Picard groups of curves over non-algebraically closed fields

Posted on June 5, 2019

If C is a curve defined over a field k we generally define the divisor group of C, denoted \text{Div}(C), as the free abelian group generated by points in C(\overline{k}). We then define the principal divisors as the subgroup \text{Prin}(C) of \text{Div}(C) consisting of all divisors of the form \text{div}(f)= \sum_{P\in C(\overline{k})}\text{ord}_P(f) with f\in\overline{k}(C)^*. We then define the Picard group \text{Pic}(C)=\text{Div}(C)/\text{Prin}(C).

Given a divisor D=\sum n_P(P)\in\text{Div}(C) and an automorphism \sigma\in\text{Gal}( \overline{k}/k) we define \sigma(D)=\sum n_P(P^{\sigma}). We say that a divisor D\in\text{Div}(C) is defined over k, denoted D\in \text{Div}_k(C), if \sigma(D)=D for all \sigma\in\text{Gal}(\overline{k}/k). We can then similarly define \text{Pic}_k(C) to be the set of classes of \text{Pic}(C) that are invariant under every element of \text{Gal}(\overline{k}/k).

From the above definitions we have the following exact sequence      0\to \overline{k}^*\to\overline{k}(C)^*\to\text{Div}^0(C)     \to\text{Pic}^0(C)^*\to 0. We would hope to have another exact sequence for the corresponding objects defined over k. In fact we will show that we always have an exact squence      0\to k^*\to k(C)^*\to\text{Div}_k^0(C)\to\text{Pic}^0_k(C). It is immediate that the above sequence is exact at k^* and k(C)^*, but it is surprisingly non-elementary to observe that the sequence is exact at \text{Div}_K^0. In order to show that the sequence is exact at \text{Div}_K^0 we need to show that if \text{div}(f)\in\text{Div}_k for some f\in\overline{k}(C)^*, then \text{div}(f)=\text{div}(g) for some g\in k(C)^*. That is, we need to show that we have the following short exact sequence:      0\to k^*\to k(C)^*\to\text{Prin}_k(C)\to 0 where \text{Prin}_k(C) is the subset of Galois invariant divisors in \text{Prin}(C). This is precisely the situation that is studied by Galois cohomology. We have \text{Gal}(\overline{k}/k)-modules k^*,k(C)^*,\text{Prin}(C), and a short exact sequence      0\to \overline{k}^*\to\overline{k}(C)^*\to\text{Prin}(C)\to 0, and we wish to know if exactness holds once consider only the submodules invariant under \text{Gal}(\overline{k}/k). Galois cohomology, or more generally group cohomology, provides a long exact sequence      0\to k^*\to k(C)^*\to\text{Prin}_k(C)\to H^1(\text{Gal}(\overline{k}/k),k^*). Finally, the cohomological version of Hilbert’s Theorem 90 tells us that H^1(\text{Gal}(\overline{k}/k),k^*)=0.

The inspiration for this problem comes from Silverman (1, Exercise 2.13).


  1. Joseph Silverman, The arithmetic of elliptic curves, 2nd ed., Springer Graduate Texts in Mathematics, 1992