nt.number theory - Definition and meaning of the conductor of an elliptic curve

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 $Deltain R$ 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.