What is the Perrin-Riou logarithm (or regulator)?

Question

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 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.

Answer 1

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.)

Answer 2

Q: Does anyone know introductory references for this topic?

An exposition of the Perrin-Riou regulator with several worked-out application examples is given in section 2 of Darmon and Lauder's contribution to Perrin-Riou's Festschrift.