By J. E. Cremona

ISBN-10: 0521598206

ISBN-13: 9780521598200

Elliptic curves are of critical and growing to be significance in computational quantity thought, with various functions in such parts as cryptography, primality checking out and factorisation. This publication, now in its moment version, provides an intensive remedy of many algorithms about the mathematics of elliptic curves, with feedback on desktop implementation. it truly is in 3 elements. First, the writer describes intimately the development of modular elliptic curves, giving an particular set of rules for his or her computation utilizing modular symbols. Secondly a set of algorithms for the mathematics of elliptic curves is gifted; a few of these haven't seemed in ebook shape prior to. They comprise: discovering torsion and non-torsion issues, computing heights, discovering isogenies and classes, and computing the rank. ultimately, an in depth set of tables is equipped giving the result of the author's implementation of the algorithms. those tables expand the commonly used 'Antwerp IV tables' in methods: the variety of conductors (up to 1000), and the extent of element given for every curve. particularly, the amounts on the subject of the Birch Swinnerton-Dyer conjecture were computed in every one case and are integrated. All researchers and graduate scholars of quantity idea will locate this publication worthy, quite these attracted to the computational aspect of the topic. That element will make it charm additionally to machine scientists and coding theorists.

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Extra info for Algorithms For Modular Elliptic Curves

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BEGIN Sum = c(1); FOR i WHILE p[i] ≤ pmax DO BEGIN add(p[i],i,ap[i],1) END END 42 II. MODULAR SYMBOL ALGORITHMS (Subroutine to add the terms dependent on p) subroutine add(n,i,a,last a) 1. BEGIN 2. IF a=0 THEN j0 = i ELSE Sum = Sum + a*c(n); j0 = 1 FI; 3. FOR j FROM j0 TO i WHILE p[j]*n ≤ nmax DO 4. BEGIN 5. next a = a*ap[j]; 6. IF j=i AND (N ≡ 0 (mod p[j])) THEN 7. next a = next a - p[j]*last a 8. FI; 9. add(p[j]*n,j,next a,a) 10. END 11. END Here the recursive function add(n,i,a,last a) is always called under the following conditions: (i) pi = p[i] is the smallest prime dividing n = n; (ii) a = a(n); (iii) last a = a(n/p i ).

We now discuss the relationship between L(f, 1) and the periods of f (by which we will always mean the periods of the differential 2πif (z)dz). 7) L(f, 1) = −2πi 0 f (z)dz = − {0, ∞}, f . The modular symbol {0, ∞} is in the rational homology, so that L(f, 1) is a rational multiple of some period of f . To find the rational factor, we use the trick of “closing the path” (see [38, page 286] or [37]). 8) (1 + p − Tp ) · {0, ∞} = k=0 {0, k/p}. Let ap be the Tp -eigenvalue of f , so that Tp f = ap f .

10). If L(f, 1) = 0 then a variation of this method may be used. 1) α, α+k p 0, α+k p k=0 p−1 = {0, pα} + k=0 − (p + 1){0, α}. 32 II. 1) lies in the integral homology H 1 (X0 (N ), Z). Hence we can express it as an integral linear combination of the generating M-symbols. 2) n(α, p, f ) Re {α, ∞}, f = Ω(f ) 2(1 + p − ap ) for some integer n(α, p, f ), where the left-hand side is independent of p. Thus we can compute each ap from n(α, p, f ), given ap0 and n(α, p0 , f ), provided that the latter in nonzero.

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