J. S. Milne, Jacobian varieties, in Arithmetic Geometry (G. Cornell and J. Silverman, Eds.), pp. 167--212, Springer, New York, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(MC ) be the 2 4 real matrix whose entries are traces from the complex matrix. The columns of MR generate a lattice in R 2 . If we make the standard identi cation between the holomorphic 1 forms on J and the holomorphic GENUS 2 BIRCH AND SWINNERTON DYER CONJECTURE 13 di erentials on C (see [Mi2]) then the notation R J(R) j 1 2 j makes sense and its value can be computed as the area of a fundamental domain for . If f 1 ; 2 g is a basis for the integral 1 forms on J , then R J(R) j 1 2 j = On the other hand, the computation of MC is simplest if we choose 1 = dX=Y , ....

J.S. Milne, Jacobian varieties, in: Arithmetic geometry, Ed. G. Cornell, G. and J.H. Silverman, Springer-Verlag, New York, 1986, pp. 167-212. MR 89b:14029

....of degree n: We will use the description of the Jacobian of a curve C as the abelian variety which coarsely represents the functor P 0 C which associates to a scheme T q Spec(K) the set P 0 C (T ) fL 2 P ic (C T ) j deg(L t ) 0 8 t 2 Tg=q P ic (T ) Theorem 1. 1 on page 168 of [6] states: Theorem There is an abelian variety J over K and a morphism of functors : P 0 C J such that : P 0 C (T ) J(T ) is an isomorphism whenever C(T ) is nonempty. Indeed the Jacobian represents the functor T 7 P 0 C (T L ) GK ; where L is any nite Galois extension of K ....

.... S is a projective at morphism whose bers are smooth genus one curves, L is a degree n line bundle on C (in particular L t is a degree n line bundle for every geometric ber C t ) and T is an S section of exact order n of the Jacobian group scheme J S. Such a J exists by Theorem 8. 1 of [6]. We de ne a 2 prepared genus one curve (over S) to be a triple (C S; L 1 ; L 2 ) T ) where C S is a projective at morphism whose bers are smooth genus one curves, L 1 and L 2 are line bundles of degree two on C which are linearly inequivalent on all bers, and T 2 J (S) has ....

Milne, J. S., Jacobian Varieties, in Cornell, G, Silverman, (eds.), Arithmetic Geometry, 167-212, Springer-Verlag, New York, 1986.

....an impossibly heavy pile of baggage on his shoulders or (ii) take a huge amount on faith and leave the heavy lifting to others, the author obstinately desiring to be self reliant has wondered how he might lighten the baggage enough so that he could pick it up himself. Milne s lucid survey [Milne 1986] gives a very good idea of how much baggage is involved here. Yet more obstinately the author has wondered: doesn t there exist some comprehensible multilinear algebra operation that the construction of Jacobians boils down to, just as the construction of Grassmannian varieties boils down to the ....

Milne, J. S.: "Jacobian varieties" in Arithmetic Geometry, G. Cornell and J. Silverman, eds., Springer-Verlag, New York, 1986, 167-212.

....Note that f D is equal to the composition C f P J [r] J t [rP D] J: But is translation invariant and [r] r ; the statement follows. Lemma 4.2. The map f = f P : J; 1 J=M ) C; 1 C=M ) is a natural isomorphism and is independent of P . Proof. From [16] we know that f P : J; 1 J=M ) C; 1 C=M ) is an isomorphism over the algebraic closure. By the previous lemma we see that this isomorphism does not depend on the choice of P , and it is clearly de ned over M . Let 2 (C; 1 C=M ) be a global 1 form on C and let = f 1 ....

Milne, J.S., Jacobian varieties, in Cornell, G. & Silverman, J.H. (eds.), Arithmetic Geometry, 167-212, SpringerVerlag, New York, 1986.

....of C=S . This proves the uniqueness of f and the asserted formula for f . To prove the existence of f , define f by this formula. Since this definition commutes with base change, we have for each s 2 S that (f ) s ffi j a;s = f s because the autoduality property holds over a field (cf. [Mi2], p. 185) and by uniqueness) But then f; f ffi j a : C A are two S morphisms whose fibres agree, and so by rigidity ( Mu1] p. 116) there is a section j of p A : A S such that f = j ffi p C ) Delta (f ffi j a ) But since f(a) 0 A = f ffi j a ) a) it follows that j = 0 A , and so f ....

....: C Theta S C J C=S be the morphism defined by the rule s C=S (x; y) cl(O(x Gamma y) where x; y 2 C(T ) are T valued points and T is any S scheme. Note that if ffi : C C Theta S C denotes the diagonal morphism, then s C=S ffi ffi = 0 (the constant map) Then by a similar argument as in [Mi2], we have the following analogue of [Mi2] Prop. 6.4: If f : C Theta S C A is any S morphism to an abelian scheme A=S such that f ffi ffi = 0, then there is a unique S homomorphism f : J C=S A such that f = f ffi s C . 7) Covers of Relative Curves: If N and g 0 1 are positive ....

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J. Milne, Jacobian Varieties. In: Arithmetic Geometry (G. Cornell, J. Silverman, eds.) Springer-Verlag, New York, 1986.

....a curve of genus g 1 defined over a number field k. The Mordell Conjecture asserts that C has only finitely many k rational points (i.e. points with coordinates in k) To any curve X of genus g 1 we can associate a g dimensional abelian variety J(X) defined over k called the Jacobian of X (see [Mi2]) The curve X is a subvariety of J(X) and J(X) is the smallest abelian variety in which X embeds. If C has genus 1, then X is an elliptic curve and J(X) X . Let C have genus g 1. Let Gamma be the k rational points of J(C) The Mordell Weil theorem (see [L] asserts that Gamma is a finitely ....

J . Milne, Jacobian varieties, in [CS]

....powers of the Frobenius, as well as curves de ned over Q having non trivial automorphisms. For genus g 1, we need a priori distinguish between automorphisms of a curve and automorphisms of its Jacobian. Automorphisms of a curve naturally de ne automorphisms on its Jacobian. By Torelli s theorem [25], all automorphisms of the Jacobian (for a xed projective embedding and a chosen zero element) arise in this way (except for multiplication by minus one on the Jacobian of non hyperelliptic curves) And we may identify the automorphisms of a curve and those of its Jacobian. If g 1, the number ....

J. S. Milne. Jacobian varieties. In G. Cornell and J. H. Silverman, editors, Arithmetic Geometry, pages 167-212. Springer-Verlag, 1986.

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J. S. Milne, Jacobian varieties, in Arithmetic Geometry (G. Cornell and J. Silverman, Eds.), pp. 167--212, Springer, New York, 1986.

No context found.

Milne, J. S., Jacobian Varieties, in: Cornell, G., Silverman, J.H.(eds.), Arithmetic geometry, 167--212, Springer-Verlag, New York, 1986.

No context found.

Milne, J. S., Jacobian Varieties, in: Cornell, G., Silverman, J.H.(eds.), Arithmetic geometry, 167--212, Springer-Verlag, New York, 1986.

No context found.

J. Milne. Jacobian varieties. In G. Cornell and J. Silverman, editors, Arithmetic Geometry. Springer-Verlag, 1986.

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