Models of Genus One Curves

Mohammad Sadek

ISBN 10: 3843353840 / ISBN 13: 9783843353847

Published by LAP LAMBERT Academic Publishing Sep 2010, 2010

About this Item

This item is printed on demand - Print on Demand Titel. Neuware -Let E be an elliptic curve defined over a number field K. An element of the n-Selmer group of E can be represented as a geometric object. Namely, as an everywhere locally soluble genus one curve defined by an equation of degree n. This equation is a generalised binary quartic when n=2, a ternary cubic when n=3, and two quadrics in four variables when n=4. By minimising these equations we mean making their invariants as small as possible. Unfortunately, the minimal (with the smallest invariants) equations of degree n are not unique in general. We exploit the theory of minimal regular models to find an alternative definition of minimality. Then we use this new definition to count the minimal equations of degree n.VDM Verlag, Dudweiler Landstraße 99, 66123 Saarbrücken 124 pp. Englisch.

Seller Inventory # 9783843353847

Bibliographic Details

• Title: Models of Genus One Curves
• Publisher: LAP LAMBERT Academic Publishing Sep 2010
• Publication Date: 2010
• Language: English
• ISBN 10: 3843353840
• ISBN 13: 9783843353847
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 220W x 150H x 9D millimeters 220 millimeters width by 150 millimeters height by 9 millimeters depth
• Weight: 203 grams

About this title

Synopsis

Let E be an elliptic curve defined over a number field K. An element of the n-Selmer group of E can be represented as a geometric object. Namely, as an everywhere locally soluble genus one curve defined by an equation of degree n. This equation is a generalised binary quartic when n=2, a ternary cubic when n=3, and two quadrics in four variables when n=4. By minimising these equations we mean making their invariants as small as possible. Unfortunately, the minimal (with the smallest invariants) equations of degree n are not unique in general. We exploit the theory of minimal regular models to find an alternative definition of minimality. Then we use this new definition to count the minimal equations of degree n.

About the Author

Mohammad Sadek received his Ph.D. from Cambridge University in 2010. He is an assistant professor at the American University in Cairo. His research interests are in arithmetic of curves and their regular models.