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 , 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