Algebraic geometry 04 Linear algebraic groups, invariant by A.N. Parshin (editor), I.R. Shafarevich (editor), V.L.

By A.N. Parshin (editor), I.R. Shafarevich (editor), V.L. Popov, T.A. Springer, E.B. Vinberg

Contributions on heavily similar matters: the speculation of linear algebraic teams and invariant concept, by means of recognized specialists within the fields. The ebook can be very important as a reference and examine advisor to graduate scholars and researchers in arithmetic and theoretical physics.

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The local L-factors are independent of l. Corollary 2. Let At. A2 be as in Corollary 1. Thefollowing are equivalent: (i) (ii) (iii) (iv) Al and A2 are isogenous. 7;(A I ) (8)z, Q, ~ 7;(A 2) (8)z, Q, as n-modules. Lv(s, AI) = Lv(s, A 2) for almost all places v of K. LJs, AI) = Lv(s, A 2) for all v. PROOF. The equivalence of (i) and (ii) follow from Theorem 4, that of (ii) and (iii) from Theorem 3 (+ Cebotarev), and that (ii) implies (iv) implies (iii) is 0 trivial. Corollary 3. Let AIK be an abelian variety, d > O.

Apr = O}. Since (A Ga(X). Xpr(X) is a subgroup of (6) The constant group functor, Jt': Let H be any group. Define a functor, Jt', by Jt'(X) = H, and for Y -+ X, let Jt'(X) -+ Jt'(Y) be the identity map. Other examples will appear below. If our group functor, F, is representable (by a scheme) and if G is the representing object, we call G a group scheme. By abuse of language, already indulged in, we also call the functor F a group scheme. This is the case with examples (1)-(5) above; example (6), however, is a non-representable functor.

Let G be a finite group scheme over a base, S, which satisfies (t). If p is a prime number, then there exists a scheme, T, which satisfies (t) and is finite and faithfully flat over S, so that GT = G x s T possesses a subgroup scheme of order pordp(G) over T. If p is not the characteristic of S, then the same statement is true for all exponents a with 0 S; a S; ordp ( G). Again, if p is not the characteristic of S, then the number of such p-Sylow subgroup schemes divides #(G) and is congruent to one modulo p.

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