In mathematics, an elliptic Gauss sum is an analog of a
Gauss sum
In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically
:G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r)
where the sum is over elements of some finite commutative ring , is a ...
depending on an
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
with complex multiplication. The
quadratic residue
In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that:
:x^2\equiv q \pmod.
Otherwise, ''q'' is called a quadratic non ...
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
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
.
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