# Picard group of a smooth projective curve

I have two (related) questions regarding the Picard group:

1) Are there examples of smooth projective curves with large Picard groups (say $Pic(X)\simeq\mathbb{Z}^n)$ for any $n$)?

2) In general, can we say what the Picard group is for a smooth projective curve of genus $g$? I was trying to compute it with the exponential sequence, but cannot quite make it work since I don't know the map $H^1(X,\mathcal{O}^*)\to H^2(X,\mathbb{Z})$.

• The Picard group of a smooth projective curve of genus $g \ge 1$ is $\mathbb{Z}$ times its Jacobian (en.wikipedia.org/wiki/Jacobian_variety), which is in particular uncountable. – Qiaochu Yuan Jun 5 '14 at 19:17
• What field are you working over? – Bruno Joyal Jun 5 '14 at 19:18
• The exponential sequence runs $1 \to H^1(X, \mathbb{Z}) \to H^1(X, \mathcal{O}_X) \to H^1(X, \mathcal{O}_X^{\times}) \to H^2(X, \mathbb{Z}) \to 1$. The last map to $H^2$ is the degree, or equivalently the first Chern class, so its kernel is the degree-$0$ line bundles and the rest of the exact sequence tells you that this can be identified with $H^1(X, \mathcal{O}_X) / H^1(X, \mathbb{Z})$. This is the Jacobian, and in particular is a complex torus $\mathbb{C}^g / \Gamma$ of complex dimension $g$, once we use Serre duality to identify $H^1(X, \mathcal{O}_X)$ with $H^0(X, \Omega^1_X)$. – Qiaochu Yuan Jun 5 '14 at 19:26
• @Qiaochu I think you are missing a dual in your statement of Serre duality – Bruno Joyal Jun 5 '14 at 19:33
• Oops. Yes, that should be $H^0(X, \Omega_1^X)^{\ast}$. Thanks. – Qiaochu Yuan Jun 5 '14 at 19:35

Over an algebraically closed field, see Qiaochu's remark. Over a number field, it is always a finitely generated abelian group. If $g=1$ and the base field is a given number field, then it is believed there exist elliptic curves of arbitrarily large rank, but this is not known (I believe the largest rank known over $\mathbf Q$ is 28, due to Elkies).
• Minor nitpick: Certainly, for all $r$, there is a number field $K$ and an elliptic curve $E$ over $K$ of rank $r$. Of course, I know you are referring to the "belief" that there are elliptic curves over $\mathbb Q$ with arbitrarily large rank (although I have the feeling that not everyone seems to agree with this "belief", right?). – Ariyan Javanpeykar Jun 7 '14 at 12:27