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, which is 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.

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) extended the theory to conductors of abelian varieties.

Definition

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

[math]\displaystyle{ \delta = \dim_{Z/lZ} \text {Hom}_{Z_l[G]}(P, M). }[/math]

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)

Ogg's formula

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

[math]\displaystyle{ f_\mathbf{p} = \nu_\mathbf{p} ( \Delta ) + 1 - n \ , }[/math]

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.

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

[math]\displaystyle{ f(E) = \prod_\mathbf{p} \mathbf{p}^{f_\mathbf{p}} \ . }[/math]

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.

References

• Cremona, John (1997). Algorithms for Modular Elliptic Curves (2nd ed.). Cambridge University Press. ISBN 0-521-59820-6. http://www.warwick.ac.uk/staff/J.E.Cremona//book/fulltext/index.html. 
• Husemöller, Dale (2004). Elliptic Curves. Graduate Texts in Mathematics. 111 (2nd ed.). Springer. ISBN 0-387-95490-2. 
• Néron, André (1964), "Modèles minimaux des variétés abéliennes sur les corps locaux et globaux" (in French), Publications Mathématiques de l'IHÉS 21: 5–128, doi:10.1007/BF02684271, ISSN 1618-1913, http://www.numdam.org/item?id=PMIHES_1964__21__5_0 
• Ogg, A. P. (1967), "Elliptic curves and wild ramification", American Journal of Mathematics 89 (1): 1–21, doi:10.2307/2373092, ISSN 0002-9327 
• 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 
• Serre, Jean-Pierre; Tate, John (1968), "Good reduction of abelian varieties", Annals of Mathematics, Second Series 88 (3): 492–517, doi:10.2307/1970722, ISSN 0003-486X 
• Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. 151. Springer-Verlag. 1994. ISBN 0-387-94328-5. 
• Rational Points on Elliptic Curves. Springer-Verlag. 1992. ISBN 0-387-97825-9. 
• John Tate (1974). "The arithmetic of elliptic curves". Inventiones Mathematicae 23 (3–4): 179–206. doi:10.1007/BF01389745. https://eudml.org/doc/142261. 
• Weil, André (1967), "Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen", Math. Ann. 168: 149–156, doi:10.1007/BF01361551 

External links

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