Definition and meaning of the conductor of an elliptic curve

Question

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 O_ℚ(√(-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 = ∏ p^f_p

where f_p = 0 if E has good reduction at p, f_p = 1 if the reduction is multiplicative, f_p = 2 if it is additive and p ≠ 2 or 3, and f_p = 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?

Answer 1

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.

Answer 2

Let me complete a little bit the story. The conductor mesures the ramification of the Galois group of the local field on the Tate module of the elliptic curves. The formal definition is given in Serre's book as said Jordan, in Buhler's text in the link given by Rob, and also in Sliverman's second volume on elliptic curves.

The conductor is a non-negative integer involved in the $L$-function of $E$. Its $p$-part $f_p$ vanishes iff the Tate module is unramified (the inertia group of ${\mathbb Q}_p$ acts trivially). Otherwise it has two part, a tame one, which depends solely on the reduction type of the Néron model of $E$ (I think this is proved in Serre-Tate's paper ''Good reduction of Abelian varieties'').

The second part (the wild one) in the conductor is the Swan conductor. It is the most headhach one. It vanishes if and only if the $p$-Sylow acts trivially on the Tate module. In very simple cases, it can be computed directly. In general, it is related to the invariants of $E$ given by Tate's algorithm: the conductor $f_p$ is given by Ogg's formula:
$$ f_p=\nu_p(\Delta) - n +1 $$ where $\Delta$ is the discriminant of a minimal Weierstrass equation of $E$, and $n$ is the number of geometric irreducible components of the fiber at $p$ of the minimal regular projective model of $E$ over $\mathbb Z$ (the fiber at $p$ is a projective, possibly reducible curve over $\mathbb F_p$, when $n$ is computed over the algebraic closure of $\mathbb F_p$). In Buhler's text, ''geometric'' is missing.

Tate's algorithm gives $\Delta$ and $n$ and computer can find them very quickly. So everybody is happy.

But, Ogg's formula, stated in his late 60's paper, was not fully proved. He checked the equality by case by case analysis. In residue characteristic 2, he said ''for the sake of simplicity, we will work in equal characteristic'' ! We know that equal characteristic is kind of limit of mixed characteristic (when the absolute ramification index tends to infinity), of course, this hypothesis simplifies a lot the computation, but does not give any crew for the mixed characteristic case (e.g. $\mathbb Q_p$). While this formula was widely used in computer programs, and often used as a definition of the conductor (!), some people were awared of the incompleteness of the proof. For example, Serre said this in seminars. This was also pointed out in the paper of Lockhart, Rosen and Silverman bounding conductor of abelian varieties (J. Alg. Geometry).

This situation is repaired in 1988 in a magistral paper of Takeshi Saito. Let $R$ be a d.v.r. with perfect residue field, let $C$ be a projective smooth and geometrically connected curve of positive genus over the field of fractions of $R$ and let $X$ be the minimal regular projective model of $C$ over $R$. One defines the Artin conductor ${\rm Art}(X/R)$ which turns out to be $f+n-1$ with the same meaning as above ($f$ is the conductor associated to the Jacobian of $C$). Saito proved that $${\rm Art}(X/R)=\nu(\Delta)$$ where $\Delta\in R$ is the ''discriminant'' of $X$ which mesures the defect of a functorial isomorphism which involves powers of the relative dualizing sheaf of $X/R$. When $C$ is an elliptic curve, one can prove that $\Delta$ is actually the discriminant of a minimal Weierstrass equation over $R$, and le tour est joué ! This paper of Saito was apparently not very known by the number theorists. Some more details are given in a text (in French).

So Ogg's formula should be called Ogg-Saito's formula. That some people do.

Answer 3

Here's one context in which the conductor is quite natural. We now know that every elliptic curve over Q is modular, which is to say it is a quotient of the modular Jacobian J_0(N) for some N. The conductor of E is precisely the minimal such N.

What's nice about the definition you give is that it can be computed directly from the elliptic curve without knowing that it's modular.

(And, of course, as Rob says, it's part of a very general story about Galois representations -- for more on this point of view you can read about Artin conductors in Serre's book on local fields.)

Answer 4

Indeed, the conductor in the context of the order in a CM field is simply the notion of a conductor of an order in a number field and completely unrelated to elliptic curves.

The conductor of an elliptic curve can be viewed as the conductor as its associated Galois representation. For a Galois representation the conductor is related to the action of inertia and breaks into a tame part and a wild part. It just turns out that the wild part in the elliptic curve case can only be non-trivial for the primes 2 and 3.

Geometrically, the wild part is related to the number of components in the special fibre of the Néron model. For both of these descriptions, you can see page 60 of the volume 9 of the IAS/Park city "proceedings" (link text)

Answer 5

I don't find the definition of a conductor clunky.

Consider any fibration f: X \to Y, just between the algebraic varieties for now. Let's say we want to study it somehow in algebraic geometry. One of the first things that will come to our mind is the locus of points L on Y where the fiber degenerates. It's an important data, and we will be able to work with f as smooth fibration outside of Y\L, e.g. we can then consider representation of \pi_1(Y\L) on cohomology of a fiber.

Now back to elliptic curves. They are in fact schemes over Spec Z (bear with me if you don't know all the words, it just means they are like X and something else is like Y in the above example) and the denegeracy locus is the submanifold of Spec Z given by equation N = 0, N being the conductor. Away from 2 and 3, this N is defined by this property and being the smallest number.

So I think from a geometric point of view it's clear that the conductor for elliptic curve was bound to appear somehow. Now there are, of course, strange and mysterious things about it, especially how to relate this definition to the one involving modular curves, but that's kind of next step.

The same procedure actually applies to number fields, since the conductor, again, is the ramification locus on Spec Z of the map of Spec R there (R being the field's integers). However, as was pointed out by some people, the conductor of a field associated to CM elliptic curve is not the same as the conductor of elliptic curve itself!