%plain TeX \normalbaselineskip=1.6\normalbaselineskip\normalbaselines \magnification=1200 \def\max{\mathop{\rm max}} \def\min{\mathop{\rm min}} \def\rk{\mathop{\rm rk}} \def\uple#1{(#1_1,\ldots,#1_n)} \def\puple#1{(#1_1:\ldots:#1_n)} \def\td{^t\!} \def\pmb#1{\setbox0=\hbox{#1}% \kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0 \kern-.025em\raise.0433em\box0 } \def\w{\pmb{$\omega$}} \def\Gm{{\bf G}_m} \def\ra{\rightarrow} \def\Z{{\bf Z}} \def\topn{\buildrel{p^n}\over\to} \def\Q{{\bf Q}} \def\C{{\bf C}} \def\F{{\bf F}} \def\zmp{\Z/p\Z} \def\O{{\cal O}} %\def\O#1{{{\cal O}_{#1}}} \def\isom{\cong} \def\mod{\ \hbox{mod}\,} \def\Ext{\mathop{\hbox{Ext}}} \def\Hom{\mathop{\rm Hom}\nolimits} \def \O{{\cal O}} \def \H{{\cal H}} \def \X{{\cal X}} \def\Spec{\mathop{\rm Spec}\nolimits} \def \bs{\bigskip} \def\Gal{\mathop{\rm Gal}\nolimits} \def\End{\mathop{\rm End}\nolimits} \def\limproj{\mathop{\oalign{\hfil$\rm lim$\hfil\cr $\longleftarrow$\cr}}} \def \L{\Lambda} \def\bk{\bar{k}} \def \ra{\rightarrow} \def \op{\frac{1}{p}} \def \st{\stackrel} \def \da{\downarrow} \def \R{{\bf Rings}} \def \d{\delta} \def \s{\sigma} \def \a{\alpha} \def \b{\beta} \def \x{\chi} \def \t{\tau} \def \con{\equiv } \def \e{\epsilon} \def \bC{\bar{C}} \def \bX{\bar{X}} \def \k{\kappa} %\def \p{^\prime} %\def \pp{^{\prime\prime} \centerline{\bf Periods of abelian varieties in characteristic $p$} \medskip \centerline{\bf Jos\'e Felipe Voloch} \bs If $A$ is an abelian variety defined over the complex numbers $\C$, then there exists a lattice $\L \subset \C^g, g = \dim A$, such that $A(\C)= \C^g/\L$. This lattice is called the period lattice because functions on $A$ will be periodic functions on $\C^g$ with periods in $\L$. In this note we give an analogue in characteristic $p$ for the period lattice $\L$ and for the parametrization $A(\C)=\C^g/\L$. Notation: For an abelian group $H$ we define $\hat H = \limproj H/p^nH$. Then $\hat H$ is a $\Z_p$-module. Let $A$ be an ordinary abelian variety over a field $K$ of characteristic $p > 0$ and let $K_s$ be the separable closure of $K$ and $G = \Gal (K_s/K)$. Let $A^{(p^n)}$ be the image of $A$ under the $n$-th power of the Frobenius map $F^n$ and $V_n: A^{(p^n)} \to A$ the dual isogeny, the $n$-th order Verschiebung, which is separable since $A$ is ordinary. Then $\ker V_n$ is the $p^n$ torsion of $A^{(p^n)}$. We define the period lattice by $\L = \limproj \ker V_n$. This definition is not new, it corresponds to the Serre-Tate parameters (see e.g. [K]), however it usually only considered when the ground field is a local field. See [K] also for the relationship between the Serre-Tate parameters and moduli. The generalization of the analytic parametrization of an abelian variety is given by the following: \proclaim Theorem 1. $\L \to \widehat {K_s^*}\otimes\L^{\otimes(-1)} \to \widehat {A(K_s)} \to 0$ as $G$-modules. A few comments are in order. As a $\Z_p$-module, $\widehat {K_s^*}\otimes\L^{\otimes(-1)}$ is isomorphic to $\widehat {(K_s^*)^g}, g = \dim A$, but they are different as $G$-modules. Secondly, the analogy with the analytic parametrization is more evident after composing it with the exponential map. This construction also generalises the Tate parametrization of elliptic curves: If $K_v$ is a local field and $E/K_v$ has split multiplicative reduction then $\exists q \in K_v$ such that $0 \to q^{\Z} \to K_v^* \to E(K_v) \to 0$ ([S], Ch. V). It is easy to show that Theorem 1 follows from the Tate parametrization in this case (see [V1], lemma 2). This generalizes to Mumford's parametrization of abelian varieties with completely multiplicative reduction [Mu]. {\it Proof:} We we have the exact sequence of group schemes $$ 0 \to \ker F^n \to \ker [p^n] \to \ker V_n \to 0.$$ Taking flat cohomology yields $$\ker V_n(K_s) \to H^1(K_s, \ker F^n) \to H^1(K_s, \ker [p^n]) \to 0.$$ On the other hand, $H^1(K_s, \ker [p^n]) = A(K_s)/p^nA(K_s)$, which follows {from} the exact sequence $0 \to \ker [p^n] \to A \to A \to 0$ and also $$H^1(K_s, \ker F^n) = H^1(K_s, \mu_{p^n}) \otimes \ker V_n^{\otimes -1} = K_s^*/(K_s^*)^{p^n} \otimes \ker V_n^{\otimes -1}.$$ Putting these together and passing to the inverse limit yields the theorem. \proclaim Corollary. If $K$ is a global field, $E/K$ an elliptic curve and $v$ a place of $K$ where $E$ has bad reduction, then $q$ is transcendental over $K$ and so is any $u \in K_v^*$ which maps to a point of infinite order in $E(K)$. This corollary is proved in detail in [V]. The proof consists in comparing the Tate parametrization and the parametrization given by Theorem 1 and using a theorem of Igusa which guarantees that the action of $G$ on $\L$ is not through a finite quotient. The transcendence of $q$ is the characteristic $p$ analogue of the recent result of Barr\'e-Sirieix et al. [B]. It would be nice to generalize the corollary to higher dimensional abelian varieties. This would require understanding the action of $G$ on $\L$. The result follows, for example if $G$ acts via the full general linear group, which is the generic case by [FC], Prop. V.7.1. Another application of theorem 1 is to local duality. It is a classical result of Tate (up to the $p$-part in characteristic $p$, which is due to Milne) that, if $K$ is a local field with finite residue field, then $A(K)$ and $H^1(G,A(K_s))$ are Pontrjagin duals. There is a conjecture of Milne ([M], III, Conjecture 10.7) which generalizes the local duality to case of algebraically closed residue field. This conjecture is known in the case of good reduction (Bester, see [M]) and for elliptic curves with split multiplicative reduction (Shatz, see [M]). We extend these results a bit in the following proposition, and we believe its proof may be extended to give further results along these lines. \proclaim Proposition. Let $K$ be a local field whose residue field is the algebraic closure of a finite field of characteristic $p > 0$ and $A/K$ an abelian variety whose reduction is a semi-abelian variety with ordinary abelian quotient. Assume also that $A[p] \cap A(K_s) = 0$, then $T_p(H^1(G,A(K_s))$ is isomorphic to $H^1(G,\L)$. {\it Proof:} Consider the exact sequence of $G$-modules $$0 \to K_s^* \topn K_s^* \to K_s^*/(K_s^*)^{p^n} \to 0.$$ Under the hypotheses of the theorem, $H^1(G,K_s^*) = H^2(G,K_s^*) = 0$, so the Galois cohomology sequence of the above exact sequence yields $(\widehat {K_s^*})^G = \widehat {K^*}$ and $H^1(G,\widehat {K_s^*}) = 0$. Now consider the exact sequence of $G$-modules $$0 \to A(K_s) \topn A(K_s) \to A(K_s)/p^nA(K_s) \to 0.$$ Taking Galois cohomology yields $0 \to \widehat {A(K)} \to \widehat {A(K_s)}^G \to T_p(H^1(G,A)) \to 0$. By our hypothesis on the reduction type of $A$ we obtain that $V_n$ is \'etale on the special fibre as well as on $A$, thus $\L = \Z_p^g$ with the trivial action of $G$. It also follows that $V_n$ is an isomorphism on the formal group of $A$. Thus, given $P \in A(K)$ in the formal group, we can find $Q \in A^{(p^n)}(K), V_n(Q) = P$ and we can then map $Q$ to $H^1(K, \ker F^n)$ using the coboundary map of the flat cohomology sequence coming from the exact sequence $0 \to \ker F^n \to A \to A^{(p^n)} \to 0$. This gives us an inverse, in the formal group, to the map $(\widehat {K^*})^g = \limproj H^1(K, \ker F^n) \to \widehat {A(K)}$ which comes from theorem 1. It follows that $\widehat {A(K)} = (\widehat {K^*})^g/\L$. We are now ready to take Galois cohomology of the exact sequence of theorem 1. Note that under our present assumptions this sequence is exact on the left also. We get $$0 \to \L \to (\widehat{K^*})^g \to \widehat {A(K_s)}^G \to H^1(G,\L) \to 0.$$ Since $(\widehat{K^*})^g$ surjects onto $\widehat {A(K)}$, we obtain that $$T_p(H^1(G,A)) = \widehat {A(K_s)}^G/\widehat {A(K)} = H^1(G,\L).$$ We say that an ordinary abelian variety $A$ is sufficiently general if $A[p^{\infty}] \cap A(K_s)$ is finite. It follows from the proof of theorem 1, that $A$ is sufficiently general if and only if the map $\L \to \widehat {K_s^*}\otimes\L^{\otimes(-1)}$ is injective. In [V2] a sufficient condition for $A$ to be sufficiently general is given which justifies the name "sufficiently general". The following theorem studies the action of the endomorphisms of $A$ on $\L$ and produces a best possible result under the hypotheses, showing that $\L$ behaves like the period lattice in this case and also like the $\ell$-adic representation. \proclaim Theorem 2. The natural map $\End(A) \otimes \Z_p \to \End (\L)$ is injective if $A$ is sufficiently general. {\it Proof:} It suffices to show, by standard arguments, that if $\phi \in \End(A)$ acts trivially (via $\phi^{(p^n)}$) on $\ker V_n$ for $n$ large, then $\phi$ factors through $[p]: A \to A$. Let $\check A$ be the dual abelian variety and fix a polarization $\a: A \to \check A$, defined over $K$. We have a dual map $\check \phi: \check A \to \check A$ and $\check \phi$ kills the Cartier dual of $\ker V_n$ which is $\ker F^n$ on $\check A$. We can thus factor $\check \phi= \psi \circ F^n, \psi: {\check A}^{(p^n)} \to \check A$. We are done if $\check \phi$ kills $\ker [p]$. But, if that is not the case there exists a cyclic subgroup $H$ of ${\check A}^{(p^n)}$ of order $p^n$ on which $\psi$ is injective. This subgroup will, moreover, be defined over $K_s$. Thus, $\a(\psi(H))$ is a large subgroup of $A$ of $p$-power order defined over $K_s$, which will be a contradiction for $n$ sufficiently large. One may conjecture, transposing a similar conjecture of Tate, that $\End(A) \otimes \Z_p$ is isomorphic to $\End_G (\L)$, if $A$ is defined over a global field $K$ with absolute Galois group $G$ and $A$ is sufficiently general. This is trivial if $A$ is an elliptic curve, since both groups are isomorphic to $\Z_p$ under the hypotheses. The first non-trivial case is when $A$ is a product of two elliptic curves and in this case the conjecture is true, being essentially equivalent to Keating's characterization of the Igusa tower [Ke]. {\bf Acknowledgements:} The author would like to thank J. Tate for many suggestions. In particular, the proofs of theorem 1 and 2 that replace more complicated proofs which I had originally follow some suggestions of his. The author would also like to thank the NSF (grant DMS-9301157), the Alfred P. Sloan Foundation and the NSA (grant MDA904-97-1-0037) for financial support. \bigskip \centerline{\bf References.} \bigskip \noindent [B] K. Barr\'e-Sirieix et al. {\it Une preuve de la conjecture de Mahler-Manin}, Inventiones Math., {\bf 124} (1996) 1-9. \medskip \noindent [FC] G. Faltings, C.L. Chai, {\it Degenerations of Abelian Varieties}, Springer Verlag, New York, 1990. \medskip \noindent [K] N. M. Katz, {\it Serre-Tate local moduli}, Springer LNM 868 (1981) 138-202. \medskip \noindent [Ke] K. Keating, {\it An abstract characterization of the Igusa tower}, Amer. J. Math. {\bf 117} (1995) 4119-440. \medskip \noindent [S] J. H. Silverman, {\it Advanced topics in the arithmetic of elliptic curves}, GTM 151, Springer, New York, 1994. \medskip \noindent [M] J. S. Milne, {\it Arithmetic duality theorems}, Academic Press, Orlando, 1986. \medskip \noindent [Mu] D. Mumford, {\it An analytic construction of degenerating Abelian Varieties over complete rings}, Compositio Math. {\bf 24} (1972) 239-272. \medskip \noindent [V1] J. F. Voloch, {\it Transcendence of elliptic modular functions in characteristic $p$}, J. Number Theory, 58 (1996) 55-59. \medskip \noindent [V2] J. F. Voloch, {\it Diophantine Approximation on Abelian varieties in in characteristic $p$}, Amer. J. Math. {\bf 117} (1995), 1089-1095. \medskip \noindent Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA \smallskip \noindent e-mail: voloch@math.utexas.edu \end