Conductor

From Elliptic Curve Crypto
Revision as of 04:09, 5 January 2025 by Rational Point (talk | contribs) (another primary reference)

The conductor [1][2][3][4] of an elliptic curve E over a field K is an integer, (or an “ideal” of the ring of integers in any algebraic field,) defined by its prime factorization:

where the exponent fp of each prime p is given by

if E has “good” reduction at p;
if E has “multiplicative” reduction at p;
if E has “additive” reduction at p > 3;
if E has “additive” reduction at p = 2 or 3;

where is some measure of “wild ramification” in the action of the inertia group on for which Silverman [5] refers to Serre and Tate [6] and Ogg [7] who in turn rely on the minimal models of Néron [8] for a precise definition.

Only a finite number of “bad” reductions are possible on an elliptic curve E.

This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.

  1. PlanetMath: conductor of an elliptic curve. https://planetmath.org/conductorofanellipticcurve
  2. Wikipedia: Conductor of an elliptic curve. https://en.wikipedia.org/wiki/Conductor_of_an_elliptic_curve
  3. LMFDB: Knowledge → ec → Conductor of an elliptic curve (reviewed). https://www.lmfdb.org/knowledge/show/ec.conductor
  4. Silverman, Joseph H., and Brumer, Armand. "The number of elliptic curves over Q with conductor N.." Manuscripta mathematica 91.1 (1996): 95-102. http://eudml.org/doc/156217.
  5. Joseph H. Silverman. The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.
  6. Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” The Annals of Mathematics, 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.
  7. A. P. Ogg. “Elliptic curves and wild ramification.” American Journal of Mathematics, vol. 89, no. 1, Jan 1967, pp. 1-21.
  8. André Néron. «Modèles minimaux des variétés abéliennes sur les corps locaux et globaux.» Publications Mathématiques de l'IHÉS, vol. 21 (1964), pp. 5-128. http://www.numdam.org/item/PMIHES_1964__21__5_0/