Siegel's half-space

In the mathematical sub-area of function theory , the Siegel half-space or the Siegel half-plane denotes a generalization of the half-plane . This room is named after the mathematician Carl Ludwig Siegel , who systematically examined this object.

definition

Siegel's (upper) half-space of degrees is defined as the set of complex symmetric matrices whose imaginary part is positive definite . ${\ displaystyle {\ mathcal {H}} ^ {g}}$ ${\ displaystyle g \ in \ mathbb {N}}$ ${\ displaystyle (g \ times g)}$

Effect of the symplectic group

The symplectic group works through Siegel's half-space ${\ displaystyle Sp (2g, \ mathbb {R})}$

{\ displaystyle \ left ({\ begin {array} {cc} A&B \\ C&D \ end {array}} \ right) Z = (AZ + B) (CZ + D) ^ {- 1}}

.

This effect is transitive , its stabilizers are conjugated to the orthogonal group . ${\ displaystyle SO (2g)}$

One can provide Siegel's half-space with a Riemannian metric , through which it becomes a symmetrical space isometric too . ${\ displaystyle Sp (2g, \ mathbb {R}) / SO (2g)}$

Remarks

In this case , Siegel's half-space is the known upper half-plane . ${\ displaystyle g = 1}$ ${\ displaystyle \ mathbb {H}: = \ {z \ in \ mathbb {C} \ mid \ operatorname {Im} (z)> 0 \}}$

Siegel's upper half-space carries an operation of the symplectic group . The quotient is the modular space of the principally polarized Abelian varieties . In the case, the quotient space parameterizes elliptic curves . The j-function indicates the j-invariant of the curve. ${\ displaystyle \ operatorname {Sp} _ {2g} (\ mathbb {Z})}$ ${\ displaystyle g = 1}$ ${\ displaystyle \ operatorname {SL} _ {2} (\ mathbb {Z}) \ backslash \ mathbb {H}}$

Ichirō Satake gave a compactification of the sealed half-space in 1957.

Multi-dimensional theta series

Siegel's half-space, as a generalization of the upper half-plane, plays an important role in defining the theta series in several complex variables. The multidimensional theta series is a function

{\ displaystyle \ theta \ colon \ mathbb {C} ^ {g} \ times {\ mathcal {H}} ^ {g} \ to \ mathbb {C},}

by

{\ displaystyle \ theta (z, T) = \ sum _ {n \ in \ mathbb {Z} ^ {g}} \ operatorname {exp} (\ pi i \ langle n, 2z + Tn \ rangle)}

is defined. This series converges normally and is therefore holomorphic .

literature

Klaus Lamotke : Riemann surfaces. 2nd supplemented and improved edition. Springer, Berlin et al. 2009, ISBN 978-3-642-01710-0 .