Fourier inversion theorem: Difference between revisions

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In [[mathematics]], a '''congruence subgroup''' of a [[matrix group]] with [[integer]] entries is a [[subgroup]] defined by congruence conditions on the entries. A very simple example would be [[invertible matrix|invertible]] 2x2 integer matrices of [[determinant]] 1, such that the off-diagonal entries are ''even''.
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An important class of congruence subgroups is given by reduction of the ring of entries: in general given a group such as the [[special linear group]] SL(n, '''Z''') we can reduce the entries to [[modular arithmetic]] in '''Z'''/N'''Z''' for any N >1, which gives a [[homomorphism]]
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:''SL''(''n'', '''Z''') &rarr; ''SL''(''n'', '''Z'''/''N''&middot;'''Z''')
 
of groups. The kernel of this reduction map is an example of a congruence subgroup – the condition is that the diagonal entries are congruent to 1 mod ''N,'' and the off-diagonal entries be congruent to 0 mod ''N'' (divisible by ''N''), and is known as a '''{{visible anchor|principal congruence subgroup}}''', Γ(''N''). Formally a congruence subgroup is one that contains Γ(''N'') for some ''N'',<ref>Lang (1976) p.26</ref> and the least such ''N'' is the ''level'' or ''Stufe'' of the subgroup.
 
In the case ''n=2'' we are talking then about a subgroup of the [[modular group Gamma|modular group]] (up to the quotient by {I,-I} taking us to the corresponding projective group): the kernel of reduction is called Γ(N) and plays a big role in the theory of [[modular form]]s. Further, we may take the inverse image of any subgroup (not just {e}) and get a congruence subgroup: the subgroups Γ<sub>0</sub>(N) important in modular form theory are defined in this way, from the subgroup of mod ''N'' ''2x2'' matrices with 1 on the diagonal and 0 below it.
 
More generally, the notion of '''congruence subgroup''' can be defined for [[arithmetic subgroup]]s of [[algebraic group]]s; that is, those for which we have a notion of 'integral structure' respected by the subgroup, and so some general idea of what 'congruence' means.
 
==Congruence subgroups and topological groups==
 
Are all subgroups of finite [[Index of a subgroup|index]] actually congruence subgroups? This is not in general true, and ''non-congruence subgroups'' exist. It is however an interesting question to understand when these examples are possible. This problem about the [[classical group]]s was resolved by {{harvtxt|Bass|Milnor|Serre|1967}}. .
 
It can be posed in topological terms: if Γ is some arithmetic group, there is a topology on Γ for which a base of neighbourhoods of {e} is the set of subgroups of finite index; and there is another topology defined in the same way using only congruence subgroups. We can ask whether those are the same topologies; equivalently, if they give rise to the same completions. The subgroups of finite index give rise to the completion of Γ as a [[pro-finite group]]. If there are essentially fewer congruence subgroups, the corresponding completion of Γ can be bigger (intuitively, there are fewer conditions for a [[Cauchy sequence]] to comply with). Therefore the problem can be posed as a relationship of two [[compact group|compact]] [[topological group]]s, with the question reduced to calculation of a possible [[kernel (algebra)|kernel]]. The solution by [[Hyman Bass]], [[Jean-Pierre Serre]] and [[John Milnor]] involved an aspect of [[algebraic number theory]] linked to [[K-theory]].
 
The use of [[adele ring|adele]] methods for [[automorphic representation]]s (for example in the [[Langlands program]]) implicitly uses that kind of completion with respect to a congruence subgroup topology - for the reason that then all congruence subgroups can then be treated within a single [[group representation]]. This approach - using a group G('''A''') and its single quotient G('''A''')/G('''Q''') rather than looking at many G/Γ as a whole system - is now normal in abstract treatments.
 
==Congruence subgroups of the modular group==
{{see also|modular curve}}
 
Detailed information about the congruence subgroups of the [[modular group|modular group Γ]] has proved basic in much research, in [[number theory]], and in other areas such as [[monstrous moonshine]].
 
===Modular group Γ(''r'')===
 
For a given positive [[integer]] ''r'', the modular group '''Γ(''r'')''' is defined as follows:<ref name=L29>Lang (1976) p.29</ref>
 
:<math>\Gamma(r) := \left\{\begin{bmatrix} a&b\\c&d \end{bmatrix} \in \Gamma : a\equiv d\equiv \pm 1,~b\equiv c\equiv 0\mod r\right\}.</math>
 
===Modular group Γ<sub>1</sub>(''r'')===
 
For a given positive [[integer]] ''r'', the modular group '''Γ<sub>1</sub>(''r'')''' is defined as follows:<ref name=L29/>
 
:<math>\Gamma_1(r) := \left\{\begin{bmatrix} a&b\\c&d \end{bmatrix} \in \Gamma :  a\equiv d\equiv 1,~c\equiv 0\mod r\right\}.</math>
 
===Modular group Γ<sub>0</sub>(''r'')===
 
For a given positive [[integer]] ''r'', the modular group '''Γ<sub>0</sub>(''r'')''' is defined as follows:<ref name=L29/>
 
:<math>\Gamma_0(r) := \left\{\begin{bmatrix} a&b\\c&d \end{bmatrix} \in \Gamma : c\equiv 0\mod r\right\}.</math>
 
It can be shown that for a [[prime number]] ''p'', the set
 
:<math>R_\Gamma \cup \bigcup_{k=0}^{p-1} ST^k(R_\Gamma)</math>
 
(where ''S''τ =&nbsp;&minus;1/τ and ''T''τ =&nbsp;τ&nbsp;+&nbsp;1) is a [[fundamental region]] of Γ<sub>0</sub>(''r'').
 
The [[normalizer]] Γ<sub>0</sub>(''p'')<sup>+</sup> of Γ<sub>0</sub>(''p'') in ''SL''(2,'''R''') has been investigated; one result from the 1970s, due to [[Jean-Pierre Serre]], [[Andrew Ogg]] and [[John G. Thompson]] is that the corresponding [[modular curve]] (the [[Riemann surface]] resulting from taking the quotient of the hyperbolic plane by Γ<sub>0</sub>(''p'')<sup>+</sup>) has [[genus (mathematics)|genus]] zero (the modular curve is an [[elliptic curve]]) [[if and only if]] ''p'' is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. When Ogg later heard about the [[monster group]], he noticed that these were precisely the [[prime factor]]s of the size of ''M'', he wrote up a paper offering a bottle of [[Jack Daniel's]] whiskey to anyone who could explain this fact – this was a starting point for the theory of [[Monstrous moonshine]], which explains deep connections between modular function theory and the monster group.
 
===Modular group Λ===
The '''modular group Λ''' (also called the '''theta subgroup''') is another [[subgroup]] of the modular group Γ. It can be characterized as the set of linear [[Möbius transformation]]s ''w'' that satisfy
 
:<math>w(t) = \frac{at + b}{ct + d}</math>
 
with ''a'' and ''d'' being [[odd number|odd]] and ''b'' and ''c'' being [[even number|even]]. That is, it is the congruence subgroup that is the kernel of reduction modulo 2, otherwise known as Γ(2).
 
==References==
{{reflist}}
*{{Citation | last1=Bass | first1=H. | last2=Milnor | first2=John Willard | author2-link=John Milnor | last3=Serre | first3=Jean-Pierre | author3-link=Jean-Pierre Serre | title=Solution of the congruence subgroup problem for SL<sub>''n''</sub> (''n''≥3) and Sp<sub>''2n''</sub> (''n''≥2) | url=http://www.numdam.org/item?id=PMIHES_1967__33__59_0 | mr=0244257 | year=1967 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=33 | pages=59–137}} ([http://www.numdam.org/item?id=PMIHES_1974__44__241_0 Erratum])
* {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Introduction to Modular Forms | volume=222 | series=Grundlehren der mathematischen Wissenschaften | publisher=[[Springer-Verlag]] | year=1976 | isbn=3-540-07833-9 }}
 
[[Category:Group theory]]
[[Category:Modular forms]]

Latest revision as of 13:34, 22 August 2014

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