Weil conjectures

In mathematics, the Weil conjectures were some highly influential proposals by André Weil (1949) on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.

A variety V over a finite field with q elements has a finite number of rational points, as well as points over every finite field with q^k elements containing that field. The generating function has coefficients derived from the numbers N_k of points over the (essentially unique) field with q^k elements.

Weil conjectured that such zeta-functions should be rational functions, should satisfy a form of functional equation, and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the Riemann zeta function and Riemann hypothesis. The rationality was proved by Dwork (1960), the functional equation by Grothendieck (1965), and the analogue of the Riemann hypothesis was proved by Deligne (1974).

The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his Disquisitiones Arithmeticae (Mazur 1974), 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 p − 1 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)^× of non-zero residues modulo p under multiplication and its unique subgroup of index three. Gauss lets ${\displaystyle {\mathfrak {R}}}$, ${\displaystyle {\mathfrak {R}}'}$, and ${\displaystyle {\mathfrak {R}}''}$ 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, ${\displaystyle ({\mathfrak {R}}{\mathfrak {R}})}$ equal to the number of elements of Z/pZ which are in ${\displaystyle {\mathfrak {R}}}$ and which, after being increased by one, are also in ${\displaystyle {\mathfrak {R}}}$. 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 α + 1 are both in ${\displaystyle {\mathfrak {R}}}$, then there exist x and y in Z/pZ such that x³ = α and y³ = α + 1; consequently, x³ + 1 = y³. Therefore ${\displaystyle ({\mathfrak {R}}{\mathfrak {R}})}$ is the number of solutions to x³ + 1 = y³ 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.

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