# If $X$ is a genus $g\geq 2$ curve over a number field $K$, then is there a bound on $X_L(L)$ for $L/K$ that depends only on $X$ and $[L:K]$?

If $X$ is a genus $g\geq 2$ curve over a number field $K$, then $X(K)$ is finite by Falting's Theorem. My question is how does $X_L(L)$ behave for finite field extensions $L/K$? In particular, is there a bound on $X_L(L)$ for $L/K$ that depends only on $X$ and $[L:K]$?

Assuming the conjecture that varieties of general type cannot have a Zariski dense set of points over a number field (which is a consequence of the more precise Vojta conjectures, for example), Patricia Pacelli  proved the following, which is much stronger than what you're asking, since the constant only depends on the genus of $X$:

Theorem Fix $d\ge1$ and $g\ge2$. There is a constant $C=C(g,d)$ such that for all number fields $L/\mathbb Q$ of degree at most $d$ and all smooth projective curves $X/L$ of genus at most $g$, one has $\#X(L)\le C(g,d)$.

Pacelli's result is a generalization of an earlier result by Caparaso, Harris, and Mazur  that wasn't quite as uniform, but that really set in motion the use of this conjecture to prove uniformity result of this sort.

But if you want unconditional results, I don't think there's anything known in general, although one might(?) be able to get something like $$\#X(L) \le C(X,[L:K])^{1+\text{rank Jac}_X(L)}.$$

 Patricia L. Pacelli, MR 1448017 Uniform boundedness for rational points, Duke Math. J. 88 (1997), no. 1, 77--102.

 Lucia Caporaso, Joe Harris, and Barry Mazur, MR 1325796 Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), no. 1, 1--35.

• Thanks! I am indeed interested in unconditional results. Just to be clear, the rank of the L-points of the Jacobian of X is not known to be bounded by $[L:K]$? Is it flat out false? – Andrew NC Oct 19 '16 at 3:26
• Sorry, I meant bounded by a constant that depends only on $[L:K]$ and $X$. – Andrew NC Oct 19 '16 at 3:27
• @AndrewNC The rank is bounded by a constant depending on $X$ and $[L:K]$ and the size of the class group of $L$ (or even on, say, $p$ and the $p$-rank of the class group of $L$ for any fixed prime $p$), but that class rank is not known to be bounded by $[L:K]$. There are some current conjectures about uniform boundedness of ranks of abelian varieties, but they are still just conjectures, and somewhat controversial at that. – Joe Silverman Oct 19 '16 at 9:56