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The distribution of the number of points modulo an integer on elliptic curves over finite fields
, 2009
"... Let Fq be a finite field and let b and N be integers. We study the probability that the number of points on a randomly chosen elliptic curve E over Fq equals b modulo N. We prove explicit formulas for the cases gcd(N, q) = 1 and N = char(Fq). In the former case, these formulas follow from a random ..."
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Cited by 5 (2 self)
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Let Fq be a finite field and let b and N be integers. We study the probability that the number of points on a randomly chosen elliptic curve E over Fq equals b modulo N. We prove explicit formulas for the cases gcd(N, q) = 1 and N = char(Fq). In the former case, these formulas follow from a random matrix theorem for Frobenius acting on the Ntorsion part of E, obtained by applying density results due to Chebotarev to the modular covering X(N) → X(1). As an additional application to this theorem, we estimate the probability that a randomly chosen elliptic curve has a point of order precisely N. 1
Fast Tabulation of Cubic Function Fields
, 2009
"... We give a general method for tabulating all cubic function fields over Fq(t) whose discriminant D has odd degree, or even degree such that the leading coefficient of −3D is a nonsquare in F ∗ q, up to a given bound on D  = q deg(D). The main theoretical ingredient is a generalization of a theore ..."
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Cited by 4 (3 self)
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We give a general method for tabulating all cubic function fields over Fq(t) whose discriminant D has odd degree, or even degree such that the leading coefficient of −3D is a nonsquare in F ∗ q, up to a given bound on D  = q deg(D). The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms, thereby providing us with an efficient way to compute equivalence classes of binary cubic forms. We present numerical data for cubic function fields over F5, F7, F11 and F13 with deg(D) odd up to various degrees. We also present numerical data for cubic function fields over F5 and F7 with deg(D) even and −3D a nonsquare in F ∗ q up to various degrees. In addition, we modify our tabulation algorithm to compute 3ranks of quadratic function fields over F5, F7, F11 and F13, by way of a generalization of a theorem of Hasse. The algorithm, whose complexity is linear in the number of reduced binary cubic forms up to some upper bound X on D, is described and numerical results are given.
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"... Abstract. We show that the number of rational points of height ≤ H on a nonrational plane curve of degree d is Od(H 2/d−δ), for some δ> 0 depending only on d. The implicit constant depends only on d. This improves a result of HeathBrown, who proved the bound O(H2/d+). We also show that one can ..."
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Abstract. We show that the number of rational points of height ≤ H on a nonrational plane curve of degree d is Od(H 2/d−δ), for some δ> 0 depending only on d. The implicit constant depends only on d. This improves a result of HeathBrown, who proved the bound O(H2/d+). We also show that one can take δ = 1/450 in the case d = 3.