Curves , Jacobians , and Zeta Functions

When we speak about a function field of one variable over a field K, we mean a finitely generated regular extension F of K of transcendence degree 1. We briefly recall the definitions of the main objects attached to F/K and their properties. See the books [Che51] or [Sti93] for details. A more comprehensive survey can be found [FrJ08, Sections 3.1-3.2]. A K-place of F is a place φ: F → K̃ ∪ {∞} such that φ(a) = a for each a ∈ F . A prime divisor p of F/K is an equivalence class of K-places of F . Let φp be a place in that class, vp the corresponding discrete valuation of F/K, and F̄p the residue field. The latter field is a finite extension of K which is uniquely determined by p up to K-conjugation. We set deg(p) = [F̄p : K]. A divisor of F/K is formal sum a = ∑ kpp, where p ranges over all prime divisors of F/K, for each p the coefficient kp is an integer, and kp = 0 for all but finitely many p′s. The degree of a is deg(a) = ∑ kp deg(p). The divisor attached to an element f ∈ F× is defined to be div(f) = ∑ vp(f)p, where p ranges over all prime divisors of F/K. This makes sense, since vp(f) = 0 for all but finitely many p’s. Further, one attaches to f the divisor of zeros div0(f) = ∑ vp(f)>0 vp(f)p and the divisor of poles div∞(f) = − ∑ vp(f)<0 vp(f)p. If f / ∈ K, the degrees of each of these divisors is equal to [F : K(f)]. Hence, deg(div(f)) = deg(div0(f))−deg(div∞(f)) = 0. If a = ∑ kpp is a divisor of F/K, we write vp(a) = kp for each prime divisor p of F/K and note that vp(div(f)) = vp(f) for each f ∈ F×. Given two divisors a, b of F/K, we write a ≤ b if vp(a) ≤ vp(b) for each prime divisor p of F/K. Finally, one attaches to each divisor a a finitely generated vector space L(a) over K consisting of all f ∈ F with div(f) + a ≥ 0 and write dim(a) for dim(L(a)).