Abelian varieties by Mumford.

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6. Proof: If Choose an (_l)n is For on V 8 V 1IJ: real-valued form definite. = h(i). ( n = p+q V ~ C vP,q e vp,q We choose a polarization a morphism polarizable then i and write the Weil operator. ) C2 acts as V (D(-n) for (C Cvp,q = is reductive. is often called i-P+q vp,q , and so is the weight of V. In H H~, ~ e v,v' V • then the real-form of a a (complex conjugation) on 0 Hover of Ha(~) • We are a connected algebraic is reductive if it has a compact real-form ]R H there is an Ha < u,v > V °= J Ha < hu,hv > dh where V.

K ~. of 1 H (X) $ Let Recall that 1 H (X) e ... X be an abelian variety over a subfield Hr (X) = Ar (HI (X) and (any cohomology theory). HI (X x X x ••• ) Let v e ~m(m} = act 26 on mB(l) as I GL(HB(X» G x v-li there is then a natural action of ~m r on n HB(X) (m) GL(H~(X» to be the subgroup of of the form cIB(Z} , for any r,n, and m. } k m 1 and by the of p .. } are bases for 1 The field respectively. p .. 1) H~R(X) k(p .. ) We can replace over generated over 1J k k k k(p .. ) < dim(G) . 1) - by its algebraic closure in and hence assume that each algebraic cycle on to an algebraic cycle on to be the functor of is an isomorphism clDR(Z) 8 I A = P(~) is not empty.

When a:, TrDR ( [a] lJ k = [B]) B are differential I-forms of the second kind, can be and computed as follows. where a or 6 Let be the finite set of points L has a pole. peL , a can be written For = a z (I a local parameter at a. zi) dz, wi th -(1)« i 1. There therefore exists a meromorphic function P where a. , such that da = a. fa We write it is defined up to a constant. /A A ~ ~ X(ct) such that the bases HI (X,:ll) , and corresponding Weierstrass ~ such that the same orientation. of in and then z .

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