Page "Weil conjectures" Paragraph 4

The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his Disquisitiones Arithmeticae, concerned with roots of unity and Gaussian periods.

In article 358, he moves on from the periods that build up towers of quadratic extensions, for the construction of regular polygons ; and assumes that p is a prime number such that is divisible by 3.

Then there is a cyclic cubic field inside the cyclotomic field of pth roots of unity, and a normal integral basis of periods for the integers of this field ( an instance of the Hilbert – Speiser theorem ).

Gauss constructs the order-3 periods, corresponding to the cyclic group ( Z / pZ )< sup >×</ sup > of non-zero residues modulo p under multiplication and its unique subgroup of index three.

Gauss lets,, and be its cosets.

Taking the periods ( sums of roots of unity ) corresponding to these cosets applied to exp ( 2πi / p ), he notes that these periods have a multiplication table that is accessible to calculation.

Products are linear combinations of the periods, and he determines the coefficients.

He sets, for example, equal to the number of elements of Z / pZ which are in and which, after being increased by one, are also in.

He proves that this number and related ones are the coefficients of the products of the periods.

To see the relation of these sets to the Weil conjectures, notice that if α and are both in, then there exist x and y in Z / pZ such that x < sup > 3 </ sup > = α and y < sup > 3 </ sup > = α + 1 ; consequently, x < sup > 3 </ sup > + 1 = y < sup > 3 </ sup >.

Therefore is the number of solutions to x < sup > 3 </ sup > + 1 = y < sup > 3 </ sup > in the finite field Z / pZ.

The other coefficients have similar interpretations.

Gauss's determination of the coefficients of the products of the periods therefore counts the number of points on these elliptic curves, and as a byproduct he proves the analog of the Riemann hypothesis.