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Serre Conjecture (other)
Serre's conjecture may refer to: * Quillen–Suslin theorem, formerly known as Serre's conjecture * Serre's conjecture II (algebra), concerning the Galois cohomology of linear algebraic groups * Serre's modularity conjecture, concerning Galois representations * Serre's multiplicity conjectures In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection n ... in commutative algebra * Ribet's theorem, formerly known as Serre's epsilon conjecture See also * Jean-Pierre Serre {{mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quillen–Suslin Theorem
The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is a statement about the triviality of vector bundles on affine space. The theorem states that every finitely generated projective module over a polynomial ring is free. History Background Geometrically, finitely generated projective modules over the ring R _1,\dots,x_n/math> correspond to vector bundles over affine space \mathbb^n_R, where free modules correspond to trivial vector bundles. This correspondence (from modules to (algebraic) vector bundles) is given by the 'globalisation' or 'twiddlification' functor, sending M\to \widetilde (cite Hartshorne II.5, page 110). Affine space is topologically contractible, so it admits no non-trivial topological vector bundles. A simple argument using the exponential exact sequence and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre's Conjecture II (algebra)
In mathematics, Jean-Pierre Serre conjectured the following statement regarding the Galois cohomology of a simply connected space, simply connected semisimple algebraic group. Namely, he conjectured that if ''G'' is such a group over a perfect field (mathematics), field ''F'' of cohomological dimension at most 2, then the Galois cohomology set ''H''1(''F'', ''G'') is zero. A converse of the conjecture holds: if the field ''F'' is perfect and if the cohomology set ''H''1(''F'', ''G'') is zero for every semisimple simply connected algebraic group ''G'' then the ''p''-cohomological dimension of ''F'' is at most 2 for every Prime number, prime ''p''. The conjecture holds in the case where ''F'' is a local field (such as p-adic number, p-adic field) or a global field with no real embeddings (such as Q()). This is a special case of the Kneser–Harder–Chernousov Hasse principle for algebraic groups over global fields. (Note that such fields do indeed have cohomological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre's Modularity Conjecture
In mathematics, Serre's modularity conjecture, introduced by , states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by Chandrashekhar Khare in 2005, and a proof of the full conjecture was completed jointly by Khare and Jean-Pierre Wintenberger in 2008. Formulation The conjecture concerns the absolute Galois group G_\mathbb of the rational number field \mathbb. Let \rho be an absolutely irreducible, continuous, two-dimensional representation of G_\mathbb over a finite field F = \mathbb_. : \rho \colon G_\mathbb \rightarrow \mathrm_2(F). Additionally, assume \rho is odd, meaning the image of complex conjugation has determinant -1. To any normalized modular eigenform : f = q+a_2q^2+a_3q^3+\cdots of level N=N(\rho) , weight k=k(\rho) , and some Nebentype character : \chi \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre's Multiplicity Conjectures
In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection numbers, around 1949, there had been a question of how to provide a more flexible and computable theory. Let ''R'' be a (Noetherian, commutative) regular local ring and ''P'' and ''Q'' be prime ideals of ''R''. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra. Serre defined the intersection multiplicity of ''R''/''P'' and ''R''/''Q'' by means of the Tor functors of homological algebra, as : \chi (R/P,R/Q):=\sum _^\infty (-1)^i\ell_R (\operatorname^R_i(R/P,R/Q)). This requires the concept of the length of a module, denoted here by \ell_R, and the assumption that : \ell _R((R/P)\otimes(R/Q)) < \infty. If this idea were to work, howe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ribet's Theorem
Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT). As shown by Serre and Ribet, the Taniyama–Shimura conjecture (whose status was unresolved at the time) and the epsilon conjecture together imply that FLT is true. In mathematical terms, Ribet's theorem shows that if the Galois representation associated with an elliptic curve has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form that gives rise to the same representation). Statement Let be a weight 2 newform on – i.e. of level where does not divide – with absolutely irreducible 2-dimensional mod Galois representation unramified at if and finite flat at . Then there exists a weigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |