# analogue of Theorem of Mattuck for Abelian varieties over $\mathbf{F}_q(\!(t)\!)$

By a theorem of Mattuck [Abelian Varieties over $$p$$-Adic Ground Fields, Annals of Mathematics, Second Series, Vol. 62, No. 1 (Jul., 1955), pp. 92-119], for an Abelian variety $$A$$ of dimension $$g$$ over a $$p$$-adic local field $$K$$ with ring of integers $$O_K$$, there is an exact sequence $$0 \to O_K^g \to A(K) \to \{\mathrm{finite}\} \to 0$$.

Is there an analogue of this theorem for Abelian varieties over $$\mathbf{F}_q(\!(t)\!)$$ (e.g. with $$O_K^g$$ replaced by a pro-$$p$$-group)? The only thing I found is a remark in [Milne, Arithmetic Duality Theorems, Remark I.3.6, bottom of p. 45]. For elliptic curves, there is Proposition 10.2.26 (Filtration of $$E(K)$$) in [Liu, Algebraic Geometry and Arithmetic Curves, 2nd ed.].

• The kernel is given, in some appropriate sense, by the formal group of $A$. This must be covered in many sources. Sep 5, 2018 at 13:28
• Following up on Jon Lubin's comment, Serre, Lie Algebras and Lie Groups, Part II, Ch IV covers formal groups. Sep 5, 2018 at 17:36
• @Lubin: Thanks (also Felipe Voloch). I proved it for $1$-dimensional formal groups and for $n$-dimensional commutative formal groups $\mathcal{F}$ of finite height over $\mathbf{F}_q[[t]]$, I need theorems on the power series $[p]_\mathcal{F}(X_1,\ldots,X_n)$. Can you recommend me some references, please?
– user19475
Sep 15, 2018 at 4:26
• Sorry, I’m no longer well-informed. In general, you handle higher-dimensional formal groups by looking at their Dieudonné module. Perhaps @FelipeVoloch can point you in the right direction. Sep 15, 2018 at 4:48
• I found [Lazard, Commutative formal groups], but it introduces a lot of notation and is therefore difficult to read.
– user19475
Sep 15, 2018 at 4:53

By [Serre, Lie Algebras and Lie Groups], p. 116, Theorem and p. 118, Corollary 2, there is an open subgroup of $$A(K)$$ which is a pro-$$p$$-group. Therefore, it suffices to show that the torsion subgroup of $$A(K)$$ is finite, because then there is an exact sequence (algebraically and topologically) $$0 \to P \to A(K) \to F \to 0$$ with $$P$$ a torsion-free pro-$$p$$-group and $$F$$ finite (shrink the pro-$$p$$-group such that it contains no torsion). By potential semistable reduction, one can assume $$A/K$$ semi-stable, and since it is clear for tori, $$A/K$$ of good reduction. Then use [Clark, Pete L. and Xarles, Xavier: Local bounds for torsion points on abelian varieties. In: Can. J. Math., 60(3) (2008), 532–555], Proposition 10 to show that the torsion subgroup of $$A(K)$$ is finite.
Note that in contrast to $$p$$-adic local fields, $$A(K)/p$$ is not finite, but uncountably infinite. (Use flat cohomology and the Kummer sequence for $$A/K$$ giving $$0 \to A(K)/p \to \mathrm{H}^1_\mathrm{fppf}(K,A[p]) \to \mathrm{H}^1(K,A)[p] \to 0$$ and [Milne, Arithmetic Duality Theorems], Theorem III.7.8.)
By [Serre, Galois cohomology], §1.1 exercise 1), $$P$$ is isomorphic to an infinite product of copies of $$\mathbf{Z}_p$$.