In mathematics, the conductor of an elliptic curve over the field of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

s, or more generally a local or

global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global function fi ...

, 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 extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...

s generated by the points of finite order in the

group law In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. The ...

of the

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

. 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 In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the ori ...

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 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(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. extended the theory to conductors of abelian varieties.

Definition

Let ''E'' be an elliptic curve defined over a

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compa ...

''K'' and p a prime ideal of the

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...

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 In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...

of the extensions of ''K'' by the

division point In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A ...

s of ''E'' by Serre's formula :

\delta = \dim_ \text _(P, M).

Here ''M'' is the group of points on the elliptic curve of order ''l'' for a prime ''l'', ''P'' is the

Swan representation In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by as an expression appearing in the functional equation of an Artin L-function. Local Artin conductors ...

, 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: :

f_\mathbf = \nu_\mathbf ( \Delta ) + 1 - n \ ,

where ''n'' is the number of components (without counting multiplicities) of the singular fibre of the

Néron minimal model Neron or Néron may refer to: * Neron (DC Comics), a fictional character in the DC Comics' universe. * An alternative name of the Roman Emperor Nero * André Néron, a mathematician, who introduced: ** Néron minimal model ** Néron differenti ...

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. 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 In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holom ...

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'' :

f(E) = \prod_\mathbf \mathbf^ \ .

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

* * * * * * * * * *{{citation, mr=0207658 , last=Weil, first= André , title=Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen , journal=Math. Ann. , volume=168 , year=1967, pages= 149–156, doi=10.1007/BF01361551, s2cid=120553723

External links

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