共查询到20条相似文献,搜索用时 0 毫秒
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The number of points on a hyperelliptic curve over a field of q elements may be expressed as q+1+S where S is a certain character sum. We study fluctuations of S as the curve varies over a large family of hyperelliptic curves of genus g. For fixed genus and growing q, Katz and Sarnak showed that is distributed as the trace of a random 2g×2g unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the limiting distribution of S is that of a sum of q independent trinomial random variables taking the values ±1 with probabilities 1/2(1+q−1) and the value 0 with probability 1/(q+1). When both the genus and the finite field grow, we find that has a standard Gaussian distribution. 相似文献
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Let q be a perfect power of a prime number p and be an elliptic curve over given by the equation . For a positive integer n we denote by the number of rational points on E (including infinity) over the extension . Under a mild technical condition, we show that the sequence contains at most 10200 perfect squares. If the mild condition is not satisfied, then is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range and . 相似文献
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We establish an upper bound for the number of points of a hypersurface without a linear component over a finite field, which is analogous to the Sziklai bound for a plane curve.Our bound is the best one for irreducible hypersurfaces that is linear on their degrees, because, for each finite field, there are at least two irreducible hypersurfaces of different degrees that reach our bound. 相似文献
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《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1999,328(12):1181-1185
Currently, the best known bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g, defined over a finite field %plane1D;53D;q, generally come either from Serre's refinement of the Weil bound if the genus is small compared to q, or from the optimization of the explicit formulae if the genus is large. We give methods for improving these bounds in both cases. Examples of improvements on the bounds include lowering them for a wide range of small genus when q = 8, 32, 213, 27, 243, 125, and when q = 2s, s > 1. For large genera, isolated improvements are obtained for q = 3, 8, 9. 相似文献
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We manage an upper bound for the number of rational points of a Frobenius nonclassical plane curve over a finite field. Together with previous results, the modified Sziklai conjecture is settled affirmatively. 相似文献
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For odd primes p and l such that the order of p modulo l is even, we determine explicitly the Jacobsthal sums l(v), ψl(v), and ψ2l(v), and the Jacobsthal–Whiteman sums and , over finite fields Fq such that . These results are obtained only in terms of q and l. We apply these results pertaining to the Jacobsthal sums, to determine, for each integer n1, the exact number of Fqn-rational points on the projective hyperelliptic curves aY2Ze−2=bXe+cZe (abc≠0) (for e=l,2l), and aY2Zl−1=X(bXl+cZl) (abc≠0), defined over such finite fields Fq. As a consequence, we obtain the exact form of the ζ-functions for these three classes of curves defined over Fq, as rational functions in the variable t, for all distinct cases that arise for the coefficients a,b,c. Further, we determine the exact cases for the coefficients a,b,c, for each class of curves, for which the corresponding non-singular models are maximal (or minimal) over Fq. 相似文献
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Christian Friesen Doug Hensley 《Proceedings of the American Mathematical Society》1996,124(9):2661-2673
Given a finite field of order and polynomials of degrees respectively, there is the continued fraction representation . Let denote the number of such pairs for which and for . We give both an exact recurrence relation, and an asymptotic analysis, for . The polynomial associated with the recurrence relation turns out to be of P-V type. We also study the distribution of . Averaged over all and as above, this presents no difficulties. The average value of is , and there is full information about the distribution. When is fixed and only is allowed to vary, we show that this is still the average. Moreover, few pairs give a value of that differs from this average by more than
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Eda Cesaratto Joachim von zur Gathen Guillermo Matera 《Journal of Number Theory》2013,133(4):1409-1434
“Most” hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by the appropriate Chow variety, and provides bounds on the probability that a random curve over a finite field is reducible. 相似文献
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We prove the following result which was conjectured by Stichtenoth and Xing: letg be the genus of a projective, irreducible non-singular algebraic curve over the finite field
and whose number of
-rational points attains the Hasse-Weil bound; then either 4g≤(q−1)2 or 2g=(q−1)q.
Supported by a grant from the International Atomic Energy and UNESCOCorrespondence to: F. Torres
This article was processed by the author using theLatex style file from Springer-Verlag. 相似文献
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Doklady Mathematics - 相似文献
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This short note is a summary of our paper with the same title [M. Homma and S. J. Kim, An elementary bound for the number of points of a hypersurface over a finite field, preprint 2012]. We establish an upper bound for the number of points of a hypersurface without a linear component over a finite field, which is analogous to the Sziklai bound for a plane curve. 相似文献
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We prove the existence of rational points on singular varieties over finite fields arising as degenerations of smooth proper
varieties with trivial Chow group of 0-cycles. We also obtain congruences for the number of rational points of singular varieties
appearing as fibres of a proper family with smooth total and base space and such that the Chow group of 0-cycles of the generic
fibre is trivial. In particular this leads to a vast generalization of the classical Chevalley-Warning theorem. The above
results are obtained as special cases of our main theorem which can be viewed as a relative version of a theorem of H. Esnault
on the number of rational points of smooth proper varieties over finite fields with trivial Chow group of 0-cycles. 相似文献
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J. Wolfmann 《代数通讯》2013,41(8):2055-2060
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We study the relationship between rational points and Galois points for a plane curve over a finite field. It is known that the set of Galois points coincides with that of rational points of the projective plane if the curve is the Hermitian, Klein quartic or Ballico–Hefez curve. The author proposes a problem: Does the converse hold true? If the curve of genus zero or one has a rational point, we have an affirmative answer. 相似文献