S. Lang, Elliptic curves: Diophantine Analysis, volume 231 of Grundlehren Math. Wiss., Springer, New York, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and symmetry problems for multivariate polynomials as problems for hypersurfaces in R or C . The full moving frame machinery can be used to effect a solution, but this work is in progress. Applications to the direct determination of symmetries of elliptic curves y = x ax b, cf. [19], would be an immediate and interesting consequence of this implementation of the general method. Acknowledgments : A part of the research discussed in this paper was conducted during the second author s visit to the Mathematical Sciences Research Institute at Berkeley in the fall of 1998. We ....

....Maple has failed to simplify the expressions 1; 2; 3; 4, and we need to make it take the square root. In the case of symmetries numbers 9; 11; 13; 15; 17; 19; 21; 23 this is done as follows. The others are handled in a similar fashion, and, for brevity, we omit the formulae here. for j in [9,11,13,15,17,19,21,23] do sq: sqrt(factor(op(op(numer(tr[j] 3] 1] I) symbolic) s[j] lf( op(numer(tr[j] 1] op(numer(tr[j] 2] sq) denom(tr[j] print(s.j=s[j] od: 2 Gamma 2 I p I ( 2) p 2) 25 As we remarked in the text, the octahedral ....

Lang, S., Elliptic Curves: Diophantine Analysis, Springer--Verlag, New York, 1978.

....First, we examine the structure of elliptic curves over finite fields. Second, we display an algorithm due to Schoof [10] which calculates the group order of an elliptic curve. One thing to keep in mind throughout this exposition is that elliptic curves have a very rich theory. In fact, Lang [6] has been quoted as saying, It is possible to write endlessly on elliptic curves. This is not a threat) Thus, in the interest of brevity, some of the less prevalent facts will be stated without proof with the emphasis on the reader to look them up from other references. Still, a great deal of ....

S. Lang. Elliptic Curves: Diophantine Analysis. Springer-Verlag, New York, 1978.

....assuming that z 1 = z 2 (which can usually be ensured at the expense of 2 units of work) a squaring then requires 12 units and a nonsquaring multiplication requires 9 units of work. The reader who is interested in learning more about the theory of elliptic curves should consult [11] 12] or [15]. 5 Lenstra s algorithm The idea of Lenstra s algorithm is to perform a sequence of pseudo random trials, where each trial uses a randomly chosen elliptic curve and has a nonzero probability of finding a factor of N . Let m and m 0 be parameters whose choice will be discussed later. To perform ....

S. Lang, Elliptic Curves -- Diophantine Analysis, Springer-Verlag, 1978.

....and symmetry problems for multivariate polynomials as problems for hypersurfaces in R m or C m . The full moving frame machinery can be used to effect a solution, but this work is in progress. Applications to the direct determination of symmetries of elliptic curves y 2 = x 3 ax b, cf. [23], would be an immediate and interesting consequence of this implementation of the general method. Acknowledgments : A part of the research discussed in this paper was conducted during the second author s visit to the Mathematical Sciences Research Institute at Berkeley in the fall of 1998. We ....

....Maple has failed to simplify the expressions 1; 2; 3; 4, and we need to make it take the square root. In the case of symmetries numbers 9; 11; 13; 15; 17; 19; 21; 23 this is done as follows. The others are handled in a similar fashion, and, for brevity, we omit the formulae here. for j in [9,11,13,15,17,19,21,23] do sq: sqrt(factor(op(op(numer(tr[j] 3] 1] I) symbolic) s[j] lf( op(numer(tr[j] 1] op(numer(tr[j] 2] sq) denom(tr[j] print(s.j=s[j] od: s9 = Gamma ( p 2 I p 2) p Gamma 2 Gamma p 2 I p 2 Gamma 2 p s11 = Gamma ( Gamma p 2 I p 2) p 2 I p 2 p 2 2 p ....

Lang, S., Elliptic Curves: Diophantine Analysis, Springer--Verlag, New York, 1978.

....the associated quadratic form hNT , which we define so that hNT (P) 1 2 #P, P#NT. WenotethatheretheNeron Tate height is normalized to take values which are independent of the chosen field of definition. As is well known, the Neron Tate height can be computed as a sum of local heights (see [7, 16]) We fix a Weierstrass equation for E, y 2 a 1 xy a 3 y = x 3 a 2 x 2 a 4 x a 6 ,a i # K. 2.1) For a p l ace w of K,weletE 0 (Kw ) # E(Kw ) denote the subgroup of points whose reduction is non singular at w, i.e. the points whose sections meet E 0 on the fiber above w. ....

....the image of (8.3) is as large as possible. Since the Verschiebung is separable, such descents behave much like prime to p descents, as opposed to the more delicate full p descent in characteristic p (see [4, 17, 18] Moreover, the proof is virtually identical to the one found in V. 5 of [7], or see [12] for further details. Proposition 8.4. Suppose Gal(K sep K) # Aut(T p (E) is surjective, and let M # E(K) be a torsion free subgroup of rank s. a) Let M = ZP.IfP # V (E (p) K) thenGal(LN KN ) # = ker VN . b) If M # V (E (p) K) pM,thenGal(LN KN ) # = ker ....

S. Lang, Elliptic curves: Diophantine analysis, Springer-Verlag, Berlin, 1978. 14 MATTHEW A. PAPANIKOLAS

....Recall the notion of the canonical logarithmic height h(P ) see x2.4) Given an elliptic curve E over Q having infinitely many rational points, let m denote the minimum of h(P ) for all nontorsion points P 2 E(Q) Let D denote the discriminant of E. Then a conjecture of Lang (see p. 92 of [12] or p. 233 of [29] states that there exists a positive absolute constant C 3 such that m C 3 log jDj. This conjecture was proved for a large class of curves in [27, 8] but it has not yet been proved unconditionally for all curves over Q. Lemma 4.1. Assume that log jDj C 1 max i=1; r ....

S. Lang, Elliptic Curves: Diophantine Analysis, Springer-Verlag, 1978.

....i ) where (n) denotes the number of distinct prime factors of n. In particular, we have rank(E(Q) 5 (e 2 Gamma e 1 ) e 3 Gamma e 2 ) e 3 Gamma e 1 ) Proof. The following elegant and explicit treatment of the Fermat descent, apart from some minor changes, can be found in Lang [L], Ch. V, Theorem 1.1; see also the survey article of Cassels [C] p. 268. We include it here for completeness. Let (s; t) ba a rational point other than a 2 division point (e 1 ; 0) e 2 ; 0) e 3 ; 0) 1, and let us write s Gamma e i = a i u 2 i with u i 2 Q and a i a non zero ....

Lang, S. Elliptic Curves--Diophantine Analysis. Springer-Verlag, Berlin-Heidelberg -New York 1978.

.... compute with such a stack: top(pop(push(1, push(2, empty) Similar examples of type constructions 6 which arise in mathematics are: ffl polynomial constructors [9] ffl localizations (special case: the field of fractions) 11] ffl tensor product [11] ffl elliptic curves over a given field [10]. 5 Conclusions and Acknowledgments We presented a framework for defining first class datatypes where algebraic structures combined with the data abstraction principle can easily be modelled. The given examples showed the usefulness of a typed language for computer algebra. It should be possible ....

Serge Lang. Elliptic Curves: Diophantine Analysis. Springer, 1978.

....and symmetry problems for multivariate polynomials as problems for hypersurfaces in R m or C m . The full moving frame machinery can be used to effect a solution, but this work is in progress. Applications to the direct determination of symmetries of elliptic curves y 2 = x 3 ax b, cf. [19], would be an immediate and interesting consequence of this implementation of the general method. Acknowledgments : A part of the research discussed in this paper was conducted during the second author s visit to the Mathematical Sciences Research Institute at Berkeley in the fall of 1998. We ....

....Maple has failed to simplify the expressions 1; 2; 3; 4, and we need to make it take the square root. In the case of symmetries numbers 9; 11; 13; 15; 17; 19; 21; 23 this is done as follows. The others are handled in a similar fashion, and, for brevity, we omit the formulae here. for j in [9,11,13,15,17,19,21,23] do sq: sqrt(factor(op(op(numer(tr[j] 3] 1] I) symbolic) s[j] lf( op(numer(tr[j] 1] op(numer(tr[j] 2] sq) denom(tr[j] print(s.j=s[j] od: s9 = Gamma ( p 2 I p 2) p Gamma 2 Gamma p 2 I p 2 Gamma 2 p s11 = Gamma ( Gamma p 2 I p 2) p 2 I p 2 p 2 2 p ....

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Lang, S., Elliptic Curves: Diophantine Analysis, Springer--Verlag, New York, 1978.

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S. Lang, Elliptic curves: Diophantine Analysis, volume 231 of Grundlehren Math. Wiss., Springer, New York, 1978.

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S. Lang, Elliptic Curves -- Diophantine Analysis, Springer-Verlag, Berlin, 1978.

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S. Lang, Elliptic Curves -- Diophantine Analysis, Springer-Verlag, Berlin, 1978.

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S. Lang, Elliptic Curves -- Diophantine Analysis, Springer-Verlag, Berlin, 1978.

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S. Lang, Elliptic Curves { Diophantine Analysis, Springer-Verlag, Berlin, 1978. 15

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S. Lang, Elliptic Curves: Diophantine Analysis, Springer-Verlag, 1978.

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