Chowla–Mordell theorem

In mathematics, the Chowla–Mordell theorem is a result in number theory determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity. It was proved and published independently by Sarvadaman Chowla and Louis Mordell, around 1951.

In detail, if ${\displaystyle p}$ is a prime number, ${\displaystyle \chi }$ a nontrivial Dirichlet character modulo ${\displaystyle p}$, and

${\displaystyle G(\chi )=\sum \chi (a)\zeta ^{a}}$

where ${\displaystyle \zeta }$ is a primitive ${\displaystyle p}$-th root of unity in the complex numbers, then

${\displaystyle {\frac {G(\chi )}{|G(\chi )|}}}$

is a root of unity if and only if ${\displaystyle \chi }$ is the quadratic residue symbol modulo ${\displaystyle p}$. The 'if' part was known to Gauss: the contribution of Chowla and Mordell was the 'only if' direction. The ratio in the theorem occurs in the functional equation of L-functions.

References

• Gauss and Jacobi Sums by Bruce C. Berndt, Ronald J. Evans and Kenneth S. Williams, Wiley-Interscience, p. 53.