Conductor of an elliptic curve
In mathematics, the conductor of an elliptic curve over the field of rational numbers, or more generally a local or global field, is an integral ideal analogous to the Artin conductor of a Galois representation. It is given as a product of prime ideals, together with associated exponents, which encode the ramification in the field extensions generated by the points of finite order in the group law of the elliptic curve. The primes involved in the conductor are precisely the primes of bad reduction of the curve: this is the Néron–Ogg–Shafarevich criterion.
Ogg's formula expresses the conductor in terms of the discriminant and the number of components of the special fiber over a local field, which can be computed using Tate's algorithm.
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 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(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.
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