew

doi:10.1090/mcom/3370

Cited by 13 publications

(12 citation statements)

References 44 publications

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“…We point out that (CL) and the method of Koutsianas [Kou15] both allow to deal with more general number fields K, while our approach currently only works in the considerably simpler case K = Q. As already mentioned, Bennett-Rechnitzer [BR15a,BR15b] substantially refined the classical Thue-Mahler approach in order to compute M ({p}) for all primes p < 2 • 10 9 . This computation is unfavorable for our method, since finding unconditionally all the required Mordell-Weil bases would (when possible) take a long time with the known techniques.…”

Section: Elliptic Curves With Good Reduction Outside a Given Set Of P...mentioning

confidence: 99%

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

Känel¹,

Matschke²

2016

Preprint

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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.

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Section: Elliptic Curves With Good Reduction Outside a Given Set Of P...mentioning

confidence: 99%

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

Känel¹,

Matschke²

2016

Preprint

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“…with the latter case occurring only if xy is odd. The isomorphism classes of elliptic curves over Q with good reduction outside {2, 3, 5, 7, 11} have recently been completely and rigorously determined using two independent approaches, by von Kanel and Matschke [37] (via computation of S-integral points on elliptic curves, based upon bounds for elliptic logarithms), and by the first author, Gherga and Rechnitzer [3] (using classical invariant theory to efficiently reduce the problem to solutions of cubic Thue-Mahler equations). One finds that there are precisely 592192 isomorphism classes of elliptic curves over Q with good reduction outside {2, 3, 5, 7, 11}; details are available at, e.g.…”

Section: (Very) Small Values Of Nmentioning

confidence: 99%

“…Suppose now that d = 231. The class of the fractional ideal P/P has order 3, and we have chosen γ to be a generator of (P/P) 3 . We may rewrite (25) as…”

Section: Frey-hellegouarch Curves and Related Objectsmentioning

confidence: 99%

Differences between perfect powers : the Lebesgue-Nagell Equation

Bennett¹,

Siksek²

2021

Preprint

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We develop a variety of new techniques to treat Diophantine equations of the shape x 2 + D = y n , based upon bounds for linear forms in p-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantine approximation. We use these to explicitly determine the set of all coprime integers x and y, and n ≥ 3, with the property that y n > x 2 and x 2 − y n has no prime divisor exceeding 11.

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“…For E pr and E all , we estimate weighted averages using the Stein-Watkins databases [SW02] consisting of over 11 milion isogeny classes of prime conductor N < 10 10 and over 115 million isogeny classes of arbitrary conductors N ≤ 10 8 . The Stein-Watkins databases do not catalogue all isogeny classes in these conductor ranges, but at least the Stein-Watkins prime conductor database appears to be nearly complete: [BGR19] estimates it contains over 99.8% of curves with prime conductor N < 10 10 . (In fact [BGR19] computed a much larger database of prime conductor elliptic curves, but that database does not include rank calculations which we require.)…”

Section: Introduction and Conjecturesmentioning

confidence: 99%

“…The Stein-Watkins databases do not catalogue all isogeny classes in these conductor ranges, but at least the Stein-Watkins prime conductor database appears to be nearly complete: [BGR19] estimates it contains over 99.8% of curves with prime conductor N < 10 10 . (In fact [BGR19] computed a much larger database of prime conductor elliptic curves, but that database does not include rank calculations which we require.) For E ht , we compute weighted averages using the height database from [BHK + 16], which contains all of the over 238 million curves with naive height H ≤ 2.7 • 10 10 .…”

Section: Introduction and Conjecturesmentioning

confidence: 99%