Y. Ihara, Some remarks on the number of rational points of algebraic curves over nite elds, J. Fac. Sci. Univ. Tokyo, 28 (1981)721-724.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....lim i 1 N(F i ) g(F i ) There are known bounds on the behaviour of function elds over a nite eld F q . Let N q (g) maxfN(F )jF a function eld over F q of genus g(F ) gg. Also, let A(q) lim sup g 1 N q (g) g; 1) then the Drinfeld Vladut bound [4] says that A(q) p q 1: 2) Ihara [10], and Tsafasman, Vladut and Zink [17] showed that this bound can be met in the case where q is a square. It is not known what the value of A(q) is for non square q, though there are results by Serre [12, 13, 14] and Schoof [11] in this direction. Clearly, for a tower of function elds F = F 1 ; F ....

Y. Ihara, Some remarks on the number of rational points of algebraic curves over nite elds, Journal of the Faculty of Science, University of Tokyo 28 (1981), 721-724.

....Since there are so many, there must be two, P 1 ; P 2 which are equivalent in J(F q ) 2J(F q ) So there is a function f with (f) P 1 P 2 2D. But this means the degree 2 cover given by y 2 = f is only rami ed at these two places; the genus of this cover is 2g d 1. For square q it is known [4, 7] that lim sup g 1 N q (g) g = p q 1. In the second paper we are able to show that lim inf g 1 N q (g) g ( p q 1) 3. A topic for future research is to close this gap. For q = p 2 we use modular curves. The simplest way to close this gap would be to show that the modular curves used in ....

Y. Ihara, Some remarks on the number of rational points of algebraic curves over nite elds, J. Fac. Sci. Univ. Tokyo 28 (1981), 721-724.

....and equating the coecients of the g 2 term computed in the two ways yields: g 2 m 2 = 1 2 ( gm) 2 (q 2 1 (q 1 gm 2a 2 ) 2gq) By reason (2.2) we must have a 2 0, so rearranging yields the desired inequality. Remark 1 Note that Proposition 1 generalizes Ihara s result [2] that the Weil bound cannot be met unless g (q p q) 2: Proposition 2 There are no defect 1 curves of genus g 2. Proof: This fact was observed in [7] due to reason (2.3) since both entries for defect 1 curves can be suitably partitioned if g 2. In the case g = 2, defect 1 is only ....

Y. Ihara, Some remarks on the number of rational points of algebraic curves over nite elds, J. Fac. Sci. Tokyo 28 (1981), p. 721-724.

.... such that and (1) It is well known (see [6] 8] that in this situation one can construct asymptotically good sequences of algebraic geometric (geometric Goppa) codes over Let is a function field of genus over and The Drinfeld Vladu t bound (see [1] tells us that and it was shown by Ihara [3] and Tsfasman, Vladu t, and Zink [7] that, if is a square For a square, and the Tsfasman Vl adu t Zink (TVZ) theorem [7] says that the parameters of the related algebraic geometric codes are better than the Gilbert Varshamov bound in a certain range of the rate. In [4] and [9] it is shown how ....

Y. Ihara, "Some remark on the number of rational points of algebraic curves over finite fields," J. Fac. Sci. Tokyo, vol. 28, pp. 721--724, 1981. VOSS AND HHOLDT: SEQUENCE OF CODES ATTAINING THE TSFASMAN--VL ADU T--ZINK BOUND 135

....in characteristic zero is a Shimura curve. Shimura curves, like classical and Drinfeld modular curves, reduce to curves over the finite field F q 2 of q 2 elements that attain the Drinfeld Vladut upper bound (q Gamma 1 o(1) g on the number of points of a curve of genus g 1 over that field [I3]. Moreover, while all three flavors of modular curves include towers that can be given by explicit formulas and thus used to construct good error correcting codes [Go1, Go2, TVZ] only the Shimura curves, precisely because of their lack of cusps, can give rise to totally unramified towers, which ....

.... equations we shall exhibit for certain choices of A and l suffice to determine explicit formulas for towers of Shimura modular curves X 0 (l r ) X 0 (l r ) towers whose reduction at any prime l 0 = 2 Sigma [ flg is known to be asymptotically optimal over the field of l 0 2 elements [I3, TVZ]. 2.4 Complex multiplication (CM) and supersingular points on Shimura curves Let F be a quadratic imaginary field, and let OF be its ring of integers. Assume that none of the primes of Sigma split in F . Then F embeds in A (in many ways) and OF embeds in O. For any embedding : F , A, the ....

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Ihara, Y.: Some remarks on the number of rational points of algebraic curves over finite fields. J. Fac. Sci. Tokyo 28 (1981), 721--724.

....of application is to coding theory: good Goppa codes [G] require curves of large genus g over a fixed finite field k = F q whose number of rational points grows as a positive multiple of g. Drinfeld and Vl adut showed that as g 1 no multiple greater than (q 1=2 Gamma 1)g is possible. Ihara [I] and, independently, Tsfasman, Vl adut, and Zink [TVZ] showed that this upper bound is attained by the supersingular points on appropriate modular curves when q is a square. For this application modular curves elliptic, Shimura, or Drinfeld are needed whose level is too high to apply the ....

Ihara, Y.: Some remarks on the number of rational points of algebraic curves over finite fields. J. Fac. Sci. Tokyo 28 (1981), 721--724.

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Y. Ihara, Some remarks on the number of rational points of algebraic curves over nite elds, J. Fac. Sci. Univ. Tokyo, 28 (1981)721-724.

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Y. Ihara, "Some remarks on the number of rational points of algebraic curves over finite fields", J. Fac. Sci. Tokyo, 28, 1981, pp. 721-724.

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Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Tokyo, 28, pp. 721-724, (1981).

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Y. Ihara, "Some remarks on the number of rational points of algebraic curves over finite fields", J. Fac. Sci. Tokyo, 28, 1981, pp. 721-724.

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Ihara, Y.: Some remarks on the number of rational points of algebraic curves over finite fields. J. Fac. Sci. Tokyo 28 (1981), 721--724.

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Yasutaka Ihara. Some remarks on the number of rational points of algebraic curves over nite elds. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28(3):721{ 724 (1982), 1981. Iwan Duursma et al.

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Ihara, Y.: Some remarks on the number of rational points of algebraic curves over finite fields. J. Fac. Sci. Tokyo 28 (1981), 721--724.

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Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo 28 (1981), 721--724.

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