Recently I've been rewatching some recordings of old talks on L-functions and explicit reciprocity laws (in particular, the series of talks by Loeffler and Zerbes given at this [workshop](https://www.crm.umontreal.ca/2020/Coleman20/horaire_e.html) at the CRM in 2020). By explicit reciprocity laws, we mean relating Euler systems or Galois cohomology to special values of L-functions (or p-adic L-functions), which are related to automorphic forms. One method of doing this mentioned in Loeffler's second talk makes use of the "Perrin-Riou logarithm", which I am unfamiliar with. Does anyone know introductory references for this topic? I am a little more familiar with the Bloch-Kato exponential from p-adic Hodge theory, which apparently this is related to.



What is the Perrin-Riou logarithm (or regulator)?
I am sure I've already written an expository account of this somewhere, but I looked over the lecture and seminar notes on my webpage and couldn't find it, so I'll write one here instead.


Suppose we start with a $p$-adic representation $V$ of $G\_{\mathbb{Q}\_p}$, and for simplicity I'll suppose $V$ is crystalline, and let $D = \mathbb{D}\_{\mathrm{cris}}(V)$; and I'm also going to suppose that no eigenvalue of $\varphi$ on $D$ is an integer power of $p$.


Then we have the Bloch--Kato exponential
$$exp\_V : D / Fil^0 D \xrightarrow{\ \cong\ } H^1\_f(\mathbb{Q}\_p, V),$$
and we also have the Bloch--Kato dual exponential
$$exp^\*\_{V^\*(1)} : H^1(\mathbb{Q}\_p, V) / H^1\_f \xrightarrow{\ \cong\ } Fil^0 D$$
defined as the dual of $exp\_{V^\*(1)}$ with respect to local Tate duality. (Here I'm using the assumption on $\varphi$-eigenvalues to avoid having to distinguish between $H^1\_{e}$, $H^1\_f$ and $H^1\_g$.)


Let me write $\log\_V$ for the *inverse* of $\exp\_V$, so both $\log$ and $\exp^\*$ go *from* some subquotient of $H^1$ and *to* some subquotient of $D$.


The idea of Perrin-Riou's map ("regulator" or "big logarithm" or "big dual exponential" depending on who's writing) is to package together the maps $\log$ and $\exp^\*$ **for all twists of $V$ simultaneously**, where the twists are by characters of $\Gamma = Gal(\mathbb{Q}\_p(\mu\_{p^\infty}) / \mathbb{Q}\_p) \cong \mathbb{Z}\_p^\times$. To make sense of this "packaging together", we note that:


* there is the *Iwasawa cohomology* group $H^1\_{\mathrm{Iw}}(\mathbb{Q}(\mu\_{p^\infty}), V)$, which is a finite-type module over $\Lambda\_{\mathbb{Q}\_p}(\Gamma)$ with the property that for every continuous character $\chi$ of $\Gamma$, there's a natural "specialisation" map
$$H^1\_{\mathrm{Iw}}(\mathbb{Q}(\mu\_{p^\infty}), V) \otimes\_{\Lambda, \chi} \mathbb{Q}\_p \xrightarrow{\ \cong\ } H^1(\mathbb{Q}\_p, V(\chi^{-1})).$$
So $H^1\_{\mathrm{Iw}}$ packages together the cohomology of all of the twists of $V$.
* For any locally algebraic (equivalently: de Rham) character $\chi$ of $\Gamma$, the space $\mathbb{D}\_{\mathrm{dR}}(V(\chi))$ is canonically isomorphic to $D$ [\*\*]. So you can think of $\Lambda\_{\mathbb{Q}\_p}(\Gamma) \otimes D$ as packaging up the $\mathbb{D}\_{\mathrm{dR}}(V(\chi))$ for all $\chi$ simultaneously.


So Perrin-Riou (almost [\*\*\*]) builds a map
$$\mathcal{L}\_V : H^1\_{\mathrm{Iw}}(\mathbb{Q}(\mu\_{p^\infty}), V) \longrightarrow \Lambda\_{\mathbb{Q}\_p}(\Gamma) \otimes D,$$
with the property that:


* if you specialise at a character $\chi: x \mapsto x^n \tau(x)$ with $\tau$ of finite order and $n \gg 0$, then the resulting map is (up to an explicit factor) the Bloch--Kato dual exponential for $V(\chi^{-1})$.
* if you specialise at $\chi : x \mapsto x^n \tau(x)$ with $\tau$ of finite order and $n \ll 0$, then the resulting map is (up to an explicit factor) the Bloch--Kato logarithm for $V(\chi^{-1})$.


This is an amazingly cool theorem: it is not so hard to build a map which has one of these two properties, but very hard to show that the *same* map interpolates both $\log$ and $\exp^\*$ -- that is (one possible statement of) Perrin-Riou's local reciprocity conjecture, which was an open problem for a good few years until it was resolved by Benois, Berger, and Colmez around 2000.


For further reading, I'd suggest my second paper with Lei and Zerbes, "Coleman maps and the p-adic regulator" (which should be a bit more digestible than our first paper, since we understood the material better by that stage).


**Footnotes**


[\*] In this answer, the cyclotomic character has Hodge--Tate weight $+1$ (the convention followed by most, but not all, of the literature).


[\*\*] That is, canonical once we have chosen a basis of $\mathbb{Z}\_p(1)$, i.e. a collection of $p^n$-th roots of unity $\zeta\_{p^n} \in \overline{\mathbb{Q}}\_p$ with $(\zeta\_{p^{n+1}})^p = \zeta\_{p^n}$.


[\*\*\*] "Almost" because it doesn't actually land in $\Lambda\_{\mathbb{Q}\_p}(\Gamma) \otimes D$ but in $\mathcal{H}\_{\mathbb{Q}\_p}(\Gamma) \otimes D$, where $\mathcal{H}\_{\mathbb{Q}\_p}(\Gamma)$ (the locally-analytic distribution algebra) is the completion of $\Lambda\_{\mathbb{Q}\_p}(\Gamma)$ in some appropriate topology. (Having spent a significant chunk of my career understanding this issue in detail, I can tell you to ignore it with a clean conscience.)


