arithmetic theory of elliptic curves - j coates

... Much of the recent research in the arithmetic of elliptic curves consists in the study of modularity properties of elliptic curves over Q, or of the structure of the Mordell-Weil group E(K) of ... USA 94 (1997), 1111 5-1 1117. J. Coates, S. Howson, Euler characteristics and elliptic curves 11, in prepara- tion. J. Coates, R. Sujatha, Galois cohomology...

Ngày tải lên: 31/03/2014, 16:21

... that W is an R-module and B is a Z p -module. Composition with τ ⊗ 1:R⊗ Z p B → B gives a bijection between the set of R-semilinear pairings W × W →R⊗ Z p B, and the set of ι-adjoint pairings W ... Hom R (W, Hom Z p (W ι ,B)).(A.2) The left-hand side is the set of R-semilinear pairings W × W →R⊗ Z p B, and the right-hand side is the set of ι-adjoint pairings W × W → B. Since compositi...

Ngày tải lên: 15/03/2014, 09:20

... universal family of lines over P 1 ? Exercise (1.6) 1) Show that the j- line M 1 is a coarse moduli space for curves of genus 1. 2) Show that a j- function J on a scheme B arises as the j- function associated ... ge- ometry of families of our moduli problem and information about the geometry of the moduli space M itself. One of the main themes of moduli theory is to bri...

Ngày tải lên: 31/03/2014, 15:08

... number of variables over any number field K (Hasse-Minkowski theorem). There is a very down-to-earth proof of the original case of the theorem in [C2]—it takes three lectures. A good exposition of the ... on affine curves Regular functions on projective curves The Riemann-Roch theorem The group law revisited Perfect base fields 5. Definition of an Elliptic Curve 19 Plane projective...

Ngày tải lên: 31/03/2014, 16:21

... ultra-short laser pulse prop- agation [3, 23], or to new particle accelerator concepts and new sources of radia- tion [42]. The concept of photon acceleration can then be seen as a kind of new theoretical ... timescale of a few tens of femtoseconds, can still be considered as slow processes in the Description of the contents 7 optical domain and stay within the range of validity...

Ngày tải lên: 24/04/2014, 17:21

... Clemens Grafe, Central Banking and the Choice of Currency Regime in Accession Countries, Vienna, 2001, ISBN 3-9 0 2-1 0 9-0 3-3 . 12. Sanjiva Prasad, Christopher J. Green, and Victor Murinde, Company Financing, ... Economies, Vienna, 2001, ISBN 3-9 0210 9-0 4-1 . 13. Martin M.G. Fase, Investment in Paintings: The interaction of monetary return and psychic income, Vienna, 2...

Ngày tải lên: 24/10/2012, 09:11

... − (A.7) c c w k = α (A.8) c y s i i i = − (A.9a) ),( , Ljjkij ryss = = (A.9b) (.,.)]1),([ jLjjj zrzss −+= υ (A.10) Lj rzr = (A.11) ∑ = −= Fbfjih hh szD ,,,, )1( (A.12a) )( − −= L dd rII µ (A.12b) ... cause of the excessive growth in short-term lending to the region. At a time of over-heating and over-optimistic expectations, short-term lenders - typically less concerned with the long...

Ngày tải lên: 24/10/2012, 09:33

... FIRST-ORDER STATISTICS OF SMALL-SCALE CHANNELS 96 4.3.3 The Three-Wave PDF Two- and Three-Wave models often are used to describe fading in microwave digital radio communications [30]. The Three-Wave ... approach. Recall from probability theory that if a real-valued random variable, Z, is the sum of a set of independent CHAPTER 4. FIRST-ORDER STATISTICS OF SMALL-SCALE CHANNELS 116 wh...

Ngày tải lên: 20/11/2012, 11:36

... numbers). For non-negative integers k j ,letT(k j ,F j ,P j ) denote the task of computing k j products of arbitrary polynomials by F j modulo P j .Let  j T(k j ,F j ,P j ) denote the task of simultaneously ... computing all of these products. If the coefficients span a vector space of dimension  j d j over φ, then the multiplicative complexity of  j...

Ngày tải lên: 25/10/2013, 02:15

... form Q i =−˜κn √ T ∂T ∂r i − ˜ λ √ T 3 ∂n ∂r i + ˜ θ 1 n 2 ∂V j ∂r j ∂T ∂r i + ˜ θ 2 n 2  2 3 ∂ ∂x i  T ∂V j ∂r j  + 2 ∂V j ∂r i ∂T ∂r j  + ˜ θ 3  2 ∂V j ∂r i ∂(nT) ∂r j + ˜ θ 4 n 2 T ∂ 2 V j ∂r i ∂r j + ˜ θ 5 n 2 ∂V j ∂r i ∂T ∂r j , (12.32) where ... P ij ∂r j = 0, (12.7) n DT Dt + 2 ∂V i ∂r j P ij + 2 ∂ Q j ∂r j =−n, (12.8) 12.3 Kinetic theory 1...

Ngày tải lên: 01/11/2013, 08:20