Elliptic and modular curves over finite fields and related computational issues

The problem of calculating the trace of an elliptic curve over a finite field has attracted considerable interest in recent years. There are many good reasons for this. The question is intrinsically compelling, being the first nontrivial case of the natural problem of counting points on a complete projective variety over a finite field, and figures in a variety of contexts, from primality proving to arithmetic algebraic geometry to applications in secure communication. It is also a difficult but rewarding challenge, in that the most successful approaches draw on some surprisingly advanced number theory and suggest new conjectures and results apart from the immediate point-counting problem.

[1]  P. Deligne.  La conjecture de Weil pour les surfacesK3 , 1971 .

[2]  P. Deligne,et al.  Les Schémas de Modules de Courbes Elliptiques , 1973 .

[3]  D. Mumford,et al.  A rank 2 vector bundle on P4 with 15,000 symmetries , 1973 .

[4]  K. Shih.  On the construction of Galois extensions of function fields and number fields , 1974 .

[5]  B. Mazur,et al.  Arithmetic of Weil curves , 1974 .

[6]  A. Ogg.  Hyperelliptic modular curves , 1974 .

[7]  Michael Pohst,et al.  Berechnung kleiner Diskriminanten total reeller algebraischer Zahlkörper. , 1975 .

[8]  G. Ligozat.  Courbes modulaires de genre 1 , 1975 .

[9]  B. Birch,et al.  Modular Functions of One Variable IV , 1975 .

[10]  J. Pollard,et al.  Monte Carlo methods for index computation () , 1978 .

[11]  K. Shih.  P-division points on certain elliptic curves , 1978 .

[12]  André Weil,et al.  Abstract versus classical algebraic geometry , 1979 .

[13]  L. V. Danilov.  Diophantine equation x3−y2=k and Hall's conjecture , 1982 .

[14]  M. Laska.  The diophantine equation x3 − y2 = r , 1983 .

[15]  Don B. Zagier,et al.  On singular moduli. , 1984 .

[16]  Paula B. Cohen.  On the coefficients of the transformation polynomials for the elliptic modular function , 1984 .

[17]  Henri Cohen,et al.  Heuristics on class groups of number fields , 1984 .

[18]  R. Schoof.  Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p , 1985 .

[19]  Joe Kilian,et al.  Almost all primes can be quickly certified , 1986, STOC '86.

[20]  N. Elkies.  The automorphism group of the modular curve $X_0(63)$ , 1990 .

[21]  J. Pila.  Frobenius maps of Abelian varieties and finding roots of unity in finite fields , 1990 .

[22]  S. Lang,et al.  Old and new conjectured diophantine inequalities , 1990 .

[23]  J. Cremona.  Algorithms for Modular Elliptic Curves , 1992 .

[24]  Jean-Pierre Serre,et al.  Topics in Galois Theory , 1992 .

[25]  Ming-Deh A. Huang,et al.  Primality Testing and Abelian Varieties over Finite Fields , 1992 .

[26]  J. Top,et al.  The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes. , 1992 .

[27]  Ming-Deh A. Huang,et al.  Counting rational points on curves over finite fields , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[28]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[29]  François Morain,et al.  Schoof's algorithm and isogeny cycles , 1994, ANTS.

[30]  Emil J. Volcheck.  Computing in the jacobian of a plane algebraic curve , 1994, ANTS.

[31]  Andreas Stein,et al.  Baby Step Giant Step in Real Quadratic Function Fields , 1995 .

[32]  Françoise Morain.  Calcul du nombre de points sur une courbe elliptique dans un corps fini : aspects algorithmiques , 1995 .

[33]  A. Wiles.  Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?) , 1995 .

[34]  A. Wiles,et al.  Ring-Theoretic Properties of Certain Hecke Algebras , 1995 .

[35]  Bjorn Poonen,et al.  Computational Aspects of Curves of Genus at Least 2 , 1996, ANTS.

[36]  Don Zagier,et al.  Supersingular j-invariants, hypergeometric series, and Atkin's orthogonal polynomials , 1998 .