Module form

the classical term of a module form is the generic term for a broad class of functions on the upper half plane (elliptical module forms) and their higher-dimension Verallgemeinerungen (e.g. Siegel module forms), which is regarded in the mathematical subsections of the function theory and number theory. The modern term of a module form is its comprehensive new formulation in terms of the representation theory (automorphe representations) and arithmetic geometry (p-adische module forms). Classical module forms are special cases of the so-called automorphen forms.

Table of contents

history

founder of the classical (purely analytic) theory of the module forms 19. Century are smelling pool of broadcasting corporations Dedekind, Felix Klein, Gotthold iron stone and Henri Poincaré. The modern theory of the module forms resulted in the first half of the twentieth century from Erich hedge and Carl Ludwig seal. Module forms in terms of the representation theory come from Robert P. Long country. p-adische module forms arise first with Nicholas Katz and Jean Pierre Serre .

elliptical module forms for <math> \ mbox {SL} _2 (\ mathbb {Z})< /math>

It is

<math> \ mathbb H= \ {\ rope \ in \ mathbb C \ avoided \ mathrm {in} \, \ rope> 0 \}< /math>

the upper half plane.

For a whole number <math> k< /math> a holomorphe is called and/or. meromorphe function <math> f< /math> on the upper half plane a holomorphe and/or. meromorphe elliptical module form of the weight <math> k< /math> to the group <math> \ mbox {SL} _2 (\ mathbb {Z})< /math>, if it

<math> f \! \ (\ frac {az+b} {cz+d} \ right) = (cz+d) ^kf (z) /math< left> for all <math> z \ in \ mathbb H< /math> and <math> A, b, C, D \ in \ mathbb Z< /math> with <math> ad-bc=1< /math>
fulfilled and
  • “holomorph and/or. meromorph in the infinite one " is: That means that the function
<math> \ tilde f (q) =f (z)< /math> with <math> q= \ mathrm e^ {2 \ pi \ mathrm i z}< /math> for <math> 0<|q|<1< /math>
with <math> q=0< /math> holomorph and/or. meromorph on the unit circular disk is continuable.

One notes that from the first condition <math> f (z+1) =f (z)< /math> follows; therefore math <\> tilde f (q) is </math> well-defined.

In case of <math> k=0< /math> one calls f a module function.

The function is <math> f (z)< /math> holomorph in the infinite one, then f is called a whole module form.

Beyond that math <f> (z) has< /math> a zero with <math> z= \ infty< /math>, then one calls f a point form.


characteristics

for odd k is math <f> = \ {0 \} /math<,> the following statements always apply therefore to straight k.

The module forms of the weight k form math <\> mathbb {C for one}< /math> - vector space, just as the whole module forms and also the point forms.

Designates one these vector spaces with <math> \ mathbb {V} _k, \ mathbb {M} _k< /math> and <math> \ mathbb {S} _k< /math>, then applies:

<math> \ mathbb {S} _k \ subset \ mathbb {M} _k \ subset \ mathbb {V} _k. </math>

To the dimension of these vector spaces applies:

<math> \ mathrm {dim} \, \ mathbb {M} _k = \ begin {cases} [\ frac {k} {12}], & \ mathrm {if} \; k \ equiv 2 \; \ mathrm {(mod} \, \ mathrm {12)} \ \ \ mathrm {[} \ frac {k} {12}] +1 & \ mathrm {if} \; k \ emergency \ equiv 2 \; \ mathrm {(mod} \, \ mathrm {12)} \ end {cases}< /math>


<math> \ mathrm {dim} \, \ mathbb {S} _k = \ mathrm {dim} \, \ mathbb {M} _ {k-12} \ quad \ mathrm {if} \ quad k \ geq 12< /math>


examples

the simplest examples of whole module forms of the weight k are the so-called iron stone rows <math> G_k< /math>, for a module function the j-function or absolute invariant and for a point form the discriminant <math> \ delta< /math>.

 

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