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52
Some integer factorization algorithms using elliptic curves
 Australian Computer Science Communications
, 1986
"... Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order ..."
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Cited by 56 (13 self)
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Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order log(p), where p is the factor which is found. In practice the speedup is significant. We mention some refinements which give greater speedup, an alternative way of implementing a second phase, and the connection with Pollard’s “p − 1” factorization algorithm. 1
Average Frobenius Distribution of Elliptic Curves
 Internat. Math. Res. Notices
, 1998
"... this paper average estimates related to the LangTrotter conjecture. The average distribution fits the one predicted by the conjecture, and the conjectural constant C E,r of Lang and Trotter is confirmed by our results, as seen in Section 2. Average estimates for the case r 0 were already obtained ..."
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Cited by 47 (7 self)
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this paper average estimates related to the LangTrotter conjecture. The average distribution fits the one predicted by the conjecture, and the conjectural constant C E,r of Lang and Trotter is confirmed by our results, as seen in Section 2. Average estimates for the case r 0 were already obtained by Fouvry and Murty [6], and we obtain a generalization of their results for any r Z. The techniques of Fouvry and Murty do not seem to extend to the general case r Z. Our proof then differs significantly from theirs. In the following, we fix r Z, and we denote by E(a, b) the elliptic curve Y b with a, b Z. Then . Following [11], we define 2 # t log t # . Theorem 1.2. Let r be an integer,A,B# 1. For every c>0, we have 1 , (2) where . (3) The constants in the Osymbol depend only on c and r. As the infinite product of (3) converges to a positive number, the constant C r is nonzero, even if some C E,r can be zero, as mentioned above. From the last theorem, we immediately obtain that the LangTrotter conjecture is true "on average." Corollary 1.3. Let #>0. If A,B>x , we have as x ##, . In analogy with the classical terminology, we can say that the average order of E(a,b) (x)isC r ( # x/ log x). Using the same techniques, we can also prove that the normal order of # E(a,b) (x)isC r ( # x/log x). Then,# C r ( # x/ log x) for "almost all" E(a, b) rather than on average (see Corollary 1.5). We are grateful to A. Granville for suggesting this application of our techniques
Sato–Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
, 2008
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Gaussian hypergeometric functions and traces
 of Hecke operators, International Mathematical Research Notices (2004
"... Abstract. We establish a simple inductive formula for the trace Trnewk (Γ0(8), p) of the pth Hecke operator on the space Snewk (Γ0(8)) of newforms of level 8 and weight k in terms of the values of 3F2hypergeometric functions over the finite field Fp. Using this formula when k = 6, we prove a conje ..."
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Cited by 17 (0 self)
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Abstract. We establish a simple inductive formula for the trace Trnewk (Γ0(8), p) of the pth Hecke operator on the space Snewk (Γ0(8)) of newforms of level 8 and weight k in terms of the values of 3F2hypergeometric functions over the finite field Fp. Using this formula when k = 6, we prove a conjecture of Koike relating Trnew 6 (Γ0(8), p) to the values 6F5(1)p and 4F3(1)p. Furthermore, we find new congruences between Tr new k (Γ0(8), p) and generalized Apéry numbers.
The Distribution Of The Eigenvalues Of Hecke Operators
"... For each prime p, we determine the distribution of the p th Fourier coefficients of the Hecke eigenforms of large weight for the full modular group. As p !1, this distribution tends to the SatoTate distribution. ..."
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Cited by 15 (1 self)
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For each prime p, we determine the distribution of the p th Fourier coefficients of the Hecke eigenforms of large weight for the full modular group. As p !1, this distribution tends to the SatoTate distribution.
Average Frobenius distributions for elliptic curves with nontrivial rational torsion
 TO APPEAR IN ACTA ARITHMETICA
, 2005
"... In this paper we consider the LangTrotter conjecture (Conjecture 1 below) for various families of elliptic curves with prescribed torsion structure. We prove that the LangTrotter conjecture holds in an average sense for these families of curves (see Theorem 3). ..."
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Cited by 14 (4 self)
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In this paper we consider the LangTrotter conjecture (Conjecture 1 below) for various families of elliptic curves with prescribed torsion structure. We prove that the LangTrotter conjecture holds in an average sense for these families of curves (see Theorem 3).
Exponential sums over finite fields and differential equations over the complex numbers: some interactions
, 1990
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On Fourier coefficients of Maass waveforms for PSL(2, Z)
, 1993
"... In this paper, we use machine experiments to test the validity of the SatoTate conjecture for Maass waveforms on PSL(2, Z)\H. We also elaborate on Stark's iterative method for calculating the Fourier coefficients of such forms. ..."
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Cited by 11 (3 self)
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In this paper, we use machine experiments to test the validity of the SatoTate conjecture for Maass waveforms on PSL(2, Z)\H. We also elaborate on Stark's iterative method for calculating the Fourier coefficients of such forms.
The Sato–Tate conjecture on average for small angles
 Trans. Am. Math. Soc
"... Abstract. We obtain average results on the SatoTate conjecture for elliptic curves for small angles. ..."
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Cited by 9 (2 self)
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Abstract. We obtain average results on the SatoTate conjecture for elliptic curves for small angles.
The Distribution of Group Structures on Elliptic Curves over Finite Prime Fields
 DOCUMENTA MATH.
, 2006
"... We determine the probability that a randomly chosen elliptic curve E/Fp over a randomly chosen prime field Fp has an ℓprimary part E(Fp)[ℓ ∞ ] isomorphic with a fixed abelian ℓgroup H (ℓ) α,β = Z/ℓα × Z/ℓ β. Probabilities for “E(Fp)  divisible by n”, “E(Fp) cyclic ” and expectations for the numb ..."
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Cited by 8 (1 self)
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We determine the probability that a randomly chosen elliptic curve E/Fp over a randomly chosen prime field Fp has an ℓprimary part E(Fp)[ℓ ∞ ] isomorphic with a fixed abelian ℓgroup H (ℓ) α,β = Z/ℓα × Z/ℓ β. Probabilities for “E(Fp)  divisible by n”, “E(Fp) cyclic ” and expectations for the number of elements of precise order n in E(Fp) are derived, both for unbiased E/Fp and for E/Fp with p ≡ 1 (ℓ r).