Conductor of an elliptic curve

The conductor of an elliptic curve over a local field was implicitly studied (but not named) by Ogg (1967) in the form of an integer invariant ε+δ which later turned out to be the exponent of the conductor.

The conductor of an elliptic curve over the rationals was introduced and named by Weil (1967) as a constant appearing in the functional equation of its L-series, analogous to the way the conductor of a global field appears in the functional equation of its zeta function. He showed that it could be written as a product over primes with exponents given by order(Δ) − μ + 1, which by Ogg's formula is equal to ε+δ. A similar definition works for any global field. Weil also suggested that the conductor was equal to the level of a modular form corresponding to the elliptic curve.

Serre & Tate (1968) extended the theory to conductors of abelian varieties.

Let E be an elliptic curve defined over a local field K and p a prime ideal of the ring of integers of K. We consider a minimal equation for E: a generalised Weierstrass equation whose coefficients are p-integral and with the valuation of the discriminant ν_p(Δ) as small as possible. If the discriminant is a p-unit then E has good reduction at p and the exponent of the conductor is zero.

We can write the exponent f of the conductor as a sum ε + δ of two terms, corresponding to the tame and wild ramification. The tame ramification part ε is defined in terms of the reduction type: ε=0 for good reduction, ε=1 for multiplicative reduction and ε=2 for additive reduction. The wild ramification term δ is zero unless p divides 2 or 3, and in the latter cases it is defined in terms of the wild ramification of the extensions of K by the division points of E by Serre's formula

${\displaystyle \delta =\dim _{Z/lZ}{\text{Hom}}_{Z_{l}[G]}(P,M).}$

Here M is the group of points on the elliptic curve of order l for a prime l, P is the Swan representation, and G the Galois group of a finite extension of K such that the points of M are defined over it (so that G acts on M)

The exponent of the conductor is related to other invariants of the elliptic curve by Ogg's formula:

${\displaystyle f_{\mathbf {p} }=\nu _{\mathbf {p} }(\Delta )+1-n\ ,}$

where n is the number of components (without counting multiplicities) of the singular fibre of the Néron minimal model for E. (This is sometimes used as a definition of the conductor).

Ogg's original proof used a lot of case by case checking, especially in characteristics 2 and 3. Saito (1988) gave a uniform proof and generalized Ogg's formula to more general arithmetic surfaces.

We can also describe ε in terms of the valuation of the j-invariant ν_p(j): it is 0 in the case of good reduction; otherwise it is 1 if ν_p(j) < 0 and 2 if ν_p(j) ≥ 0.

Let E be an elliptic curve defined over a number field K. The global conductor is the ideal given by the product over primes of K

${\displaystyle f(E)=\prod _{\mathbf {p} }\mathbf {p} ^{f_{\mathbf {p} }}\ .}$

This is a finite product as the primes of bad reduction are contained in the set of primes divisors of the discriminant of any model for E with global integral coefficients.

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• Elliptic Curve Data - tables of elliptic curves over Q listed by conductor, computed by John Cremona