Schoof’s algorithm computes the number m of points on an elliptic curve E defined over a finite field Fq. Schoof determines m modulo small primes ` using the characteristic equation of the Frobenius of E and polynomials of degree O(`2). With the works of Elkies and Atkin, we have just to compute, when ` is a “good” prime, an eigenvalue of the Frobenius using polynomials of degree O(`). In this ...

The present paper is concerned with exploring some problems in analyzing the structure of friendship choices. The data come from a sociometric study of the impact of a computer conference on affective ties among some social scientists and mathematicians. They have been repor ted elsewhere (L. Freeman & S. Freeman, 1979; S. Freeman & L. Freeman, 1979). The analytic devices that will be brought t...

cancer. N Engl J Med 326:653-657, 1992 (9) Kelly RB, Shank C: Adherence to screening flexible sigmoidoscopy in asymptomatic patients. Med Care 30:1029-1042, 1992 (10) Atkin WS, Morson BC, Cuzick J: Long-term risk of colorectal cancer after excision of rectosigmoid adenomas. N Engl J Med 326:658—662, 1992 (//) Atkin WS, Cuzick J, Northover JMA, et al: Prevention of colorectal cancer by once-only...

LetBD be the indefinite quaternion algebra overQ of reduced discriminantD=p1· · · · ·p2r for pairwise different prime numbers pi and let XD/Q be the Shimura curve attached to BD. As it was shown by Shimura [23], XD is the coarse moduli space of abelian surfaces with quaternionic multiplication by BD. Let W = {ωm : m | D} ⊆ Aut Q(XD) be the group of Atkin-Lehner involutions. For any m | D, we wi...

A Q-curve is an elliptic curve defined over Q that is isogenous to all its Galois conjugates. The term Q-curve was first used by Gross to denote a special class of elliptic curves with complex multiplication having that property, and later generalized by Ribet to denote any elliptic curve isogenous to its conjugates. In this paper we deal only with Q-curves with no complex multiplication, the c...

We describe how to generate large primes using the primality proving algorithm of Atkin. Figure 1: The Titanic .

1 Drosophila S2R+ cells 2 3 4 Peter V. Lidsky , Frank Sprenger, Christian F. Lehner 5 6 7 1) Institute of Molecular Life Sciences (IMLS), University of Zurich, Zurich, Switzerland 8 2) Institute of Genetics, University of Regensburg, Regensburg, Germany 9 3) Current address: Institute of Gene Biology, Russian Academy of Sciences, Moscow, 10 Russia. 11 12 13 *) corresponding author: Christian Le...

We show how Rank-Crank type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank-Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watso...

Abstract. This is the third and final installment in our series of papers applying the method of Atkin and Swinnerton-Dyer to deduce formulas for rank differences. The study of rank differences was initiated by Atkin and Swinnerton-Dyer in their proof of Dyson’s conjectures concerning Ramanujan’s congruences for the partition function. Since then, other types of rank differences for statistics ...

In 1986, following the work of Schoof on counting points on elliptic curves over finite fields, new algorithms for primality proving emerged, due to Goldwasser and Kilian on the one hand, and Atkin on the other. The latter algorithm uses the theory of complex multiplication. The algorithm, now called ECPP, has been used for nearly ten years. The purpose of this paper is to give an account of th...