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Consider the following simple situation: We work over the ring $R=\mathbb{Z}_p[[t]]$, over which we consider a rank $2$ free module $M$ with basis $(e,f)$. On $M$, we define a flat (topologically nilpotent) connection with log poles by the formula $$\nabla: f\mapsto e\otimes\frac{dt}{t}.$$ In other words, we have $t\frac{df}{dt}=e$.

Now, fix some finite extension $K/\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ and choose two non-units $q,q'\in\mathcal{O}_K$. Identifying these elements with continuous maps $q,q':R\to\mathcal{O}_K$, we get two specializations $M_q$ and $M_{q'}$ of $M$ over $K$. We can now try to perform ($p$-adic) parallel transport of sections from $M_{q'}$ to $M_q$ using the usual Taylor series: $$f(q')\mapsto\sum_{i=0}^\infty\frac{d^if}{dt^i}(q)\frac{(q'-q)^i}{i!}.$$

Here, by $f(q)$, I of course mean the pull-back of sections $q^*f$.

It is easily checked that the above series simplifies to $$f(q)+\left(-\sum_{i=1}^\infty\frac{(1-q'q^{-1})^i}{i}\right)e(q).$$ The series in parentheses is immediately seen to be the one for $\log(q'q^{-1})$. This, as is well known, will converge only if $q'q^{-1}$ is a principal unit (that is, it is integral and has residue class $1$).

But, of course, we can go ahead and fix some branch of the $p$-adic logarithm, say, the one such that $\log(p)=0$. And then we can simply interpolate this isomorphism, which is given above by parallel transport in the cases where $q'q^{-1}$ is a principal unit, to all pairs $(q,q')$ by the formula $$f(q')\mapsto f(q)+\log(q'q^{-1})e(q).$$

Question: What is in fact going on here? Is there an intrinsic description of this interpolation (without the choice of bases, etc.)? In particular, can there be some general result along the following lines: for any $R$-module $M$ with flat connection with log poles, and any choice of $p$-adic logarithm, there is an interpolation as above of parallel transport between specializations at $K$-valued points, with certain properties? I'm sure somewhere in Dwork, Katz, et al, there is something about this, but it seems to have evaded me so far.

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Dear Keerthi: The extension of log using $\log(p) = 0$ doesn't mesh well with the rigid-analytic viewpoint (as opposed to $p$-adic analytic manifolds as totally disconnected spaces). To have a good notion of analytic continuation and parallel transport, should use the rigid-analytic viewpoint (admissible covers, etc.) At first glance it seems that your proposed extension does not treat the open unit disc $D$ as the connected rigid-analytic space it "should" be, but rather as the disjoint union of certain "open residue discs". So not really "parallel transport". Do I misunderstand the intent? – BCnrd Oct 29 '10 at 17:26
I don't know much about p-adic differential equations but in your case I don't think it matters as what you are looking at is actually the canonical nilpotent connection over Gm/Z. I think what you're looking for is the tannakian equivalence between nilpotent connections and unipotent representations of the étale or rigid fundamental group as explained for example in this paper by Furusho – AFK Oct 29 '10 at 17:26
What does $\nabla$ do to the basis vector $e$? – S. Carnahan Oct 30 '10 at 6:28
Sorry, didn't realize I don't get notifications for comments, so never checked. Brian--I'm not saying that this is parallel transport. For my purposes, I only care about transport within a 'residue disc' (an open sub-space of points with the same reduction in the special fiber?). But even this cannot always be defined using the naive Taylor series. YBL--Thanks for the link. I found the precise statement I need in a paper of Vologodsky's: See Theorem B on page 5. Scott--Sorry for the omission. $e$ is parallel for $\nabla$. – Keerthi Madapusi Pera Nov 1 '10 at 2:28

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