Conductor

Caveat emptor: This is an incomplete definition which has never been fully elicited or nailed down outside of a certain highly insular “monstrous moonshine groupie” community of “soft math” researchers. The working goal here would be to develop a simple, self-contained and rigorous definition suitable for outside work.

The conductor 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:

$N=\prod _{\mathrm {all\;primes\;} p}p^{f_{p}}$

with the exponent f_p of each prime p given by

$f_{p}=0$ if E has “good reduction” at p;
$f_{p}=1$ if E has “multiplicative reduction” at p;
$f_{p}=2$ if E has “additive reduction” at p > 3;
$f_{p}=2+\delta _{p}$ if E has “additive reduction” at p = 2 or 3;

where $\delta _{p}$ is some measure of “wild ramification” in the action of the inertia group on $T_{\ell }(E)$ for which Silverman refers to Serre and Tate and Ogg who in turn rely on the minimal models of Néron among other “monstrous moonshine” literature of the period for a “precise” definition.

Only a finite number of “bad reductions” are possible on any one elliptic curve E. A particular algorithm of Tate is supposed to find the minimal Weierstraß form of the elliptic curve and calculate f₂ and f₃ for the last few special cases of bad reduction with wild ramification, in order to fully complete the definition of the conductor, assuming an impartial outsider can ascertain exactly what is meant by vague terms such as “good and bad reduction.“

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.