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WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB

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In mathematics, an '''elliptic Gauss sum''' is an analog of a [[Gauss sum]] depending on an [[elliptic curve]] with complex mutliplication. 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 ''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 ''i''.

:<math>-\sum_t\chi(t)\phi\left ( \frac{t}{\pi} \right )^{(p-1)/m}</math>
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'')ω''z'') 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 '''Q'''[''i'']
*ζ is a primitive 4''n''-th root of 1
*π is a primary prime in the Gaussian integers '''Z'''[''i''] with norm ''p''
*''P'' is a prime in the ring of integers of ''K'' lying above π with inertia degree 1

==References==

*{{Citation | last1=Asai | first1=Tetsuya | title=Proceedings of the Symposium on Algebraic Number Theory and Related Topics | url=http://arxiv.org/abs/0707.3711 | publisher=Res. Inst. Math. Sci. (RIMS), Kyoto | series=RIMS Kôkyûroku Bessatsu, B4 | id={{MR|2402004}} | year=2007 | chapter=Elliptic Gauss sums and Hecke L-values at ''s'' = 1 | pages=79–121}}
*{{Citation | last1=Cassou-Noguès | first1=Ph. | last2=Taylor | first2=M. J. | title=Un élément de Stickelberger quadratique | url=http://dx.doi.org/10.1016/S0022-314X(05)80046-0 | doi=10.1016/S0022-314X(05)80046-0 | id={{MR|1096447}} | year=1991 | journal=[[Journal of Number Theory]] | issn=0022-314X | volume=37 | issue=3 | pages=307–342}}
*{{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=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0039 | id=Reprinted in Math. Werke II, 556–619 | year=1850 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=39 | pages=224–287}}
*{{Citation | last1=Lemmermeyer | first1=Franz | title=Reciprocity laws | url=http://books.google.com/books?id=EwjpPeK6GpEC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-66957-9 | id={{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) | url=http://books.google.com/books?id=SraEAAAAIAAJ | publisher=[[Oxford University Press]] | series=Inst. Math. Appl. Conf. Ser. New Ser. | isbn=978-0-19-853620-8 | id={{MR|960495}} | year=1988 | volume=14 | chapter=Galois module structure of elliptic functions | pages=69–91}}

[[Category:Algebraic number theory]]
[[Category:Elliptic curves]]

Neovius surface - Revision history

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB