In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Schottky–Klein prime form ''E''(''x'',''y'') of a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
''X'' depends on two elements ''x'' and ''y'' of ''X'', and vanishes if and only if ''x'' = ''y''. The prime form ''E'' is not quite a holomorphic function on ''X'' × ''X'', but is a section of a holomorphic line bundle over this space. Prime forms were introduced by
Friedrich Schottky
Friedrich Hermann Schottky (24 July 1851 – 12 August 1935) was a German mathematician who worked on elliptic, abelian, and theta functions and introduced Schottky groups and Schottky's theorem. He was born in Breslau, Germany (now Wrocław, ...
and
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
.
Prime forms can be used to construct meromorphic functions on ''X'' with given poles and zeros. If Σ''n''
''i''''a''
''i'' is a divisor linearly equivalent to 0, then Π''E''(''x'',''a''
''i'')
''n''''i'' is a meromorphic function with given poles and zeros.
See also
*
Fay's trisecant identity
In algebraic geometry, Fay's trisecant identity is an identity between theta functions of Riemann surfaces introduced by . Fay's identity holds for theta functions of Jacobians of curves, but not for theta functions of general abelian varieties.
...
References
*
*
*{{Citation , last1=Mumford , first1=David , author1-link=David Mumford , title=Tata lectures on theta. II , publisher=Birkhäuser Boston , location=Boston, MA , series=Progress in Mathematics , isbn=978-0-8176-3110-9 , mr=742776 , year=1984 , volume=43 , doi=10.1007/978-0-8176-4578-6
Riemann surfaces