Euler-Poincaré characteristics of abelian varieties

Coates, John ; Sujatha, Ramdorai (1999) Euler-Poincaré characteristics of abelian varieties Comptes Rendus de l'Academie des Sciences - Series I: Mathematics, 329 (4). pp. 309-313. ISSN 0764-4442

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Let F be a finite extension of Q , and let A be an abelian variety defined over F. Let p be a prime, and let Ap be the Galois module of points of A of order a power of p. Let G denote the Galois group of the extension F(Ap)/F. Suppose that G contains no element of order p. Then the cohomology groups Hi(G, Ap) are finite for all i ≥ 0, and zero for i sufficiently large; let ×(G, Ap) be the alternating product of their orders. The principal result of this Note is that we have x(G, Ap) = 1.

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