Many problems in modern number theory revolve around the following theme: relate arithmetic objects to special values of $L$-functions.

$\left\{ \text{arithmetic objects} \right\} \leftrightarrow \{ \text{complex $L$-functions} \}$

For example, on the left side, we might want to study the class group of a number field and on the right side, we might want to study values of Dirichlet $L$-functions. The bridge linking them is the class number formula.

As an another example, on the left side, we might want to study the rank of an elliptic curve and on the right side, we might want to study the Hasse Weil $L$-function of that elliptic curve. The bridge linking them is the BSD formula.

The problem with proving these theorems is that the left side is arithmetic, while the right side is analytic. So they are very "far" and it is hard to relate them. A crucial tool in relating them is to use an intermediary object called a "$p$-adic $L$-function".

$\left\{ \text{arithmetic objects} \right\} \leftrightarrow \{ \text{p-adic L-functions} \} \leftrightarrow \{ \text{complex L-functions}. \}$

This $p$-adic $L$-function (whatever it is) should straddle the worlds of arithmetic and analysis. On one hand, it should know "analytic" information from the complex function. On the other hand, it should also be "algebraic" in nature, which would make it easier to relate to arithmetic objects than the purely complex $L$-functions.

This approach has been immensely successful in proving, for example, big results in the direction of the Birch and Swinnerton Dyer Conjecture. The breakthrough results of Kato and Skinner-Urban about BSD rely critically on the notion of $p$-adic $L$-functions. And it's almost certain that $p$-adic $L$-functions will play a huge role in proving "arithmetic-analytic" theorems in the future.

In this post, I'll explain the simplest example of a $p$-adic $L$-function, the Kubota-Leopoldt $p$-adic $L$-function, to illustrate the basic ideas behind it. I'll also give a hint about its relation to arithmetic, which is the so-called Iwasawa main conjecture.

The Kubota-Leopoldt $p$-adic $L$-function (whatever it is) "$p$-adically interpolates" special values of the Riemann zeta function. Consider the Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}}$. The fundamental observation, first observed by Kummer, is that the values of $\zeta(s)$ at negative integers satisfy interesting congruences.

In these two examples, we set $p=5$. Set $s_1 = -1$ and $s_2 = -21$. We have a congruence

$-1 \equiv -21 \mod 5.$

We can calculate that $\zeta_5(s_1) = \frac{1}{3}$ and $\zeta_5(s_2) = \frac{9260535240173320423}{69}$, and it turns out that

$\zeta_5(-1) \equiv \zeta_5(-21) \mod 5^2.$

As another example, put $s_1 = -1$ and $s_2 = -101$. Then $-1 \equiv -101 \mod 5^2$. And we can calculate that

$\zeta(-1) \equiv \zeta(-101) \mod 5^3.$

These can be phrased as a rigorous theorem.

**Theorem: (Kummer)** Let $p$ be a prime and let $\zeta_p(s) = (1-p^{-s}) \, \zeta(s)$ be the Reimann zeta function with the Euler factor at $p$ removed. Define the set

$S = \{ s \in \mathbb{Z}_{<0}: s \equiv -1 \mod (p-1) \}.$

Then for any $s_1, s_2 \in S$, we have

$s_1 \equiv s_2 \mod p^n \implies \zeta_p(s_1) \equiv \zeta_p(s_2) \mod p^{n+1}.$

We can phrase this in a more suggestive way: *the function $\zeta_p(s)$ is $p$-adically continuous on the set $S \subset \mathbb{Z}$.*

Here is a very simple observation: the set $S$ defined above is *dense* in $\mathbb{Z}_p$. So we have a continuous function

$\zeta_p: S \to \mathbb{Z}_p$

defined on a dense subset $S$ of $\mathbb{Z}_p$. So we can ask: can we extend $\zeta_p$ to a continuous function to *all* of $\mathbb{Z}_p$? That is, can we define a continuous function $\mathbb{Z}_p \to \mathbb{Z}_p$ whose restriction to $S$ agrees with $\zeta_p$?

It turns out that the answer is *yes*! And furthermore, not only does such an extension exist, it is also *unique*. This was proven by Kubota and Leopoldt.

**Theorem. (Kubota-Leopoldt)** There is a unique continuous function $\zeta_p: \mathbb{Z}_p \to \mathbb{Z}_p$ which extends the function $\zeta_p$ defined on $S$ defined earlier. That is, for any $s \in S$, we have $\zeta_p(s) = (1-p^{-s}) \zeta(s)$.

**Definition.** This function $\zeta_p: \mathbf{Z}_p \to \mathbf{Z}_p$ in the above theorem is called the *$p$-adic Reimann zeta function*, or the Kubota-Leopoldt $p$-adic zeta function.

What we have just done is a $p$-adic analogue of the process of analytic continuation for complex $L$-functions. In the complex story, we define an $L$-function on a right half of the complex plane, and we show that we can *extend* it via analytic continuation to the entire complex plane. We then show that this extension is unique.

In our case, we define a $p$-adically continuous function $\zeta_p$ on a set $S$, and then we "analytically continue" $\zeta_p$ to all of $\Z_p$. We then show that this extension is unique. We call this function the Kubota-Leopoldt $p$-adic $L$-function.

You might have noticed that when we wrote down Kummer's Theorem about congruences of the zeta function, we defined the set $S$ as

$S = \{ s \in \mathbb{Z}_{<0}: s \equiv -1 \mod (p-1) \}.$

But why $-1$ in particular in the above definition? There is nothing special about $-1$ here; we could define for any $i=-1, -3, -5, \dots$ the set

$S_i = \{ s \in \mathbb{Z}_{<0}: s \equiv i \mod (p-1) \}.$

Note that what we originally called $S$ is just $S_i$ in the case where $i=-1$. For the sets $S_i$, we have an analogous form of Kummer's theorem.

**Theorem. (Kummer)** Let $p$ be a prime and let $\zeta_p(s) = (1-p^{-s}) \, \zeta(s)$ be the Reimann zeta function with the Euler factor at $p$ removed. Then for any $i = -1, -3, -5, \dots$, the function $\zeta_p: S_i \to \mathbf{Z}_p$ is $p$-adically continuous on $S_i$.

And just like before, $S_i$ is dense subset of $\mathbf{Z}_p$, and we can extend $\zeta_p$ to a continuous function on all of $\mathbf{Z}_p$.

**Theorem. (Kubota-Leopoldt)** There is a unique continuous function $\zeta_p^{(i)}: \mathbb{Z}_p \to \mathbb{Z}_p$ which extends the function $\zeta_p$ defined on $S_i$ defined earlier.

This function $\zeta_p^{(i)}$ is called the *$i$-th branch* of the Kubota-Leopoldt zeta function. The zeta function $\zeta_p$ we defined in the previous section was the branch corresponding to $i=-1$.

So who cares about $p$-adic zeta functions? A priori, they are just numerical curiosities. It turns out, however, that $p$-adic zeta functions encode very deep arithmetic information: they "know" about the class groups of cyclotomic fields. This is the content of a very deep theorem called the Iwasawa Main Conjecture.

$\{ p\text{-adic zeta function} \} \xleftrightarrow{\text{Iwasawa Main Conjecture}} \, \left\{ \text{class groups of cyclotomic fields} \right\}$

The left side is *analytic*; it is constructed by $p$-adically interpolating special values of the Riemann zeta function. The right side is *algebraic*; it is group that measures the failure of unique factorization in a number ring. The Iwasawa main conjecture (whatever it is) gives a profound connection between the analytic and algebraic sides of number theory.

It turns out that $\zeta_p^{(i)}$, the i-th branch of the Kubota-Leopoldt $p$-adic zeta function, encodes information about the $\chi^{i}$-th eigenspaces of the class groups of cyclotomic fields, where $\chi$ is the cyclotomic character. (This is why we need all the branches of the $p$-adic $L$-function in the first place.)

I won't say more now, because this connection deserves its own post. So in a future post, I'll properly motivate and state the main conjecture.