View source for Gamma-invariant in the theory of modular forms

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'' $  \Gamma $-
invariant''

Let  $  G ( \Gamma ) $
be a [[Fuchsian group|Fuchsian group]] of the first kind, acting on the real hyperbolic plane  $  H $. 
Suppose that  $  G $
is a group of genus zero, i.e. the number of hyperbolic generators of  $  G $
is equal to zero. Let  $  J $
be a [[Conformal mapping|conformal mapping]] of the [[Fundamental domain|fundamental domain]]  $  F $
of  $  G $
onto the complex plane  $  \mathbf C $
that is extended as an [[Automorphic function|automorphic function]] to the whole of  $  H $. 
In other words, there exists an isomorphism of complex manifolds between the compactification of  $  F $
and the Riemann sphere which induces an univalent automorphic function  $  J $
on  $  H $. 
The function  $  J $
is defined up to a [[Fractional-linear mapping|fractional-linear mapping]] and satisfies the Schwarz differential equation (see [[#References|[a1]]]; see also [[Schwarz differential|Schwarz differential]]; [[Schwarz equation|Schwarz equation]]).

The function  $  J $
is called a gamma- or automorphic (modular) invariant.

The first example of a gamma-invariant,  $  k ( 2 ) $, 
was discovered and investigated in the Abel–Gauss–Jacobi and Eisenstein–Weierstrass theory of elliptic and elliptic modular functions between 1820– 1850 (see [[#References|[a2]]]). The function  $  k ( 2 ) $
was obtained as a function of the quotient of the periods of a certain [[Elliptic integral|elliptic integral]]. It corresponds to the level-two congruence subgroup of the [[Modular group|modular group]]  $  { \mathop{\rm PSL} } ( 2, \mathbf Z ) $. 
For the modular group itself, the invariant  $  J $
was constructed in 1877 by R. Dedekind and, one year later and independently, by F. Klein. In this definition,  $  J $
is a certain rational function of the two main modular holomorphic [[Eisenstein series]]  $  E ( 4 ) $
and  $  E ( 6 ) $. 
These two classical invariants  $  k ( 2 ) $
and  $  J $
satisfy the modular equation.

Gamma-invariants play an important role in various domains of mathematics and mathematical physics. For example, in [[Algebraic number theory|algebraic number theory]] the modular equation and its generalizations were used by Ch. Hermite and L. Kronecker to obtain the transcendental solution of the general [[Algebraic equation|algebraic equation]] of the fifth degree (see [[#References|[a2]]]). The modular invariant is important in the extension of the [[Kronecker theorem|Kronecker theorem]] on Abelian fields for imaginary quadratic ground fields and, more generally, in [[Class field theory|class field theory]] (see [[#References|[a3]]], [[#References|[a4]]]). Other important interpretations and developments of this theory are in [[Algebraic geometry|algebraic geometry]] (the theory of elliptic and modular curves) and in the theory of Dirichlet  $  L $-
series (see [[#References|[a5]]], [[#References|[a6]]]; see also [[Dirichlet L-function|Dirichlet  $  L $-
function]]).

Automorphic invariants participate also in Selberg's theory of the trace formula for constructing resolvents of automorphic Laplacians (automorphic Green functions) (see [[#References|[a1]]], [[#References|[a6]]]).

In [[#References|[a7]]] modular equations are used for investigating non-congruence subgroups of the modular group.

There is a mysterious relationship between simple finite groups (the Monster, cf. [[Simple finite group|Simple finite group]]) and automorphic invariants (see [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]]). In particular, via this relation there is a connection between modular invariants and [[Mathematical physics|mathematical physics]] (string theory [[#References|[a11]]], vertex operator algebra and quantum field theory [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a15]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.B. Venkov, "Spectral theory of automorphic functions" , Kluwer Acad. Publ. (1990) {{MR|1135112}} {{ZBL|0719.11030}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Lehner, "Discontinuous groups and automorphic functions" , Amer. Math. Soc. (1964) {{MR|0164033}} {{ZBL|0178.42902}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Weber, "Lehrbuch der Algebra" , Chelsea (1961) {{MR|}} {{ZBL|48.0074.04}} {{ZBL|43.0143.01}} {{ZBL|30.0093.01}} {{ZBL|29.0064.01}} {{ZBL|39.0508.06}} {{ZBL|27.0056.01}} {{ZBL|26.0102.01}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> G. Shimura, "Complex multiplication in terms of elliptic curves" , ''Lecture Notes in Mathematics'' , '''320''' , Springer (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> H. Stark, "Complex multiplication" , ''Lecture Notes in Mathematics'' , '''320''' , Springer (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> D. Zagier, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110060/g11006029.png" />-series and the Green functions of modular curves" , ''Proc. Internat. Congress Mathematicians, Berkeley'' , Amer. Math. Soc. (1986) pp. 689–698 {{MR|0934271}} {{ZBL|0684.14005}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A.O.L. Atkin, H.P.P. Swinnerton-Dyer, "Modular forms on non-congruence subgroups" , ''Proc. Symp. Pure Math.'' , '''19''' , Amer. Math. Soc. (1971) pp. 1–26 {{MR|337781}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J.G. Thompson, "Finite groups and modular functions" ''Bull. London Math. Soc.'' , '''11''' (1979) pp. 347–351 {{MR|0554401}} {{ZBL|0424.20011}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> J.H. Conway, S.P. Norton, "Monstrous moonshine" ''Bull. London Math. Soc.'' , '''11''' (1979) pp. 308–339 {{MR|0554399}} {{ZBL|0424.20010}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Koike, "Moonshine: a mysterious relationship between simple groups and automorphic functions" ''Amer. Math. Soc. Transl. Ser. 2'' , '''160''' (1994) pp. 33–45 {{MR|1308539}} {{ZBL|0829.11025}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> R.E. Borchereds, "Sporadic groups and string theory" , ''First European Congress of Math.'' , '''1''' (1992) pp. 411–421</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> I.B. Frenkel, J. Lepowsky, A. Meurman, "Vertex operator algebras and the monster" , Acad. Press (1988) {{MR|0996026}} {{ZBL|0674.17001}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> E. Verlinde, H. Verlinde, "Chiral bosonization,determinants,and the string partition function" ''Nucl. Phys.'' , '''B288''' (1987) pp. 357 {{MR|0896667}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> D.J. Smit, "String theory and the algebraic geometry of moduli spaces" ''Comm. Math. Phys.'' , '''114''' (1988) pp. 645 {{MR|929134}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> E. Witten, "Elliptic genera and quantum field theory" ''Comm. Math. Phys.'' , '''109''' (1987) pp. 525–536 {{MR|0885560}} {{ZBL|0625.57008}} </TD></TR></table>