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 , at least in the lemniscate case when the elliptic curve has complex multiplication by , 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 .
where The sum is over residues mod whose representatives are Gaussian integers is a positive integer is a positive integer dividing is a rational prime congruent to 1 mod 4 where is the sine lemniscate function, an elliptic function. is the th power residue symbol in with respect to the prime of is the field is the field is a primitive th root of 1 is a primary prime in the Gaussian integers with norm is a prime in the ring of integers of lying above with inertia degree 1
References