S. David, Minorations de formes lineaires de logarithmes elliptiques, Memoires Soc. Math. France (N.S) 62 (1995).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to the chosen Mordell Weil basis. This is a positive definite form, hence c 5 is positive. Combining this with (13) 14) and (15) we obtain exp( c 6 N c 7 ) with c 6 = 2c 5 # , c 7 = log c 1 log # c 2 2c 4 # . 16) The lower bound for ) is provided by S. David s Theorem [7], namely, exp( c 8 (log N 0 c 9 ) log log N 0 c 10 ) 17) where N 0 is as in (12) and k = r 2 if #(P r ) or k = r 3, otherwise. Because of (12) the lower bound is expressed in terms of N . This lower bound is valid, provided N 0 is not less than a certain small ....

S. David, Minorations de formes lineaires de logarithmes elliptiques, Memoires Soc. Math. France (N.S) 62 (1995).

....when the integrand is continuous. Note that the torsion points occur in pairs usually. Working with # n (Q) they only occur with multiplicity 1, hence the formula di#ers from the usual elliptic Jensen formula in this respect. The only potential problem arises from torsion points close to Q: by [Dav95], for x = x(Q) with nQ = 0, x(Q) v n C for some C 0 which depends on E and Q only. This inequality is enough to imply that the Riemann sum given by the n torsion points for log x(Q) v converges, which gives (15) and the explicit error term gives the estimate in (14) Assume now that ....

.... q u v = 1 then the analogous sum to (18) is close to by the same argument. Assume therefore that there is a j with this property. Then the first sum in (18) is replaced by u# v , 19) where r = j n only depends on u. By v adic elliptic transcendence theory (see [Dav95]) there is a lower bound for log v of the form (log n) where A depends on E and u = u(P ) only. It follows that the first sum vanishes in the limit as before. The second sum in (18) is simply rearranged under rotation by u, so converges to as before. This proves (17) The ....

Sinnou David, Minorations de formes lineaires de logarithmes elliptiques, Mem. Soc. Math. France (N.S.) (1995), no. 62, iv+143.

....any # ball around the identity contains the image of B under an automorphism of Q A , X = Q A , and T n (x) # n x where # n = a n b n = x(2 Q) Then h h(Q) for non torsion Q. Proof. At the infinite place, a bound on max 1#n#N is provided by elliptic transcendence theory (see [7]) The minimum distance of nQ from the identity on C L is bounded below by n A for some A = A(E, Q) 0. The size of the x coordinate is approximately the inverse square of this quantity. Since we are running through the powers of 2 only, this gives an upper bound for max 1#n#N of the shape ....

....G is the normalized Haar measure on G (see [9] The points of N torsion are dense and uniformly distributed in E(C) as N ##, the limit sum over the torsion points will tend to the integral when the integrand is continuous. The only potential problem arises from torsion points close to Q: by [7], for x = x(P ) with NP = 0, x x(Q) N C for some C 0 which depends on E and Q only. This inequality is enough to imply that the Riemann sum given by the N torsion points for log x(Q) converges, which gives (11) Now q n = n (q) so log e N log #, 14 M. ....

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Sinnou David. Minorations de formes lineaires de logarithmes elliptiques. Mem. Soc. Math. France (N.S.), (62):iv+143, 1995.

.... be computed quite quickly and accurately via the algorithm given in [22] Related fundamental periods are 1 = 0:4247668014 : and 2 = i 1 , so that = 2 = 1 = i: Applying Theorem 2 of [9] which depends crucially upon the lower bound for linear forms in elliptic logarithms of David [7]) if P = n 1 P 1 P 2 is an integral point on E 1 over Q, for P 2 a torsion point, then jn 1 j 1:22 Theta 10 15 : To dispose of the remaining small cases, we need only apply the continued fraction algorithm to u 1 or, roughly equivalently, lattice basis reduction a la de Weger [21] to ....

S. David. Minorations de formes lin'eaires de logarithmes elliptiques. Publ. Math. Univ. Pierre et Marie Curie 106, Probl`emes diophantiens

....when the integrand is continuous. Note that the torsion points occur in pairs usually. Working with n (Q) they only occur with multiplicity 1, hence the formula differs from the usual elliptic Jensen formula in this respect. The only potential problem arises from torsion points close to Q: by [Dav95], for x = x(Q) with nQ = 0, jx Gamma x(Q)j v n GammaC for some C 0 which depends on E and Q only. This inequality is enough to imply that the Riemann sum given by the n torsion points for log jx Gamma x(Q)j v converges, which gives (15) and the explicit error term gives the estimate in ....

....therefore that there is a j with this property. Then the first sum in (18) is replaced by Gamman Gamma2 n Gamma1 X i=0 log j1 Gamma q j=n ui i j v Gamma n Gamma2 log j1 Gamma (q r u) n j v ; 19) where r = j=n only depends on u. By v adic elliptic transcendence theory (see [Dav95]) there is a lower bound for log j1 Gamma (q r u) n j v of the form Gamma(log n) A , where A depends on E and u = u(P ) only. It follows that the first sum vanishes in the limit as before. The second sum in (18) is simply rearranged under rotation by u, so converges to Gamma k 12 as ....

Sinnou David, Minorations de formes lin'eaires de logarithmes elliptiques, M'em. Soc. Math. France (N.S.) (1995), no. 62, MR 98f:11078, Zbl 859.11048.

....[ST97] The overall picture before 1994 was that of a field with many individual results but lacking a comprehensive approach to effectively settle the elliptic equation problem in some generality. But after S. David obtained an explicit lower bound for linear forms in elliptic logarithms (cf. [D95]) the elliptic logarithm method, independently developed in [ST94] and [GPZ94] and based upon David s result, provided a more generally applicable approach for solving elliptic equations. First Weierstra equations were tackled (see [ST94] Sm94] St95] BST97] then in [Tz96] Tzanakis ....

....inequality requires u and v to be integral. Moreover we have h(P ) c 1 M 2 : 35) Recall that c 1 = 0:478212. Putting it all together we obtain by (33) 34) 27) and (35) subject to v 5 or v Gamma2, that jL(P )j exp(13:5713 Gamma 0:956424M 2 ) 36) Next we apply David s result [D95] to the linear form in elliptic logarithms L(P ) As max 0i5 jm i j 5M , we obtain the lower bound jL(P )j exp ( Gammac 4 (log(5M) c 5 ) log log(5M) c 6 ) 8 : Here we use Tzanakis notation of [Tz96] We computed c 4 = 3:98179 Delta Delta Delta Theta 10 280 , c 5 = 2:09861 : ....

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S. David, "Minorations de formes lin'eaires de logarithmes elliptiques", M'emoires Soc. Math. France (N.S), 62 (1995), iv + 143 pp.

....we need the fact that U and V are integers. Finally we have h(P ) c 1 M 2 : 14) Recall that c 1 = 0:259202. Putting it all together, by (12) 13) 7) and (14) provided V Gamma17 or V 8, we deduce jL(P )j exp(19:5267 Gamma 0:518404M 2 ) 15) At this point we apply David s result [D95] to the linear form in elliptic logarithms L(P ) Observe that OE(P i ) and OE(Q 0;i ) are indeed elliptic logarithms, though the (X; Y ) coordinates of these points are in a field of degree 9. We thus obtain jL(P )j exp ( Gammac 4 (log(4M) c 5 ) log log(4M) c 6 ) 7 : The computed ....

S. David, "Minorations de formes lin'eaires de logarithmes elliptiques", M'emoires Soc. Math. France (N.S), 62 (1995), iv + 143 pp. On integral zeroes of binary Krawtchouk polynomials 12

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S. David, Minorations de formes lin'eaires de logarithmes elliptiques, Publ. Math. de l'Un. Pierre et Marie Curie no. 106, Probl`emes diophantiens

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