[Paper Review] Period, index and potential sha

Pete L. Clark, Shahed Sharif|ArXiv.org|Nov 18, 2008

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper advances the theory of O'Neil's period-index obstruction map to resolve arithmetic questions about genus one curves over global fields. It proves that for any pair (P, I) with P dividing I and I dividing P², there exists a number field K and a genus one curve C over K with period P and index I. It further constructs infinitely many such curves with Jacobian E and index P², yielding strong consequences for the structure of Shafarevich-Tate groups under field extensions.

ABSTRACT

In this paper we advance the theory of O'Neil's period-index obstruction map and derive consequences for the arithmetic of genus one curves over global fields. Our first result implies that for every pair of positive integers (P,I) with P dividing I and I dividing P^2, there exists a number field K and a genus one curve C over K with period P and index I. Second, let E be any elliptic curve over a global field K, and let P > 1 be any integer indivisible by the characteristic of K. We construct infinitely many genus one curves C over K with period P, index P^2, and Jacobian E. We deduce strong consequences on the structure of Sharevich-Tate groups under field extension.

Motivation & Objective

• To extend O'Neil's period-index obstruction map to global fields and apply it to the arithmetic of genus one curves.
• To resolve the possible range of period and index pairs (P, I) for genus one curves over global fields.
• To construct infinite families of genus one curves with prescribed period P, index P², and given Jacobian elliptic curve E.
• To deduce structural consequences for the Shafarevich-Tate group under field extensions.
• To explore the generic behavior of index relative to period in moduli spaces of genus one curves.

Proposed method

• Utilizes Lichtenbaum-Tate duality and theta group functoriality to analyze the period-index obstruction map.
• Applies Galois cohomology techniques to H¹(K, E) and constructs classes with controlled local behavior.
• Employs corestriction maps and prime ideal selection to build cohomology classes with specified period and index.
• Uses the existence of full P-torsion on elliptic curves and conditions on P*-torsion to control the obstruction map.
• Constructs explicit examples via field extensions and local conditions to realize desired period and index pairs.
• Applies results on Brauer groups and cohomological symbols to analyze the obstruction map in higher-dimensional and positive characteristic settings.

Experimental results

Research questions

• RQ1 For which pairs (P, I) with P | I | P² does there exist a genus one curve over a global field with period P and index I?
• RQ2 Can one construct infinitely many genus one curves over a global field with fixed Jacobian E, period P, and index P²?
• RQ3 How does the Shafarevich-Tate group behave under field extensions, particularly in relation to the period-index discrepancy?
• RQ4 Is the period-index obstruction map surjective or computable in positive characteristic?
• RQ5 Can the generic index of a moduli space of genus one curves of period P be shown to be P²?

Key findings

• For every pair (P, I) with P dividing I and I dividing P², there exists a number field K and a genus one curve C over K with period P and index I.
• For any elliptic curve E over a global field K and integer P > 1 not divisible by the characteristic, there exist infinitely many genus one curves over K with period P, index P², and Jacobian E.
• The Shafarevich-Tate group of E over some degree P extension L/K contains at least r elements of order P, for any given r.
• The obstruction map ΔP is shown to control the ratio I/P, and the conjecture that I/P = |ΔP(H¹(K, E[P]))| is equivalent to a tight cohomological bound.
• The paper constructs examples with I = P² more easily than with I < P², suggesting that I = P² is generic in a moduli-theoretic sense.
• The paper formulates open problems on extending the obstruction map to positive characteristic and relating period-index problems on curves to their Jacobians.