{{Short description|Gauss sum on an elliptic curve}} {{No footnotes|date=June 2020}} 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 {{harvs|txt|last=Eisenstein|authorlink=Gotthold Eisenstein||year=1850}}, at least in the lemniscate case when the elliptic curve has complex multiplication by {{mvar|i}}, but seem to have been forgotten or ignored until the paper {{harv|Pinch|1988}}.
==Example==
{{harv|Lemmermeyer|2000|loc=9.3}} gives the following example of an elliptic Gauss sum, for the case of an elliptic curve with complex multiplication by {{mvar|i}}.
:<math>-\sum_t\chi(t)\varphi\left ( \frac{t}{\pi} \right )^\frac{p-1}{m}</math> where *The sum is over residues mod {{mvar|P}} whose representatives are Gaussian integers *{{mvar|n}} is a positive integer *{{mvar|m}} is a positive integer dividing {{math|4''n''}} *{{math|''p'' {{=}} 4''n'' + 1}} is a rational prime congruent to 1 mod 4 *{{math|''φ''(''z'') {{=}} sl((1 – ''i'')''ωz'')}} where {{math|sl}} is the sine lemniscate function, an elliptic function. *{{mvar|χ}} is the {{mvar|m}}th power residue symbol in {{mvar|K}} with respect to the prime {{mvar|P}} of {{mvar|K}} *{{mvar|K}} is the field {{math|''k''[''ζ'']}} *{{mvar|k}} is the field <math>\mathbb{Q}[i]</math> *{{mvar|ζ}} is a primitive {{math|4''n''}}th root of 1 *{{mvar|π}} is a primary prime in the Gaussian integers <math>\mathbb{Z}[i]</math> with norm {{mvar|p}} *{{mvar|P}} is a prime in the ring of integers of {{mvar|K}} lying above {{mvar|π}} with inertia degree 1
==References==
*{{Citation | last1=Asai | first1=Tetsuya | title=Proceedings of the Symposium on Algebraic Number Theory and Related Topics | arxiv=0707.3711| publisher=Res. Inst. Math. Sci. (RIMS), Kyoto | series=RIMS Kôkyûroku Bessatsu, B4 |mr=2402004 | year=2007 | chapter=Elliptic Gauss sums and Hecke ''L''-values at ''s'' = 1 | pages=79–121| bibcode=2007arXiv0707.3711A }} *{{Citation | last1=Cassou-Noguès | first1=Ph. | last2=Taylor | first2=M. J. | title=Un élément de Stickelberger quadratique | doi=10.1016/S0022-314X(05)80046-0 |mr=1096447 | year=1991 | journal=Journal of Number Theory | issn=0022-314X | volume=37 | issue=3 | pages=307–342| doi-access=free }} *{{Citation | last1=Eisenstein | first1=Gotthold | title=Über einige allgemeine Eigenschaften der Gleichung, von welcher die Teilung der ganzen Lemniskate abhängt, nebst Anwendungen derselben auf die Zahlentheorie | url=https://zenodo.org/record/1844062/files/article.pdf | id=Reprinted in Math. Werke II, 556–619 | year=1850 | journal=Journal für die Reine und Angewandte Mathematik | issn=0075-4102 | volume=1850 | issue=39 | pages=224–287| doi=10.1515/crll.1850.39.224 | s2cid=123157985 }} *{{Citation | last1=Lemmermeyer | first1=Franz | title=Reciprocity laws | url=https://books.google.com/books?id=EwjpPeK6GpEC | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-66957-9 |mr=1761696 | year=2000}} *{{Citation | last1=Pinch | first1=R. | editor1-last=Stephens | editor1-first=Nelson M. | editor2-last=Thorne. | editor2-first=M. P. | title=Computers in mathematical research (Cardiff, 1986) | chapter-url=https://books.google.com/books?id=SraEAAAAIAAJ | publisher=Oxford University Press | series=Inst. Math. Appl. Conf. Ser. New Ser. | isbn=978-0-19-853620-8 | mr=960495 | year=1988 | volume=14 | chapter=Galois module structure of elliptic functions | pages=[https://archive.org/details/computersinmathe0000unse_e9v1/page/69 69–91] | url=https://archive.org/details/computersinmathe0000unse_e9v1/page/69 }}
Category:Algebraic number theory Category:Elliptic curves