Eisenstein–Kronecker number explained

In mathematics, Eisenstein–Kronecker numbers are an analogue for imaginary quadratic fields of generalized Bernoulli numbers.^[1] They are defined in terms of classical Eisenstein–Kronecker series, which were studied by Kenichi Bannai and Shinichi Kobayashi using the Poincaré bundle.^{[1] [2]}

Eisenstein–Kronecker numbers are algebraic and satisfy congruences that can be used in the construction of two-variable p-adic L-functions.^{[1] [3]} They are related to critical L-values of Hecke characters.^[3]

Definition

When is the area of the fundamental domain of

\Gamma divided by \pi , where \Gamma is a lattice in C :^[3] $$e\_^(z\_0,w\_0):=\backslash sum\_\backslash frac\backslash langle\backslash gamma,w\_0\backslash rangle\_\backslash Gamma,$$when N _{0:=N\cup\{0\}, \{a,b\inN} _{0:b} > a +2 \},z _{0,w} _{0\inC,}
where \langle z,w\rangle ^{(z\overline{w)/(- w\overline{z}}{A})} _{\Gamma: = e} and \overline{z} is the complex conjugate of .

Notes and References

1. Charollois . Pierre . Sczech . Robert . 2016 . Elliptic Functions According to Eisenstein and Kronecker: An Update . EMS Newsletter . en . 2016-9 . 101 . 8–14 . 10.4171/NEWS/101/4 . 1027-488X. free .
2. Sprang . Johannes . 2019 . Eisenstein–Kronecker Series via the Poincaré bundle . . en . 7 . e34 . 10.1017/fms.2019.29 . 2050-5094. free . 1801.05677 .
3. Bannai . Kenichi . Kobayashi . Shinichi . 2010 . Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers . . 153 . 2 . 10.1215/00127094-2010-024 . 0012-7094. math/0610163 .