By Kevin Walker

This publication describes an invariant, l, of orientated rational homology 3-spheres that is a generalization of labor of Andrew Casson within the integer homology sphere case. allow R(X) denote the gap of conjugacy sessions of representations of p(X) into SU(2). allow (W, W, F) be a Heegaard splitting of a rational homology sphere M. Then l(M) is asserted to be an accurately outlined intersection variety of R(W) and R(W) within R(F). The definition of this intersection quantity is a fragile activity, because the areas concerned have singularities. A formulation describing how l transforms below Dehn surgical procedure is proved. The formulation comprises Alexander polynomials and Dedekind sums, and will be used to provide a slightly hassle-free facts of the lifestyles of l. it's also proven that after M is a Z-homology sphere, l(M) determines the Rochlin invariant of M

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Therefore, every abelian variety admits a polarization. 7. The theory of symmetric homomorphisms for abelian varieties is analogous to the theory of symmetric bilinear forms for real vector spaces. Let B(−, −) be a symmetric bilinear form on a real vector space V . The symmetric homomorphism λL associated to a line bundle L is analogous to the symmetric bilinear form B(x, y) = Q(x + y) − Q(x) − Q(y) 22 4. POLARIZATIONS associated to a quadratic form Q. 3 is an analog to the fact that every form B arises this way, with 1 Q(x) = B(x, x).

The functor X classiﬁes OB -linear polarized abelian varieties up to isomorphism (with certain restrictions on the slopes of the p-divisible group and the Weil pairings). We shall now introduce a diﬀerent functor X which classiﬁes OB,(p) -linear polarized abelian varieties with rational level structure up to isogeny. The functor X will be shown to be equivalent to X . 4. EQUIVALENCE OF THE MODULI PROBLEMS 39 Assume that S is a locally noetherian connected scheme on which p is locally nilpotent.

Since the image η(Lp ) is compact, the index [η(Lp ) : N · T p (A)] must be ﬁnite. Therefore η(Lp ) is a virtual subgroup κ(η) ∈ VSubp (A). Automorphisms of Lp do not alter the image η(Lp ), so the virtual subgroup η(Lp ) depends only on the K0p -orbit of η. 1. A rational level structure η : V p − level structure if and only if the virtual subgroup κ(η) is T p (A). 4. The Tate representation Let k be a ﬁeld, and let be distinct from the characteristic of k. If A is an abelian variety over k, the -adic Tate module of A is deﬁned by T (A) = T (A ⊗k k).