Picard groups of curves over non-algebraically closed fields

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).

References

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