# An Introduction to the Theory of Numbers by Vinogradov I. M. By Vinogradov I. M.

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Ii) E PROOF. By Proposition 1 . 9 ( 1 ) , it suffices to show the following. Let n and m be integers satisfying condition ( 1 . 6) . 26) . 26) n, m , and n - m are all £th power times a power of 2. Condition ( 1 . 26) means that the p-adic valuation of n, m , and n - m are all divisible by £ for all odd prime p. Let p be an odd prime. At least two of the p-adic valuations are 0, and let ep be the valuation of the remaining one. If ep = 0, E has good reduction modulo p. 32, E has multiplicative reduction modulo p and the number of irreducible components of the geometric closed fiber of the semistable model of E over Z (p ) is 2ep .

Suppose E is defined by the equation y2 = f (x) . Let L be an extension of K. We prove that a point P = ( s, t) # 0 in E(L) is of order 2 if and only if f ( s ) = t = 0. By the definition of the group law, P is of order 2 if and only if the tangent line at P passes through 0 = (0 : 1 : 0) . Since a line in P2 passes through (0 : 1 : 0) if and only if it is parallel to the y-axis, the tangent line must be the line x = s . This is equivalent to the fact that the system of equation x = s , y2 = f (x) has a multiple root at (x, y) = (s, t) .

35. 34. The kernel E ( N ) [NJ of the restriction of the multiplication­ by-N morphism, [NJ : E ( N ) -+ £ ( 1 ) , is a finite fiat commutative group scheme over S of degree N 2 . Moreover, if N is invertible in S, E ( N ) [NJ is a finite etale commutative group scheme over S. 36 . Let E be an elliptic over Q . Suppose E has multiplicative reduction modulo a prime p, and let Ez < v l be its semi­ stable model. If N divides the number of irreducible components of the geometric closed fiber of Ez < p l , then, the kernel E�N( �) [NJ of the multiplication-by-N morphism [NJ : Ef: l -+ E�1 ) is a finite fiat ( p) ( p) commutative group scheme over Z ( p ) · If p does not divide N, then E [NJ is a finite etale commutative group scheme over Z ( p ) .