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010 a| 2023031265;

020 a| 9781470464165q| (paperback);

020 z| 9781470475130q| (pdf);

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084 a| 11D25a| 11D59a| 11Y502| msc;

100 1 a| Känel, Rafael von,d| 1985-?| UNAUTHORIZED;

245 10 a| Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via the Shimura-Taniyama conjectureh| [electronic resource] /c| Rafael von Kanel, Benjamin Matschke.;

260 a| Providence, RI :b| American Mathematical Society,c| 2023.;

300 a| vi, 142 pages :b| illustrations ;c| 26 cm.;

490 0 a| Memoirs of the American Mathematical Society,x| 0065-9266 ;v| Volume 286, Number 1419;

504 a| Includes bibliographical references (pages 135-142).;

505 0 a| The Shimura-Taniyama conjecture -- Algorithms for S-unit equations -- Algorithms for Mordell equations -- Algorithms for Thue and Thue-Mahler equations -- Algorithms for generalized Ramanujan-Nagell equations -- Mordell equations and almost primitive solutions -- Height bounds for Thue and Thue-Mahler equations -- Height bounds for Ramanujan-Nagell equations -- Height bounds for Mordell and S-unit equations -- The elliptic logarithm sieve.;

506 a| Available only to authorized users.;

520 a| "In the first part we construct algorithms (over Q) which we apply to solve Sunit, 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-Sprindzuk. 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 thefirst part. Our results and algorithms all ultimately rely on the method of Faltings (Arakelov, Parsin, 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 Sintegral 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"--c| Provided by publisher.;

538 a| Mode of access: World Wide Web;

650 0 a| Diophantine equations.=| ^A352074;

650 7 a| Number theory -- Diophantine equations -- Cubic and quartic equations.2| msc;

650 7 a| Number theory -- Diophantine equations -- Thue-Mahler equations.2| msc;

650 7 a| Number theory -- Computational number theory -- Computer solution of Diophantine equations.2| msc;

655 0 a| Electronic books.=| ^A491897;

700 1 a| Matschke, Benjamin.?| UNAUTHORIZED;

856 40 z| Full text available from Ebook Central - Academic Completeu| https://ebookcentral.proquest.com/lib/eastcarolina/detail.action?docID=30627623;

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Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via the Shimura-Taniyama conjecture / Rafael von Kanel, Benjamin Matschke.

Author/creator	Känel, Rafael von, 1985-
Other author	Matschke, Benjamin.
Format	Electronic
Publication Info	Providence, RI : American Mathematical Society, 2023.
Description	vi, 142 pages : illustrations ; 26 cm.
Supplemental Content	Full text available from Ebook Central - Academic Complete
Subjects	Diophantine equations.

Series	Memoirs of the American Mathematical Society, 0065-9266 ; Volume 286, Number 1419
Contents	The Shimura-Taniyama conjecture -- Algorithms for S-unit equations -- Algorithms for Mordell equations -- Algorithms for Thue and Thue-Mahler equations -- Algorithms for generalized Ramanujan-Nagell equations -- Mordell equations and almost primitive solutions -- Height bounds for Thue and Thue-Mahler equations -- Height bounds for Ramanujan-Nagell equations -- Height bounds for Mordell and S-unit equations -- The elliptic logarithm sieve.
Abstract	"In the first part we construct algorithms (over Q) which we apply to solve Sunit, 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-Sprindzuk. 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 thefirst part. Our results and algorithms all ultimately rely on the method of Faltings (Arakelov, Parsin, 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 Sintegral 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"-- Provided by publisher.
Bibliography note	Includes bibliographical references (pages 135-142).
Access restriction	Available only to authorized users.
Technical details	Mode of access: World Wide Web
Genre/form	Electronic books.
LCCN	2023031265
ISBN	9781470464165 (paperback)
ISBN	(pdf)

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Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via the Shimura-Taniyama conjecture / Rafael von Kanel, Benjamin Matschke.

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