Chowla–Mordell theorem

Short description: When a Gauss sum is the square root of a prime number, multiplied by a root of unity

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 [math]\displaystyle{ p }[/math] is a prime number, [math]\displaystyle{ \chi }[/math] a nontrivial Dirichlet character modulo [math]\displaystyle{ p }[/math], and

[math]\displaystyle{ G(\chi)=\sum \chi(a) \zeta^a }[/math]

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

[math]\displaystyle{ \frac{G(\chi)}{|G(\chi)|} }[/math]

is a root of unity if and only if [math]\displaystyle{ \chi }[/math] is the quadratic residue symbol modulo [math]\displaystyle{ p }[/math]. 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.

fi:Chowlan–Mordellin lause