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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**Example text**

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.