By David Mumford

ISBN-10: 0195605284

ISBN-13: 9780195605280

Now again in print, the revised variation of this renowned examine offers a scientific account of the fundamental effects approximately abelian kinds. Mumford describes the analytic equipment and effects acceptable whilst the floor box ok is the advanced box C and discusses the scheme-theoretic equipment and effects used to house inseparable isogenies whilst the floor box ok has attribute p. the writer additionally presents a self-contained facts of the lifestyles of a twin abeilan sort, reports the constitution of the hoop of endormorphisms, and contains in appendices "The Theorem of Tate" and the "Mordell-Weil Thorem." this is often a longtime paintings via an eminent mathematician and the one e-book in this topic.

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For convenience suppose first that L = L o. Consider the decomposition A = Al +A2 with \, E Av. 2 a) one checks that iJo(v + Ai) = aLo(Ai,V)iJO(v) for i = 1,2 and all v E V. Then using the cocycle relation the assertion follows for iJo. In order to prove the assertion in the case of an arbitrary characteristic c E V notice that by definition iJc and iJo are related as follows (3) Using this we obtain iJC(v + A) = e( -7rH(v + A, c) - ~H(c, c))aLo (A, v + c)iJo(v + c) = e(27riE(c,A))aLo(A,v)iJC(v) = at;Lo(A,V)iJC(v) This completes the proof.

For every L E Pic(X) and nEll. n"xL c::!. L n 2 tn 2 n 2 _n ® (-1)"xL-2- Proof. Recall the analytic representation nv: V L = L(H, X) we get: L n 2 tn 2 n 2 _n 2 ® (-1)"xL-2- = L(n in H -+ V, V I-t nv of n x. For + n ;n(-1)irH,x-2- . 1 a)) = L(nirH,nAX) =n"xL(H,x) . 4) o A line bundle L on X is called symmetric, if (-l)"xL c::!. L. There are several reasons why it is convenient to work with symmetric line bundles. One reason is that they are easier to handle. 6) Corollary. For every symmetric line bundle L E Pic(X) and nEll.

D 38 Chapter 2. Line Bundles on Complex Tori Denote by K(L) the kernel of (h. In order to describe K(L), define A(L) = {v Obviously A(L) E V I ImH(v, A) <;;; Z} . =

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Abelian Varieties by David Mumford

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