Picard groups of curves over non-algebraically closed fieldsPosted on June 5, 2019
If is a curve defined over a field we generally define the divisor group of , denoted , as the free abelian group generated by points in . We then define the principal divisors as the subgroup of consisting of all divisors of the form with . We then define the Picard group .
Given a divisor and an automorphism we define We say that a divisor is defined over , denoted , if for all We can then similarly define to be the set of classes of that are invariant under every element of
From the above definitions we have the following exact sequence We would hope to have another exact sequence for the corresponding objects defined over . In fact we will show that we always have an exact squence It is immediate that the above sequence is exact at and , but it is surprisingly non-elementary to observe that the sequence is exact at . In order to show that the sequence is exact at we need to show that if for some , then for some . That is, we need to show that we have the following short exact sequence: where is the subset of Galois invariant divisors in . This is precisely the situation that is studied by Galois cohomology. We have -modules , and a short exact sequence and we wish to know if exactness holds once consider only the submodules invariant under . Galois cohomology, or more generally group cohomology, provides a long exact sequence Finally, the cohomological version of Hilbert’s Theorem 90 tells us that .
The inspiration for this problem comes from Silverman (1, Exercise 2.13).
- Joseph Silverman, The arithmetic of elliptic curves, 2nd ed., Springer Graduate Texts in Mathematics, 1992