Eisenstein–Kronecker number

In mathematics, Eisenstein–Kronecker numbers are an analogue for imaginary quadratic fields of generalized Bernoulli numbers.^[1][2][3] They are defined in terms of classical Eisenstein–Kronecker series, which were studied by Kenichi Bannai and Shinichi Kobayashi using the Poincaré bundle.^[3][4]

Eisenstein–Kronecker numbers are algebraic and satisfy congruences that can be used in the construction of two-variable p-adic L-functions.^[3][5] They are related to critical L-values of Hecke characters.^[1][5]

Definition

When A is the area of the fundamental domain of ${\displaystyle \Gamma }$ divided by ${\displaystyle \pi }$, where ${\displaystyle \Gamma }$ is a lattice in ${\displaystyle \mathbb {C} }$:^[5] $${\displaystyle e_{a,b}^{*}(z_{0},w_{0}):=\sum _{\gamma \in \Gamma \setminus \{-z_{0}\}}{\frac {({\bar {z_{0}}}+{\bar {\gamma }})^{a}}{(z_{0}+\gamma )^{b}}}\langle \gamma ,w_{0}\rangle _{\Gamma },}$$ when ${\displaystyle \mathbb {N} _{0}:=\mathbb {N} \cup \{0\},\,\{a,b\in \mathbb {N} _{0}:b>a+2\},\,z_{0},w_{0}\in \mathbb {C} ,}$
where ${\displaystyle \langle z,w\rangle _{\Gamma }:=e^{\frac {z{\overline {w}}-w{\overline {z}}}{A}}}$ and ${\displaystyle {\overline {z}}}$ is the complex conjugate of z.