Semistable elliptic curve
Encyclopedia
In algebraic geometry
, a semistable abelian variety is an abelian variety
defined over a global
or local field
, which is characterized by how it reduces at the primes of the field.
For an Abelian variety A defined over a field F with ring of integers R, consider the Néron model
of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme
over
(cf. spectrum of a ring
) for which the generic fibre constructed by means of the morphism
gives back A. Let A0 denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a residue field
k, A0k is a group variety over k, hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus
, so that A0k is a semiabelian variety, then A has semistable reduction at the prime corresponding to k. If F is global, then A is semistable if it has good or semistable reduction at all primes.
The semistable reduction theorem of Alexander Grothendieck
states that an abelian variety acquires semistable reduction over a finite extension of F.
that has bad reduction only of multiplicative type. Suppose E is an elliptic curve defined over the rational number
field Q. It is known that there is a finite, non-empty set S of prime number
s p for which E has bad reduction modulo
p. The latter means that the curve Ep obtained by reduction of E to the prime field with p elements has a singular point
. Roughly speaking, the condition of multiplicative reduction amounts to saying that the singular point is a double point, rather than a cusp
. Deciding whether this condition holds is effectively computable according to Tate's algorithm. Therefore in a given case it is decidable whether or not the reduction is semistable, namely multiplicative reduction at worst.
The semistable reduction theorem for E may also be made explicit: E acquires semistable reduction over the extension of F generated by the coordinates of the points of order 12.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a semistable abelian variety is an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
defined over a global
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
or local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
, which is characterized by how it reduces at the primes of the field.
For an Abelian variety A defined over a field F with ring of integers R, consider the Néron model
Néron model
In algebraic geometry, a Néron model for an abelian variety AK defined over a local field K is the "best possible" group scheme AO defined over the ring of integers R of the local field K that becomes isomorphic to AK after base change from R to K.They were introduced by André Néron...
of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
over
- Spec(R)
(cf. spectrum of a ring
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
) for which the generic fibre constructed by means of the morphism
- Spec(F) → Spec(R)
gives back A. Let A0 denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a residue field
Residue field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field...
k, A0k is a group variety over k, hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus
Algebraic torus
In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory...
, so that A0k is a semiabelian variety, then A has semistable reduction at the prime corresponding to k. If F is global, then A is semistable if it has good or semistable reduction at all primes.
The semistable reduction theorem of Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
states that an abelian variety acquires semistable reduction over a finite extension of F.
Semistable elliptic curve
A semistable elliptic curve may be described more concretely as an elliptic curveElliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
that has bad reduction only of multiplicative type. Suppose E is an elliptic curve defined over the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
field Q. It is known that there is a finite, non-empty set S of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s p for which E has bad reduction modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
p. The latter means that the curve Ep obtained by reduction of E to the prime field with p elements has a singular point
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
. Roughly speaking, the condition of multiplicative reduction amounts to saying that the singular point is a double point, rather than a cusp
Cusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
. Deciding whether this condition holds is effectively computable according to Tate's algorithm. Therefore in a given case it is decidable whether or not the reduction is semistable, namely multiplicative reduction at worst.
The semistable reduction theorem for E may also be made explicit: E acquires semistable reduction over the extension of F generated by the coordinates of the points of order 12.