Then for every m Z we have from b \u02c6 h m P m 2 \u02c6 h P 0 Now e implies that there

Then for every m z we have from b ˆ h m p m 2 ˆ h p

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Then, for every m Z , we have (from (b)) ˆ h ([ m ] P ) = m - 2 ˆ h ( P ) = 0 . Now (e) implies that there exists a constant C such that for each m Z , we have h f ([ m ] P ) = | deg( f ) ˆ h ([ m ] P ) - h f ([ m ] P ) | ≤ C. So { P, 2 P, 3 P, . . . } ⊆ { Q E ( K ) | h f ( Q ) C } , and therefore P has finite order since this last set is finite. (f) Suppose that ˆ h satisfies ˆ h [ m ] = m 2 ˆ h and (deg f ) ˆ h = h f + O (1) for some m 2 . Then ˆ h [ m N ] = m 2 N ˆ h and ˆ h = m - 2 N ˆ h [ m N ] = m - 2 N ( ˆ h [ m N ] + O (1)) = ˆ h + m - 2 N O (1) (since ˆ h satisfies (b)). Now let N → ∞ to obtain ˆ h = ˆ h . Lemma 8.29 Suppose that V is a finite-dimensional R -vector space, and let L V be a lattice. Let q : V R be a positive definite quadratic form satisfying: (i) If P L , then q ( P ) = 0 iff P = 0 . (ii) For every constant C , # { P L | q ( P ) C } < . Then q is positive definite on V . Proof. We may choose a basis of V such that for any X = ( x 1 , . . . , x n ) V , we have q ( X ) = s i =1 x 2 i - t i =1 x 2 s + i , where s + t n = dim( V ) . We may view V R n via this choice of basis. Suppose that s = n . Let λ be the length of the shortest vector in L , i.e. λ = inf { q ( P ) | P L, P = 0 } . Then (i) and (ii) imply that λ > 0 . Now consider the set B ( δ ) := ( x 1 , . . . , x n ) R n x 2 1 + · · · + x 2 s λ 2 , x 2 s +1 + · · · + x 2 t δ . 107
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Then length (using q !) of any vector in B ( δ ) is at most λ/ 2 , and so B ( δ ) L = { 0 } . Now B ( δ ) is compact, convex, and symmetric about the origin, and Vol( B ( δ )) → ∞ as δ → ∞ . This contradicts Minkowski’s convex body theorem. Theorem 8.30 (Minkowski.) Let L be a lattice in R n with fundamental paral- lelepiped D , and suppose that B R n is compact, convex, and symmetric about the origin. If Vol( B ) 2 n Vol( D ) , then B contains a nonzero point of L . Proof. We claim that if S is a measurable set in R n with Vol( S ) > Vol( D ) , then S contains distinct points α and β with α, β L . Note that Vol( S ) = L Vol( S ( D + )) , D will contain a unique translate (by an element of L ) of each set S ( D + ) . Since Vol( S ) > Vol( D ) , at least two of these sets will overlap, so there exist α, β S such that α - λ = β - λ for distinct λ, λ L , so α - β = λ - λ L \ { 0 } , as claimed. Now take S = 1 2 B = x 2 x B . Then Vol( S ) = 1 2 n Vol( B ) > Vol( D ) , so there exist α, β B such that α 2 - β 2 L . Since B is symmetric about the origin, - β B . Since B is convex, 1 2 ( α + ( - β )) B . Theorem 8.31 The Néron-Tate height is a positive definite quadratic form on R E ( K ) . Proof. Apply Lemma 8.29 to the lattice E ( K ) /E ( K ) tors in E ( K ) R . Definition 8.32 The Néron-Tate height pairing on E/K is defined by , : E ( ¯ K ) × E ( ¯ K ) R , given by P, Q = ˆ h ( P + Q ) - ˆ h ( P ) - ˆ h ( Q ) . Definition 8.33 The elliptic regulator R E/K is the volume of the fundamental 108
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domain of E ( K ) /E ( K ) tors with respect to ˆ h , i.e. if P 1 , . . . , P r E ( K ) form a basis of E ( K ) /E ( K ) tors , then R E/K := det( P i , P j ) . (If r = 0 , set R E/K = 1 .) Corollary 8.34 R E/K > 0 .
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  • Fall '13
  • SimonRubinstein-Salzedo
  • Math, Algebra, Number Theory, Integral domain, Algebraic geometry, deg, Elliptic curve cryptography, Algebraic curve, elliptic curves

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