By Gisbert Wüstholz

Alan Baker's sixtieth birthday in August 1999 provided an amazing chance to prepare a convention at ETH Zurich with the objective of featuring the state-of-the-art in quantity concept and geometry. a few of the leaders within the topic have been introduced jointly to provide an account of study within the final century in addition to speculations for attainable additional examine. The papers during this quantity disguise a wide spectrum of quantity idea together with geometric, algebrao-geometric and analytic features. This quantity will attract quantity theorists, algebraic geometers, and geometers with a bunch theoretic history. notwithstanding, it is going to even be priceless for mathematicians (in specific study scholars) who're attracted to being educated within the nation of quantity concept firstly of the twenty first century and in attainable advancements for the longer term.

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**Sample text**

Let E be an elliptic curve over a scheme S, and let N be a positive integer. The multiplication-by-N morphism [NJ : E --+ E is finite and fiat, and its degree is N 2 . If, furthermore, N is invertible in S, then [NJ : E --+ E is etale. P ROOF. 12, it suffices to show it for each geometric fiber. 16 and Corollary 1. 17. Let E be an elliptic curve over a scheme S, and let N be a pos itive integer. The kernel E[NJ = [NJ - 1 0 of the multiplication-by-N morphism [NJ : E --+ E is the fibered product over E of [NJ : E --+ E and the 0-section : S --+ E.

20. Let E be an elliptic curve over a field K, let f be an endomorphism of E, and let C =f:. char(K) be a prime. Then, the determinant det(f : TtE) of the action of f on Tt (E) z; equals the degree of f : deg f = det(f : TtE) . ( 1 . 19) PROOF . Let N = en in ( 1 . 18), and take the projective limit. :::::'. We now prove Theorem 1 . 15 . To do so, we first show the follow ing proposition. Let E be an elliptic curve over F and let C be a prime different from p. Taking a matrix expression of the action of the geometric Frobenius Frp on the Tate module TtE, we define the polynomial det(X - Frp · Y : TtE) E Zi [X, Y] .

We can make a change of coordinates such that each coefficient of the equation belongs to Z(p). There are many ways to do so, and the reduction modulo p of E depends on how good we can make it. 5. Let E be an elliptic curve over Q. (1) Let p be an odd prime. 7) = ax3 +bx2 +ex +d of E a E Z(P)'b,c,d E Z(p) and the cubic ax3 +bx2 +ex+ d mod p E Fp [x] obtained by the reduction modulo p of the right-hand side does not have a multiple root. (2) Let p be an odd prime. 8) a E Z(p)'b, c,d E Z(p) and the cubic ax3 +bx2 +ex+ d mod p E Fp [x] obtained by the reduction modulo p of the right-hand side does not have a triple root.