Difference between revisions of "User:Richard Pinch/sandbox-WP"
(Starr article: Manin obstruction) |
(Start article: Conductor of an abelian variety) |
||
| Line 1: | Line 1: | ||
| + | |||
| Line 18: | Line 19: | ||
==References== | ==References== | ||
* {{cite book | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 | pages=1, 111-112}} | * {{cite book | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 | pages=1, 111-112}} | ||
| + | |||
| + | =Conductor of an abelian variety= | ||
| + | In [[mathematics]], in [[Diophantine geometry]], the '''conductor of an abelian variety''' defined over a [[local field|local]] or [[global field]] ''F'' is a measure of how "bad" the [[bad reduction]] at some prime is. It is connected to the [[ramification]] in the field generated by the [[division point]]s. | ||
| + | |||
| + | 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 (mathematics)|scheme]] over | ||
| + | |||
| + | :Spec(''R'') | ||
| + | |||
| + | (cf. [[spectrum of a ring]]) for which the [[generic fibre]] constructed by means of the morphism | ||
| + | |||
| + | :Spec(''F'') → Spec(''R'') | ||
| + | |||
| + | gives back ''A''. Let ''A''<sup>0</sup> denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a [[residue field]] ''k'', ''A''<sup>0</sup><sub>''k''</sub> is a group variety over ''k'', hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a [[unipotent group]]. Let ''u'' be the dimension of the unipotent group and ''t'' the dimension of the torus. The order of the conductor is | ||
| + | |||
| + | :<math> f = 2u + t + \delta , \, </math> | ||
| + | |||
| + | where δ is a measure of wild ramification. | ||
| + | |||
| + | ==Properties== | ||
| + | * If ''A'' has [[good reduction]] then ''f'' = ''u'' = ''t'' = δ = 0. | ||
| + | * If ''A'' has [[semistable abelian variety|semistable reduction]] or, more generally, acquires semistable reduction over a Galois extension of ''F'' of degree prime to ''p'', the residue characteristic, then δ = 0. | ||
| + | * If ''p'' > 2''d'' + 1, where ''d'' is the dimension of ''A'', then δ = 0. | ||
| + | |||
| + | ==References== | ||
| + | * {{cite book | author=S. Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=70–71 }} | ||
| + | * {{cite journal | author=J.-P. Serre | coauthors=J. Tate | title=Good reduction of Abelian varieties | journal=Ann. Math. | volume=88 | year=1968 | pages=492–517 | doi=10.2307/1970722 | issue=3 | publisher=The Annals of Mathematics, Vol. 88, No. 3 | jstor=1970722 }} | ||
| + | |||
=Descendant subgroup= | =Descendant subgroup= | ||
Revision as of 07:54, 26 August 2013
Baer–Specker group
An example of an infinite Abelian group which is a building block in the structure theory of such groups.
Definition
The Baer-Specker group is the group B = ZN of all integer sequences with componentwise addition, that is, the direct product of countably many copies of Z.
Properties
Reinhold Baer proved in 1937 that this group is not free abelian; Specker proved in 1950 that every countable subgroup of B is free abelian.
See also
References
Conductor of an abelian variety
In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the division points.
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
- Spec(R)
(cf. spectrum of a ring) 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 k, A0k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let u be the dimension of the unipotent group and t the dimension of the torus. The order of the conductor is
\[ f = 2u + t + \delta , \, \]
where δ is a measure of wild ramification.
Properties
- If A has good reduction then f = u = t = δ = 0.
- If A has semistable reduction or, more generally, acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic, then δ = 0.
- If p > 2d + 1, where d is the dimension of A, then δ = 0.
References
Descendant subgroup
A subgroup of a group for which there is an descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predecessor.
The series may be infinite. If the series is finite, then the subgroup is subnormal.
See also
References
Essential subgroup
A subgroup that determines much of the structure of its containing group. The concept may be generalized to essential submodules.
Definition
A subgroup \(S\) of a (typically abelian) group \(G\) is said to be essential if whenever H is a non-trivial subgroup of G, the intersection of S and H is non-trivial: here "non-trivial" means "containing an element other than the identity".
References
Manin obstruction
In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction is attached to a geometric object X which measures the failure of the Hasse principle for X: that is, if the value of the obstruction is non-trivial, then X may have points over all local fields but not over a global field.
For abelian varieties the Manin obstruction is just the Tate-Shafarevich group and fully accounts for the failure of the local-to-global principle. There are however examples, due to Skorobogatov, of varieties with trivial Manin obstruction which have points everywhere locally and yet no global points.
References
Pinch point
A pinch point or cuspidal point is a type of singular point on an algebraic surface. It is one of the three types of ordinary singularity of a surface.
The equation for the surface near a pinch point may be put in the form
\[ f(u,v,w) = u^2 - vw^2 + [4] \, \]
where [4] denotes terms of degree 4 or more.
References
Residual property
In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X".
Formally, a group G is residually X if for every non-trivial element g there is a homomorphism h from G to a group with property X such that \(h(g)\neq e\).
More categorically, a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of \(\phi\colon G \to H\) where H is a group with property X.
Examples
Important examples include:
- Residually finite
- Residually nilpotent
- Residually solvable
- Residually free
References
Stably free module
A module which is close to being free.
Definition
A module M over a ring R is stably free if there exist free modules F and G over R such that
\[ M \oplus F = G . \, \]
Properties
- A projective module is stably free if and only if it possesses a finite free resolution.
See also
References
Richard Pinch/sandbox-WP. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP&oldid=30251