By J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

This quantity includes the improved types of the lectures given via the authors on the C. I. M. E. tutorial convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers accumulated listed here are huge surveys of the present examine within the mathematics of elliptic curves, and likewise include a number of new effects which can't be discovered in other places within the literature. due to readability and magnificence of exposition, and to the history fabric explicitly integrated within the textual content or quoted within the references, the quantity is definitely suited for learn scholars in addition to to senior mathematicians.

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Bravais class An arithmetic crystal class determined by (L, B(L)), where L is a lattice and B(L) is the Bravis group of L. See Bravais group. bounded torsion group A torsion group T where there is an integer n ≥ 0 such that t n = 1 for all t ∈ T . Bravais group The group of all orthogonal transformations that leave invariant a given lattice L. c 2001 by CRC Press LLC Bravais lattice A representative of a Bravais type. See Bravais type. Bravais type An equivalence class of arithmetically equivalent lattices.

K). canonical homomorphism (1) Let R be a commutative ring with identity and let L, M be algebras over R. Then, the tensor product L ⊗R M of R-modules is an algebra over R. The mappings l → l ⊗ 1 (l ∈ L) and m → m ⊗ 1 (m ∈ M) give algebra homorphisms L → L ⊗R M and M → L ⊗R M. Each one of these homomorphisms is called a canonical homomorphism (on tensor products of algebras). (2) Let the ring R = i∈I Ri be the direct product of rings Ri . The mapping φi : R → Ri that assigns to each element r of R its ith component ri is called a canonical homomorphism (of direct product of rings).

See Lie group. complex multiplication (1) The multiplication of two complex numbers a + ib and c + id (where a, b, c, d are real) using the rule (a + ib)(c + id) = (ac − bd) + i(ad + bc) , which is simply the usual rule for multiplying binomials, coupled with the property that i 2 = −1. (2) The multiplication of two complexes of a group, which is defined as follows. , subsets of G. Then AB = {ab : a ∈ A, b ∈ B}. , A(BC) = (AB)C for all complexes A, B, C of G. complex number A number of the form z = x + iy where x and y are real and i 2 = −1.