# is there a p-adic Borel theorem?

1. Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. The class number formula says that the leading term of $\zeta_F$ at $s=0$ equals $$\zeta_F^\ast(0):=\lim_{s \to 0} \zeta_F(s)s^{-(r_1+r_2-1)}=-\frac{h}{w} R, \quad (1)$$ where $R$ is the covolume of the image of Dirichlet's regulator $\mathcal{O}_F^\ast \to \mathbb{R}^{r_1+r_2}.$ By the functional equation, this statement is equivalent to the computation of the residue of $\zeta_F$ at $s=1$.

2. Borel introduced higher regulators $$K_{2n-1}(\mathcal{O}_F) \to \mathbb{R}^{d_n}$$ where $n \geq 1$ and $d_n=r_1+r_2$ if $n$ is odd and $r_2$ if $n$ is even. When $n=1$, $K_1(\mathcal{O}_F)\sim \mathcal{O}_F^\ast$ and one recovers the first regulator. Then he proved that the leading term of $\zeta_F(s)$ at $s=1-n$ is $$\zeta_F^\ast(1-n)=\alpha \pi^{d_n} R_n \quad (2)$$ where $\alpha \in \mathbb{Q}^\times$ and $R_n$ is the covolume of the image of $K_{2n-1}(\mathcal{O}_F)$ in $\mathbb{R}^{d_n}$.

3. Given a prime number $p$, one has at disposal a $p$-adic Dedekind zeta function $\zeta_{F, p}$ and an equivalent of Dirichlet's regulator, constructed by Leopoldt. It is known that the equivalent of (1) holds in this case (I think for arbitrary $F$ this is due to Colmez).

Question: Does there exist a $p$-adic version of (2)?

Yes, there are $p$-adic analogues of (2). The case where $F$ is abelian over $\mathbf{Q}$ is known: see the paper

Manfred Kolster and Thong Nguyen Quang Do, Syntomic regulators and special values of p-adic L-functions, Invent. math. 133, 417-447 (1998).

From the abstract: "In this paper p-adic analogs of the Lichtenbaum Conjectures are proven for abelian number fields $F$ and odd prime numbers $p$, which generalize Leopoldt's $p$-adic class number formula, and express special values of $p$-adic $L$-functions in terms of orders of $K$-groups and higher $p$-adic regulators."

There's also been a lot of more recent work on this. For example, see Besser, Buckingham, De Jeu and Roblot, On the p-adic Beilinson conjecture for number fields, Pure and Applied Math Quarterly 5 (2009), number 1, 375-434

There are some more things that could be said concerning $p$-adic analogues of Borel's results. First, there is an interesting paper on $p$-adic Borel regulators by Huber and Kings:

• A. Huber and G. Kings. A p-adic analogue of the Borel regulator and the Bloch-Kato exponential map. J. Inst. Math. Jussieu, 10, 2011, no. 1, 149-190.

The result of the paper is a construction of a Borel regulator $K_{2n-1}(R)\to K$ for $K/\mathbb{Q}_p$ a $p$-adic field and $R$ its valuation ring. The construction is actually pretty close to Borel's construction in the number field case. In the paper, you can also find a comparison to Soulé's regulator $K_{2n-1}(R)\to H^1_{et}(K,\mathbb{Q}_p(n))$ via the Bloch-Kato exponential map.

The $p$-adic Borel regulator of Huber and Kings has been compared to Karoubi's regulator (which is constructed via cyclic homology) by Tamme, in the following paper:

• G. Tamme Comparison of Karoubi's regulator and the p-adic Borel regulator, Journal of K-Theory 9 (2012), no. 3, 579-600.

Concerning the comparison between regulators and leading terms of zeta-functions, I do not know much about this beyond relations to the Leopoldt conjecture as in the question. Probably a full $p$-adic analogue of (2) is not known at the moment, but I do not know about the state of the art.