In (Canning et al. 2024), recently accepted for publication in Acta Mathematica, Canning, Larson, Payne, and Willwacher proved a striking result: the number of genus- curves over a finite field , counted up to geometric isomorphism, is given by a polynomial in if and only if .
Arithmetic algebraic geometry studies polynomial equations over arbitrary fields, with finite fields playing a particularly important role. If is a prime power, then, up to isomorphism, there is a unique finite field with elements, denoted . An algebraic closure of is an algebraically closed field containing such that every element of is a root of some nonzero polynomial with coefficients in .
For any field , the projective space is the set of nonzero vectors in modulo multiplication by nonzero scalars. We write the class of as . If is algebraically closed and are homogeneous polynomials over , their common zero set in is called a projective algebraic set. If is a projective algebraic set, a closed algebraic subset of is a subset of the form , where is a projective algebraic set. A subset of is called open if its complement is closed. The set is irreducible if it cannot be written as the union of two proper closed algebraic subsets.
A projective curve over is an irreducible projective algebraic set of dimension , meaning that it is not a point and that all of its proper irreducible closed algebraic subsets are points.
Let be a projective curve, and let . We say that is nonsingular at if, in some affine chart containing , there is an open neighborhood of and polynomials on such that is the common zero set of on , and the Jacobian matrix has rank . The curve is called smooth if it is nonsingular at every point.
Let and be projective algebraic sets. A map is regular if for every point there is an open subset containing and homogeneous polynomials of the same degree such that for every point , the values , , are not all zero, and A bijection is an isomorphism if both and are regular.
A smooth projective curve is said to be defined over if it can be cut out by homogeneous equations with coefficients in . Two smooth projective curves defined over are geometrically isomorphic if they are isomorphic over .
Now let be a smooth projective curve defined over . For each integer , let denote the field with elements, and let be the number of points of whose homogeneous coordinates may be chosen in . The zeta function of is the formal power series Weil proved that there is a unique polynomial such that The degree of is even, and the genus of is defined by
For a fixed integer , let denote the number of geometric isomorphism classes of smooth projective curves defined over with . It is known that These examples suggest a natural question: for which genera is always given by a polynomial in ? We say that is polynomial if there exists a polynomial such that for every prime power .
In 2024, Canning, Larson, Payne, and Willwacher posted the preprint (Canning et al. 2024), proving the following theorem.
Thus is precisely the largest genus for which the number of smooth projective curves over , up to geometric isomorphism, is given by a single polynomial in for all finite fields.