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

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

Definition

When A is the area of the fundamental domain of ${\displaystyle \Gamma }$ divided by ${\displaystyle \pi }$, where ${\displaystyle \Gamma }$ is a lattice in ${\displaystyle \mathbb {C} }$: $${\displaystyle e_{a,b}^{*}(z_{0},w_{0}):=\sum _{\gamma \in \Gamma \setminus \{-z_{0}\}}{\frac {({\bar {z_{0}}}+{\bar {\gamma }})^{a}}{(z_{0}+\gamma )^{b}}}\langle \gamma ,w_{0}\rangle _{\Gamma },}$$ 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.

• Number theory