## Hodge-Tate proliferation

Let ${C/\mathbf{Q}_{p}}$ be a complete algebraically closed extension, and let ${A/C}$ be an abelian variety, with p-adic Tate module ${T_{p}A}$ and dual abelian variety ${A^{\ast}}$. There are then at least three natural candidates for a “Hodge-Tate map”

$\displaystyle \mathrm{HT}_{A}:T_{p}A\rightarrow\omega_{A^{\ast}}.$

To wit:

1. (Scholze) For any smooth proper rigid space ${X/C}$, we have the Hodge-Tate spectral sequence

$\displaystyle E_{2}^{i,j}=H^{i}(X,\Omega_{X/C}^{j})(-j)\Rightarrow H_{\mathrm{et}}^{i}(X,\mathbf{Q}_{p})\otimes C.$

Taking ${X=A}$, we get canonical identifications ${H^{1}(A,\mathcal{O}_{A})=\mathrm{Lie}A^{\ast}}$ and ${H_{\mathrm{et}}^{1}(A,\mathbf{Q}_{p})\otimes C=\mathrm{Hom}_{\mathbf{Z}_{p}}(T_{p}A,C)}$, so the edge map ${E_{2}^{1,0}\rightarrow H^{1}}$ gives a map ${\mathrm{Lie}A^{\ast}\rightarrow\mathrm{Hom}_{\mathbf{Z}_{p}}(T_{p}A,C)}$, and we may define ${\mathrm{HT}_{A}}$ as its ${C}$-dual.

2. (Tate, Fargues) Supposing ${G/\mathcal{O}_{C}}$ is a p-divisible group, there is a very direct definition of a Hodge-Tate map for ${G}$: writing ${G^{D}}$ for the Cartier dual, we have

$\displaystyle \begin{array}{rcl} T_{p}G & \cong & \mathrm{Hom}_{\mathrm{pdiv}/\mathcal{O}_{C}}(G^{D},\mu_{p^{\infty}})\\ x & \mapsto & \lambda_{x},\end{array}$

and we define

$\displaystyle \begin{array}{rcl} \mathrm{HT}_{G}:T_{p}G & \rightarrow & \omega_{G^{D}}\\ x & \mapsto & \lambda_{x}^{\ast}\frac{dT}{T}.\end{array}$

For ${A}$ with good reduction, we can apply this right away to ${G=A[p^{\infty}]}$ – but this really only works over ${\mathcal{O}_{C}}$, since ${\mu_{p^{n}}}$ doesn’t have any differentials over ${C}$! To push this through in general, let ${\mathcal{A}^{\ast}}$ be the formal semiabelian scheme over ${\mathcal{O}_{C}}$ obtained by completing the Neron model of ${A^{\ast}}$ along the identity component of its special fiber. The rigid generic fiber of ${\mathcal{A}^{\ast}}$ is naturally a rigid analytic subgroup of (the rigid space associated with) ${A^{\ast}}$, and in particular we get a canonical inclusion ${\mathcal{A}^{\ast}[p^{n}](\mathcal{O}_{C})\subset A^{\ast}[p^{n}](C)}$. This dualizes and compiles into a surjection ${\tau:T_{p}A\rightarrow T_{p}\mathcal{A}^{\ast D}}$. Now ${\mathcal{A}^{\ast}[p^{\infty}]}$ is an honest p-divisible group over ${\mathcal{O}_{C}}$, so the discussion above gives a map ${\mathrm{HT}':T_{p}\mathcal{A}^{\ast D}\rightarrow\omega_{\mathcal{A}^{\ast}}}$, and finally ${\omega_{\mathcal{A}^{\ast}}}$ is naturally an ${\mathcal{O}_{C}}$-lattice in ${\omega_{A^{\ast}}}$, and we may define ${\mathrm{HT}_{A}}$ as the composite map

$\displaystyle T_{p}A\overset{\mathrm{HT}'\circ\tau}{\rightarrow}\omega_{\mathcal{A}^{\ast}}\overset{\mathrm{incl}}{\rightarrow}\omega_{A^{\ast}}.$

3. (Coleman, Hodge-Tate periods and p-adic abelian integrals) Given an element ${x=(0,x_{1},\dots,x_{n},\dots)\in T_{p}A}$ with ${x_{n}\in A[p^{n}](C)}$, choose ${D_{n}}$ a divisor on ${A^{\ast}}$ corresponding to ${x_{n}\in A=\mathrm{Pic}^{0}(A^{\ast})}$. Since ${p^{n}x_{n}=0}$, we may choose rational functions ${f_{n}}$ on ${A^{\ast}}$ such that ${(f_{n})=p^{n}D_{n}}$. The limit

$\displaystyle \lim_{n\rightarrow\infty}\mathrm{dlog}f_{n}$

exists and defines an element of ${H^{0}(A^{\ast},\Omega_{A^{\ast}/C}^{1})\cong\omega_{A^{\ast}}}$ depending only on ${x}$ (the nonholomorphic bits of the individual ${\mathrm{dlog}f_{n}}$‘s go to zero p-adically).

In practice, definitions 1 and 2 are both pretty useful. Definition 1 varies well in families, and e.g. plays a crucial role in Scholze’s torsion paper. Definition 2 captures more delicate integral behavior and simultaneously allows one to import information from the p-divisible groups universe; for example, the proof of Theorem D in this post uses definition 2 crucially. Fortunately, Scholze proves the equivalence of these two definitions in his CDM article (and it requires a pretty ridiculous diagram chase!). Coleman’s definition is a lot crazier, and I don’t think it’s been proven to agree with the others in general. To quote Coleman’s paper, “It is also possible to make sense of the limit…when ${A}$ has bad reduction; however, we have not established its connection with Hodge-Tate.” Anyone up for it?