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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 , 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 . ==Example== gives the following example of an elliptic Gauss sum, for the case of an elliptic curve with complex multiplication by ''i''. : 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''() *''k'' is the field Q() *ζ is a primitive 4''n''-th root of 1 *π is a primary prime in the Gaussian integers Z() with norm ''p'' *''P'' is a prime in the ring of integers of ''K'' lying above π with inertia degree 1 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Elliptic Gauss sum」の詳細全文を読む スポンサード リンク
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