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Galois representations attached to Picard curves

M.G. Upton
- 15 Aug 2009 - 
- Vol. 322, Iss: 4, pp 1038-1059
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TLDR
For a Picard curve C with endomorphism ring Z[ζ3] the images of these representations are full for all but finitely many primes l in the reduction modulo l as discussed by the authors.
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This article is published in Journal of Algebra.The article was published on 2009-08-15 and is currently open access. It has received 9 citations till now. The article focuses on the topics: Galois cohomology & Galois group.

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Galois representations with open image

TL;DR: In this paper, an approach to constructing Galois extensions of Q with Galois group isomorphic to an open subgroup of GL_n({\mathbf{Z}}_p) for various values of n and primes p.
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Sato-Tate groups of abelian threefolds: a preview of the classification

TL;DR: In this article, the classification of Sato-Tate groups of abelian threefolds over number fields is presented, and the key points of the "upper bound" aspect of the classification are summarized.
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A note on Galois representations with big image

TL;DR: In this article, the authors presented a motivic galois representation of Gal(Q/Q) with open image in GL(n,Q`) for any even n ≥ 6 and any ε which is ≡ 1 mod 3 or mod 4.
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Sato-Tate groups of abelian threefolds

TL;DR: In this paper, it was shown that for abelian three-folds, there are 410 possible Sato-Tate groups, of which 33 are maximal with respect to inclusions of finite index.
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Computing L-Polynomials of Picard curves from Cartier-Manin matrices

TL;DR: In this paper, the authors studied the sequence of zeta functions of a generic Picard curve with genus greater than 2, and showed that for all but a density zero subset of primes, the zeta function is uniquely determined by the Cartier-Manin matrix of the primes modulo $p$ and the splitting behavior modulo of $f$ and $\psi_f$.
References
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Book

Algebraic Number Theory

Serge Lang
TL;DR: The second edition of Lang's well-known textbook as mentioned in this paper contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties.
Book

Abelian l̳-adic representations and elliptic curves

TL;DR: This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem.
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