Elliptic Gauss sum

Template:Short description Template:No footnotes 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 Template:Harvs, at least in the lemniscate case when the elliptic curve has complex multiplication by Template:Mvar, but seem to have been forgotten or ignored until the paper Template:Harv.

Example

Template:Harv gives the following example of an elliptic Gauss sum, for the case of an elliptic curve with complex multiplication by Template:Mvar.

$-\sum _t\chi (t)\phi {\left(\frac{t}{\pi }\right)}^{\frac{p-1}{m}}$

where

• The sum is over residues mod Template:Mvar whose representatives are Gaussian integers
• Template:Mvar is a positive integer
• Template:Mvar is a positive integer dividing Template:Math
• Template:Math is a rational prime congruent to 1 mod 4
• Template:Math where Template:Math is the sine lemniscate function, an elliptic function.
• Template:Mvar is the Template:Mvarth power residue symbol in Template:Mvar with respect to the prime Template:Mvar of Template:Mvar
• Template:Mvar is the field Template:Math
• Template:Mvar is the field $\mathbb{Q}[i]$
• Template:Mvar is a primitive Template:Mathth root of 1
• Template:Mvar is a primary prime in the Gaussian integers $\mathbb{Z}[i]$ with norm Template:Mvar
• Template:Mvar is a prime in the ring of integers of Template:Mvar lying above Template:Mvar with inertia degree 1

References

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