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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.
References
- Serre, Jean-Pierre; Tate, John (1968). "Good reduction of abelian varieties". The Annals of Mathematics . Second. 88 (3): 492–517. doi:10.2307/1970722. JSTOR 1970722. Zbl 0172.46101.