[jl2718]

This is not my field, and from loose understanding, so I’ll type out a hot take off the top of my head and prepare to be corrected about it, but… some of the things in the article seemed a bit in need of clarification and relevance. Hope I get it right.

Weil is most often encountered in the field of elliptic curves, which they alluded to by the form of the polynomial, but they are mixing the reference to two types of polynomials. The equivalence of finite fields and polynomials was already established by Galois, and is maybe the central theory of abstract algebra. Weil connected this to algebraic geometry through ‘torsion points’ on an elliptic curve, which is a specific form of polynomial itself defined on a (finite) field, that have degree (adding a point to itself until it equals itself again) equal to a prime power in elliptic algebra (addition is the inverse of the intersection of a secant defined by the two points with the curve). This defines a group of prime order, and to get a field, he defined a ‘pairing’ function that works like multiplication between two group elements and returns an element in an ‘extended’ curve that belongs to a group of the same order and has the same coefficient as if the two original group coefficients were multiplied, which is known as the bilinearity property. The extension here is I think what they mean by ‘complex’ in that it doubles the degree of the field polynomial, which gives a Cartesian product of groups from the original definition, and this can be used to divide through ‘torsion points’ back to the field element that defines the product point. This opens up an elliptic form of the zeta function, which was used to prove the Riemann hypothesis on elliptic curves, although not yet extended to the integers, and Fermat’s last theorem. The most common use of this is probably pairing-based cryptography, which is based on bilinearity and computational asymmetry of the pairing operation, which is to say that you can ‘multiply’ through the pairing operation, but you cannot efficiently factor.

So now you have some context of why it’s relevant and some basic terms to look up and you’ll probably find a lot more precise but convoluted language about these things to correct me on.