I never really understood the definition of the conductor of an elliptic curve.
What I understand is that for an elliptic curve E over ℚ, End(E) is going to be (isomorphic to) ℤ or an order in a imaginary quadratic field ℚ(√(-d)), and that this order is uniquely determined by an integer f, the conductor, so that End(E) ≅ ℤ + f Oℚ(√(-d)) (where O just means ring of integers).
However I feel that this is not very convenient; this definition does not say anything about elliptic curves without complex multiplication.
The other definition I have come across gives the conductor as the product of primes at which the elliptic curve does not have good reduction:
N = ∏ pfp
where fp = 0 if E has good reduction at p, fp = 1 if the reduction is multiplicative, fp = 2 if it is additive and p ≠ 2 or 3, and fp = 2 + δ if p = 2 or 3, where δ is some (seemingly complicated) measure of how bad the reduction is.
I've never been able to make much sense of the second definition, nor have I seen any relation with the first. How did the idea initially appear, and why is this particular definition more useful (or "natural") than other similar definitions?
Best Answer
The conductor of the curve and the conductor of the order in the endomorphism ring are not equal in the CM case; it's just unfortunate terminology. For example, y^2 = x^3 - x has complex multiplication by the maximal order Z[i] (conductor = 1) of Q(i), but it certainly doesn't have everywhere good reduction.
The conductor N defined in the rather clunky way, prime by prime, is useful for organizing the information that's packed into the L-function of the elliptic curve. More specifically, it shows up in the functional equation that relates the L-function in the right half-plane to its values in the left half-plane. (Which is conjectural unless E is modular-- including all curves defined over Q-- or E has complex multiplication.) The conceptual reason the funny business shows up at the primes 2 and 3 is that the L-function is a product of local L-functions counting points on reductions, and this counting is harder to do mod 2 or mod 3. This is all sketched in sections 15 and 16 of appendix C of Silverman's first book on elliptic curves and spelled out in his second book.