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導手：Conductor of an elliptic curve

https://en.wikipedia.org/wiki/Conductor_of_an_elliptic_curve
Conductor of an elliptic curve
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Contents
1 History
2 Definition
3 Ogg's formula
4 Global conductor
5 References
6 Further reading

History
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.

Ogg's formula

Saito (1988) gave a uniform proof and generalized Ogg's formula to more general arithmetic surfaces.

References
・Saito, Takeshi (1988), "Conductor, discriminant, and the Noether formula of arithmetic surfaces", Duke Math. J., 57 (1): 151?173, doi:10.1215/S0012-7094-88-05706-7, MR 0952229