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Division Fields of Elliptic Functions

Division Fields of Elliptic Functions D . W. MASSER Let p(z) be a Weierstrass elliptic function with algebraic invariants g and g 2 3 contained in the algebraic number field F. Let A denote its lattice of periods. For an integer n ^ 2 we define an n-division point of p (z) as a complex number 8 not in A such that nd is in A. As 5 runs over all such division points, the expressions p(S) and p'{8) clearly take only finitely many values, and these generate over F what we shall call the n-division field F of p (z). It has long been recognized that F is finite over F, and in fact quite precise upper and lower bounds for the relative degree d(n) = [F : F] are known. It is comparatively easy to deduce upper bounds; if p(z) has complex multiplication we have d(n)<cn , (1) while otherwise d(n) < en <K«) (2) where 0(n) is Euler's function and c is an effectively computable constant depending only on g and g . Lower bounds are much more interesting, however. In the case 2 3 of complex multiplication d(n) is known exactly (see Part 2 of [2], especially Theorem 2 on http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Division Fields of Elliptic Functions

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/9.1.49
Publisher site
See Article on Publisher Site

Abstract

D . W. MASSER Let p(z) be a Weierstrass elliptic function with algebraic invariants g and g 2 3 contained in the algebraic number field F. Let A denote its lattice of periods. For an integer n ^ 2 we define an n-division point of p (z) as a complex number 8 not in A such that nd is in A. As 5 runs over all such division points, the expressions p(S) and p'{8) clearly take only finitely many values, and these generate over F what we shall call the n-division field F of p (z). It has long been recognized that F is finite over F, and in fact quite precise upper and lower bounds for the relative degree d(n) = [F : F] are known. It is comparatively easy to deduce upper bounds; if p(z) has complex multiplication we have d(n)<cn , (1) while otherwise d(n) < en <K«) (2) where 0(n) is Euler's function and c is an effectively computable constant depending only on g and g . Lower bounds are much more interesting, however. In the case 2 3 of complex multiplication d(n) is known exactly (see Part 2 of [2], especially Theorem 2 on

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Mar 1, 1977

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