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In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms. The general case, for general groups, is reviewed in the article 'factor of automorphy'.

Definition

An automorphic factor of weight k is a function

ν:Γ×H→C{displaystyle nu :Gamma times mathbb {H} to mathbb {C} }

satisfying the four properties given below. Here, the notation H{displaystyle mathbb {H} } and C{displaystyle mathbb {C} } refer to the upper half-plane and the complex plane, respectively. The notation Γ{displaystyle Gamma } is a subgroup of SL(2,R), such as, for example, a Fuchsian group. An element γ∈Γ{displaystyle gamma in Gamma } is a 2x2 matrix

γ=[abcd]{displaystyle gamma =left[{begin{matrix}a&b\c&dend{matrix}}right]}

with a, b, c, d real numbers, satisfying ad−bc=1.

An automorphic factor must satisfy:

1. For a fixed γ∈Γ{displaystyle gamma in Gamma }, the function ν(γ,z){displaystyle nu (gamma ,z)} is a holomorphic function of z∈H{displaystyle zin mathbb {H} }.

2. For all z∈H{displaystyle zin mathbb {H} } and γ∈Γ{displaystyle gamma in Gamma }, one has

|ν(γ,z)|=|cz+d|k{displaystyle vert nu (gamma ,z)vert =vert cz+dvert ^{k}}

for a fixed real number k.

3. For all z∈H{displaystyle zin mathbb {H} } and γ,δ∈Γ{displaystyle gamma ,delta in Gamma }, one has

ν(γδ,z)=ν(γ,δz)ν(δ,z){displaystyle nu (gamma delta ,z)=nu (gamma ,delta z)nu (delta ,z)}

Here, δz{displaystyle delta z} is the fractional linear transform of z{displaystyle z} by δ{displaystyle delta }.

4.If −I∈Γ{displaystyle -Iin Gamma }, then for all z∈H{displaystyle zin mathbb {H} } and γ∈Γ{displaystyle gamma in Gamma }, one has

ν(−γ,z)=ν(γ,z){displaystyle nu (-gamma ,z)=nu (gamma ,z)}

Here, I denotes the identity matrix.

Properties

Every automorphic factor may be written as

ν(γ,z)=υ(γ)(cz+d)k{displaystyle nu (gamma ,z)=upsilon (gamma )(cz+d)^{k}}

with

|υ(γ)|=1{displaystyle vert upsilon (gamma )vert =1}

The function υ:Γ→S1{displaystyle upsilon :Gamma to S^{1}} is called a multiplier system. Clearly,

υ(I)=1{displaystyle upsilon (I)=1},

while, if −I∈Γ{displaystyle -Iin Gamma }, then

υ(−I)=e−iπk{displaystyle upsilon (-I)=e^{-ipi k}}

References

• 
  Robert Rankin, Modular Forms and Functions, (1977) Cambridge University Press .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  ISBN 0-521-21212-X. (Chapter 3 is entirely devoted to automorphic factors for the modular group.)