The heart of Schoof's algorithm for computing the cardinal-ity m of an elliptic curve over a nite eld is the computation of m modulo small primes`. Elkies and Atkin have designed practical improvements to the basic algorithm, that make use of \good" primes`. We show how to use powers of good primes in an eecient way. This is done by computing isogenies between curves over the ground eld. A new ...

We shall first briefly review some general facts on elliptic curves over finite fields, and the algorithms of Schoof and Schoof Elkies Atkin (SEA) for counting points. We refer to text books as Cox, Silverman or Washington [Cox, Si, Wa] for the general topics and to the original papers of Schoof [Sch, Sch1] for a presentation of the algorithms. Let p be an odd prime. Consider the elliptic curve:

In Situ IonicÕElectric Conductivity Measurement of La0.55Li0.35TiO3 Ceramic at Different Li Insertion Levels Chunsheng Wang,* Prashanth Patil, A. John Appleby,* Frank. E. Little, Mehmet Kesmez, and David L. Cocke* Center for Manufacturing Research and Department of Chemical Engineering, Tennessee Technological University, Cookeville, Tennessee 38505, USA Gill Chair Group of Chemistry and Chemic...

D. Gopalakrishnan1*, Ravichandra Juluri2, J. Srihari3 and Vidya Viswanathan4 1Department of Periodontology and Oral Implantology, Dr. D.Y Patil Dental College & Hospital, Dr. D.Y Patil Vidyapeeth, Pune, India 2Adjunct Faculty, OSF Dental residency, Peoria, IL, USA & Private practice, Dunlap, IL, USA 3Department of Periodontology, Sri Ramakrishna Dental College & Hospital, Coimbatore, India 4Dep...

In 1988 P. Erdös asked if the prime divisors of x−1 for all n = 1, 2, . . . determine the given integer x; the problem was affirmatively answered by Corrales-Rodorigáñez and R. Schoof [2] in 1997 together with its elliptic version. Analogously, K. Yamanoi [14] proved in 2004 that the support of the pull-backed divisor fD of an ample divisor on an abelian variety A by an algebraically non-degene...

A natural variant of Lehmer’s conjecture that the Ramanujan $$\tau $$ -function never vanishes asks whether, for any given integer $$\alpha , there exist $$n \in \mathbb {Z}^+$$ such (n) = \alpha . series recent papers excludes many integers as possible values using theory primitive divisors Lucas numbers, computations points on curves, and congruences (n)$$ We synthesize these results methods ...

Let A be an abelian variety over a number field K. If P and Q are K-rational points of A such that the order of the (mod p) reduction of Q divides the order of the (mod p) reduction of Q for almost all primes (mod p), then there exists a K-endomorphism φ of A and a positive integer k such that kQ = φ(P). In this paper, we solve the support problem for abelian varieties over number fields, thus ...

Let ℓ≥5 be prime. For the partition function p(n) and 5≤ℓ≤31, Atkin found a number of examples primes Q≥5 such that there exist congruences form p(ℓQ3n+β)≡0(modℓ). Recently, Ahlgren, Allen, Tang proved are infinitely many for every ℓ. In this paper, wide range c∈Fℓ, we prove p(ℓQ3n+β0)≡c⋅p(ℓQn+β1)(modℓ) Q. positive integer r, let pr(n) r-colored function. Our methods yield similar pr(n). partic...

In a 1998 paper [2], Kaneko and Zagier explain unpublished work of Atkin which exhibits an infinite sequence of polynomials with the property that when suitable polynomials are reduced mod p for a prime p, one gets the locus of supersingular elliptic curves. Here we generalize this phenomenon by considering the continued fraction expansions of modular and quasimodular forms.

Genetic Programming Application Marc S. Atkin and Paul R. Cohen Experimental Knowledge Systems Laboratory Department of Computer Science, LGRC, Box 34610 University of Massachusetts, Amherst, MA 01003 [email protected] Abstract| Finding optimal or at least good monitoring strategies is an important consideration when designing an agent. We have applied genetic programming to this task, with mi...