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. The initial chapters are devoted to the Abelian case (complex multiplication), where one finds a nice correspondence between the l-adic representations and the linear representations of some algebraic groups (now called Taniyama groups). The last chapter handles the case of elliptic curves with no complex multiplication, the main result of which is that the image of the Galois group (in the corresponding l-adic representation) is "large."
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4 of 6 people found the following review helpful:
5.0 out of 5 stars
Serre is great,
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This review is from: Abelian l-adic Representations and Elliptic Curves (Research Notes in Mathematics) (Hardcover)
This book, as all books by Serre, is very good. He writes with precision, always giving the reader the central ideas. I can only think of Milnor as an author to compare to him. Buy the book if you are interested at all in the material. I am buying all books I can get by Serre.
4 of 7 people found the following review helpful:
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excellent,
By skeezer "skeezer" (Salem, Massachusetts) - See all my reviews
This review is from: Abelian l-adic Representations and Elliptic Curves (Research Notes in Mathematics) (Hardcover)
great reference if you are doing work in algebraic representations of geometric structures at the graduate level. There aren't very many exercises, but as a graduate reference text, exercises don't do a whole lot of good. great explanations, and clear proofs.
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Let K be a field, and let Ks be a separable algebraic closure of K. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
strictly compatible system, uniformizing element, commuting algebra, algebraic morphism, abelian representations, inertia group, inertia subgroup, arithmetic subgroup, open subgroup, good reduction, abelian varieties, class field theory, norm map, decomposition group, complex multiplication, finite extension, continuous homomorphism, quadratic extension, exceptional set, projective limit, elliptic curves, closed subgroup, finite index, algebraic group, abelian variety Key Phrases - Capitalized Phrases (CAPs): (learn more)
Exercise Let, Remark Let New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Let K be a field, and let Ks be a separable algebraic closure of K. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
strictly compatible system, uniformizing element, commuting algebra, algebraic morphism, abelian representations, inertia group, inertia subgroup, arithmetic subgroup, open subgroup, good reduction, abelian varieties, class field theory, norm map, decomposition group, complex multiplication, finite extension, continuous homomorphism, quadratic extension, exceptional set, projective limit, elliptic curves, closed subgroup, finite index, algebraic group, abelian variety Key Phrases - Capitalized Phrases (CAPs): (learn more)
Exercise Let, Remark Let New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 8
books:
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- Abelian Varieties by S. Lang in Back Matter
- Algebraic Number Theory (Graduate Texts in Mathematics) by Serge Lang in Back Matter
- Number Theory: An approach through history from Hammurapi to Legendre by André Weil in Back Matter
- Number Theory III: Diophantine Geometry (Encyclopaedia of Mathematical Sciences) by Serge Lang in Back Matter
- Basic Number Theory (Classics in Mathematics) by Andre Weil in Back Matter
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