The conductor of an elliptic curve over a local field was implicitly studied by 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 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 − 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. extended the theory to conductors of abelian varieties.
Definition
Let E be an elliptic curve defined over a local field K and p the 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 νp 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 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
Ogg's formula
The exponent of the conductor is related to other invariants of the elliptic curve by Ogg's formula: where n is the number of components of the singular fibre of the Néron minimal model for E.. Ogg's original proof used a lot of case by case checking, especially in characteristics 2 and 3. 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 νp: it is 0 in the case of good reduction; otherwise it is 1 if νp < 0 and 2 if νp ≥ 0.
Global conductor
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 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.
