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.

References

1. Bannai, Kenichi; Kobayashi, Shinichi (2007), "Algebraic theta functions and Eisenstein-Kronecker numbers", in Hashimoto, Kiichiro (ed.), Proceedings of the Symposium on Algebraic Number Theory and Related Topics, RIMS Kôkyuroku Bessatsu, B4, Res. Inst. Math. Sci. (RIMS), Kyoto, pp. 63–77, arXiv:0709.0640, Bibcode:2007arXiv0709.0640B, MR 2402003
2. Bannai, Kenichi; Kobayashi, Shinichi; Tsuji, Takeshi (2009), "Realizations of the elliptic polylogarithm for CM elliptic curves", in Asada, Mamoru; Nakamura, Hiroaki; Takahashi, Hiroki (eds.), Algebraic number theory and related topics 2007, RIMS Kôkyuroku Bessatsu, B12, Res. Inst. Math. Sci. (RIMS), Kyoto, pp. 33–50, MR 2605771
3. Charollois, Pierre; Sczech, Robert (2016). "Elliptic Functions According to Eisenstein and Kronecker: An Update". EMS Newsletter. 2016–9 (101): 8–14. doi:10.4171/NEWS/101/4. ISSN 1027-488X.
4. Sprang, Johannes (2019). "Eisenstein–Kronecker Series via the Poincaré bundle". Forum of Mathematics, Sigma. 7: e34. arXiv:1801.05677. doi:10.1017/fms.2019.29. ISSN 2050-5094.
5. Bannai, Kenichi; Kobayashi, Shinichi (2010). "Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers". Duke Mathematical Journal. 153 (2). arXiv:math/0610163. doi:10.1215/00127094-2010-024. ISSN 0012-7094.