On the symplectic type of isomorphisms of the p-torsion of elliptic curves / Nuno Freitas, Alain Kraus.

Format:

Book

Author/Creator:

Freitas, Nuno.

Contributor:

Kraus, Alain.

Series:

Memoirs of the American Mathematical Society ; 1361

Memoirs of the American Mathematical Society ; Volume 277, Number 1361 (second of 6 numbers)

Language:

English

Subjects (All):

Curves, Elliptic.

Number theory.

Physical Description:

1 online resource (118 pages)

Edition:

1st ed.

Other Title:

On the Symplectic Type of Isomorphisms of the p- Torsion of Elliptic Curves.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"Let be a prime. Let and be elliptic curves with isomorphic torsion modules and . Assume further that either every modules isomorphism admits a multiple with preserving the Weil pairing; or no isomorphism preserves the Weil pairing. This paper considers the problem of deciding if we are in case . Our approach is to consider the problem locally at a prime . Firstly, we determine the primes for which the local curves and contain enough information to decide between . Secondly, we establish a collection of criteria, in terms of the standard invariants associated to minimal Weierstrass models of and , to decide between . We show that our results give a complete solution to the problem by local methods away from . We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form and where is a prime; we also give incremental results on the Fermat equation . As a different application, we discuss variants of a question raised by Mazur concerning the existence of symplectic isomorphisms between the torsion of two non-isogenous elliptic curves defined over "-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Motivation and results

1.1. Introduction

1.2. A double motivation

1.3. Our approach to the problem of determining the symplectic type

1.4. A complete list of local symplectic criteria at ℓ≠

Chapter 2. The existence of local symplectic criteria

2.1. Existence of symplectic criteria in terms of the image of \overline{ }_{ , }

2.2. Symplectic criteria with \overline{ }_{ , }( _{ℚ_{ℓ}}) abelian

Chapter 3. The criterion in the case of good reduction

3.1. The action of Frobenius

3.2. Proof of Theorem 1.18

3.3. A more general theorem

Chapter 4. Elliptic curves with potentially good reduction

4.1. An useful Weierstrass model

4.2. The field of good reduction

4.3. The Galois group of the -torsion field in the cases =3,4

4.4. Proof of Theorem 1.4

4.5. Proof of Lemmas 1.7, 1.8, 1.16 and 1.17

4.6. The completeness of Table 1.1

Chapter 5. The morphism _{ }

5.1. Explicit description of _{ }

5.2. The morphism _{ } in the tame case =3

5.3. The morphism _{ } in the wild case =3

5.4. The morphism _{ } in the tame case =4

5.5. The morphism _{ } in the wild case =4

5.6. The morphism _{ } in the wild case =8

5.7. The morphism _{ } in the wild case =12

5.8. Tables with coordinate changes

Chapter 6. Proof of the criteria

6.1. Proof of Theorem 1.3

6.2. Proof of Theorem 1.5

6.3. Proof of Theorem 1.6

6.4. Proof of Theorem 1.9

6.5. Proof of Theorem 1.10

6.6. Proof of Theorem 1.22

6.7. Proof of Theorems 1.13 and 1.15

Chapter 7. Applications

7.1. Revisiting a question of Mazur

7.2. The Generalized Fermat equation ²+ ³= ^{ }

7.3. On the hyperelliptic curves ²= ^{ }-ℓ and ²= ^{ }-2ℓ

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9781470470937

1470470934

OCLC:

1338694970