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Infinite multiplicity of roots of unity of the Galois group in the representation on elliptic curves

โœ Scribed by Bo-Hae Im

Publisher
Elsevier Science
Year
2005
Tongue
English
Weight
221 KB
Volume
114
Category
Article
ISSN
0022-314X

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โœฆ Synopsis

Let K be a number field, K an algebraic closure of K and E/K an elliptic curve defined over K. Let G K be the absolute Galois group Gal(K/K) of K over K. This paper proves that there is a subset โŠ† G K of Haar measure 1 such that for every โˆˆ , the spectrum of in the natural representation E(K) โŠ— C of G K consists of all roots of unity, each of infinite multiplicity. Also, this paper proves that any complex conjugation automorphism in G K has the eigenvalue -1 with infinite multiplicity in the representation space E(K) โŠ— C of G K .

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Let \(X\) be a smooth proper connected algebraic curve defined over an algebraic number field \(K\). Let \(\pi_{1}(\bar{X})\), be the pro-l completion of the geometric fundamental group of \(\bar{X}=X \otimes_{k} \bar{K}\). Let \(p\) be a prime of \(K\), which is coprime to l. Assuming that \(X\) ha