In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a semistable abelian variety is an
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic 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 functi ...
defined over a
global
Global means of or referring to a globe and may also refer to:
Entertainment
* ''Global'' (Paul van Dyk album), 2003
* ''Global'' (Bunji Garlin album), 2007
* ''Global'' (Humanoid album), 1989
* ''Global'' (Todd Rundgren album), 2015
* Bruno ...
or
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 compact ...
, which is characterized by how it reduces at the primes of the field.
For an abelian variety
defined over a field
with
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 deno ...
, consider the
Néron model In algebraic geometry, the Néron model (or Néron minimal model, or minimal model)
for an abelian variety ''AK'' defined over the field of fractions ''K'' of a Dedekind domain ''R'' is the "push-forward" of ''AK'' from Spec(''K'') to Spec(''R''), ...
of
, which is a 'best possible' model of
defined over
. This model may be represented as a
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
over
(cf.
spectrum of a ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
) for which the
generic fibre constructed by means of the
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
gives back
. The Néron model is a smooth
group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have ...
, so we can consider
, the connected component of the Néron model which contains the identity for the group law. This is an open subgroup scheme of the Néron model. For a
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. Frequently, ''R'' is a ...
,
is a
group variety over
, hence an extension of an abelian variety by a linear group. If this linear group is an
algebraic torus In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine algebraic group commonly found in Projective scheme, projective algebraic geometry and toric ...
, so that
is a
semiabelian variety, then
has ''semistable reduction'' at the prime corresponding to
. If
is a
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 f ...
, then
is semistable if it has good or semistable reduction at all primes.
The fundamental
semistable reduction theorem of
Alexander Grothendieck states that an abelian variety acquires semistable reduction over a finite extension of
.
Semistable elliptic curve
A semistable elliptic curve may be described more concretely as an
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. I ...
that has
bad reduction only of
multiplicative type. Suppose is an elliptic curve defined over the
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 ...
field
. It is known that there is a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
,
non-empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
''S'' of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s for which has ''bad reduction'' ''
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
'' . The latter means that the curve
obtained by reduction of to the
prime field
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
with elements has a
singular point
Singularity or singular point may refer to:
Science, technology, and mathematics Mathematics
* Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
. Roughly speaking, the condition of multiplicative reduction amounts to saying that the singular point is a
double point
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.
Algebraic curves in the plane
Algebraic cur ...
, rather than a
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
. Deciding whether this condition holds is effectively computable by
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 may also be made explicit: acquires semistable reduction over the extension of generated by the coordinates of the points of order 12.][This is implicit in Husemöller (1987) pp.117-118][
]
References
*
*
* {{cite book , first=Serge , last=Lang , authorlink=Serge Lang , title=Survey of Diophantine geometry , url=https://archive.org/details/surveydiophantin00lang_347 , url-access=limited , publisher=Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, year=1997 , isbn=3-540-61223-8 , zbl=0869.11051 , pag
70
Abelian varieties
Diophantine geometry
Elliptic curves