Elliptic Gauss sum

In mathematics, an elliptic Gauss sum is an analog of a Gauss sum depending on an elliptic curve with complex multiplication. The quadratic residue symbol in a Gauss sum is replaced by a higher residue symbol such as a cubic or quartic residue symbol, and the exponential function in a Gauss sum is replaced by an elliptic function. They were introduced by Eisenstein (1850), at least in the lemniscate case when the elliptic curve has complex multiplication by $i$ , but seem to have been forgotten or ignored until the paper (Pinch 1988).

Example

(Lemmermeyer 2000, 9.3) gives the following example of an elliptic Gauss sum, for the case of an elliptic curve with complex multiplication by $i$ .

-\sum _{t}\chi (t)\varphi \left({\frac {t}{\pi }}\right)^{\frac {p-1}{m}}

where

The sum is over residues mod $P$ whose representatives are Gaussian integers
$n$ is a positive integer
$m$ is a positive integer dividing $4 n$
$p = 4 n + 1$ is a rational prime congruent to 1 mod 4
$φ (z) = sl((1 - i) ωz)$ where $sl$ is the sine lemniscate function, an elliptic function.
$χ$ is the $m$ th power residue symbol in $K$ with respect to the prime $P$ of $K$
$K$ is the field $k [ζ]$
$k$ is the field $\mathbb {Q} [i]$
$ζ$ is a primitive $4 n$ th root of 1
$π$ is a primary prime in the Gaussian integers $\mathbb {Z} [i]$ with norm $p$
$P$ is a prime in the ring of integers of $K$ lying above $π$ with inertia degree 1

References

Asai, Tetsuya (2007), "Elliptic Gauss sums and Hecke L-values at s = 1", Proceedings of the Symposium on Algebraic Number Theory and Related Topics, RIMS Kôkyûroku Bessatsu, B4, Res. Inst. Math. Sci. (RIMS), Kyoto, pp. 79–121, arXiv:0707.3711, Bibcode:2007arXiv0707.3711A, MR 2402004
Cassou-Noguès, Ph.; Taylor, M. J. (1991), "Un élément de Stickelberger quadratique", Journal of Number Theory, 37 (3): 307–342, doi:10.1016/S0022-314X(05)80046-0, ISSN 0022-314X, MR 1096447
Eisenstein, Gotthold (1850), "Über einige allgemeine Eigenschaften der Gleichung, von welcher die Teilung der ganzen Lemniskate abhängt, nebst Anwendungen derselben auf die Zahlentheorie" (PDF), Journal für die Reine und Angewandte Mathematik, 1850 (39): 224–287, doi:10.1515/crll.1850.39.224, ISSN 0075-4102, S2CID 123157985, Reprinted in Math. Werke II, 556–619
Lemmermeyer, Franz (2000), Reciprocity laws, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66957-9, MR 1761696
Pinch, R. (1988), "Galois module structure of elliptic functions", in Stephens, Nelson M.; Thorne., M. P. (eds.), Computers in mathematical research (Cardiff, 1986), Inst. Math. Appl. Conf. Ser. New Ser., vol. 14, Oxford University Press, pp. 69–91, ISBN 978-0-19-853620-8, MR 0960495