Period and index for higher genus curves

Abstract

Period and indexLet K be a field; usually, we will consider it to be a number field. By a curve C over K, we shall mean a smooth projective geometrically integral curve. The period and index of C over K are two integer invariants which measure the failure of C to have rational points. Specifically, the index is the gcd of degrees of all effective divisors D ∈ Div C-that is, effective divisors which are rational over K. Equivalently, the index is the gcd of degreesσP is a rational effective divisor, where σ ranges over the embeddings of L into an algebraic closure of K; and conversely any minimal rational effective divisor is of this form. Also observe that if C already has K-points, then its index is 1.The period is less stringent: here we look at the smallest positive degree of rational divisor classes; these are given by divisors which are linearly equivalent to their Galois conjugates. To see that the two invariants need not be the same, consider any conic over R without rational points, say the curve with affine piece x 2 + y 2 = −1. Certainly the index is 2, but as our curve is genus 0, there is a single divisor class of degree 1; namely, the class of a point. Therefore this class must be rational over R, and so the period is 1.As a rational divisor automatically belongs to a rational divisor class, we have P | I. Furthermore, the canonical class gives a rational divisor of degree 2(g − 1), so I | 2(g − 1). Lichtenbaum in [Lic69] found further conditions on the possible values of the period and index: Theorem 1.1 (Theorem 8,[Lic69]). Let C/K be a curve over a field with genus g, period P , and index I. Then2 , and(ii) if either 2(g − 1)/I or P is even, then I | P 2 .We say a triple of integers (g, P, I) is admissible if they satisfy the divisibility conditions of Lichtenbaum's theorem; that is, if they are possible values for the genus, period, and index of a curve. The goal of this paper is to determine whether every admissible triple indeed occurs as the invariants of some curve over a number field. Our main result is the following: 1

Fields generated by points on superelliptic curves

“…The index of a curve C, denoted I(C), is the greatest common divisor of degrees [L : K], where L/K ranges over algebraic extensions such that C(L) = ∅. See [GLL13,Sha18] for more on the index of a curve. In terms of the index, we can ask whether the index of our superelliptic curve C/Q is related to gcd(m, d).…”