| 词条 | Schottky form |
| 释义 |
In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form J of degree 4 and weight 8, introduced by {{harvs|txt|authorlink=Friedrich Schottky|first=Friedrich |last=Schottky|year=1888|year2=1903}} as a degree 16 polynomial in the Thetanullwerte of genus 4. He showed that it vanished at all Jacobian points (the points of the degree 4 Siegel upper half-space corresponding to 4-dimensional abelian varieties that are the Jacobian varieties of genus 4 curves). {{harvtxt|Igusa|1981}} showed that it is a multiple of the difference θ4(E8 ⊕ E8) − θ4(E16) of the two genus 4 theta functions of the two 16-dimensional even unimodular lattices and that its divisor of zeros is irreducible. {{harvtxt|Poor|Yuen|1996}} showed that it generates the 1-dimensional space of level 1 genus 4 weight 8 Siegel cusp forms. Ikeda showed that the Schottky form is the image of the Dedekind Delta function under the Ikeda lift. References
|last=Igusa|first= Jun-ichi |chapter=Schottky's invariant and quadratic forms|title= E. B. Christoffel (Aachen/Monschau, 1979)|pages= 352–362|publisher=Birkhäuser|place= Basel-Boston, Mass.|year= 1981|isbn=978-3-7643-1162-9|doi=10.1007/978-3-0348-5452-8_24 }}
|last=Igusa|first= Jun-ichi |title=On the irreducibility of Schottky's divisor |journal=J. Fac. Sci. Univ. Tokyo Sect. IA Math.|volume= 28 |orig-year=1981|issue= 3|pages= 531–545 |year=1982|url=http://jairo.nii.ac.jp/0021/00004593/en}}
|last=Poor|first= Cris|last2= Yuen|first2= David S. |title=Dimensions of spaces of Siegel modular forms of low weight in degree four |journal=Bull. Austral. Math. Soc.|volume= 54 |year=1996|issue= 2|pages= 309–315|doi=10.1017/s0004972700017779}}
|title=Zur Theorie der Abel'schen Functionen von vier Variabeln| jfm =20.0488.02 |journal=Journal für die Reine und Angewandte Mathematik|volume= 102|pages= 304–352 |year=1888|url= http://resolver.sub.uni-goettingen.de/purl?GDZPPN002160641}}
|title=Über die Moduln der Thetafunktionen|jfm= 34.0506.03 |journal=Acta Math.|volume= 27|pages= 235–288 |year=1903|doi=10.1007/bf02421309}} 1 : Automorphic forms |
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