What is a pp-adic LL-function?

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

{arithmetic objects}{complex 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 LL-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 LL-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 "pp-adic LL-function".

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

This pp-adic LL-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 LL-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 pp-adic LL-functions. And it's almost certain that pp-adic LL-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 pp-adic LL-function, the Kubota-Leopoldt pp-adic LL-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.

Congruences between zeta values

The Kubota-Leopoldt pp-adic LL-function (whatever it is) "pp-adically interpolates" special values of the Riemann zeta function. Consider the Riemann zeta function ζ(s)=n=11ns\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}}. The fundamental observation, first observed by Kummer, is that the values of ζ(s)\zeta(s) at negative integers satisfy interesting congruences.

Examples

In these two examples, we set p=5p=5. Set s1=1s_1 = -1 and s2=21s_2 = -21. We have a congruence

121mod5.-1 \equiv -21 \mod 5.

We can calculate that ζ5(s1)=13\zeta_5(s_1) = \frac{1}{3} and ζ5(s2)=926053524017332042369\zeta_5(s_2) = \frac{9260535240173320423}{69}, and it turns out that

ζ5(1)ζ5(21)mod52.\zeta_5(-1) \equiv \zeta_5(-21) \mod 5^2.

As another example, put s1=1s_1 = -1 and s2=101s_2 = -101. Then 1101mod52-1 \equiv -101 \mod 5^2. And we can calculate that

ζ(1)ζ(101)mod53.\zeta(-1) \equiv \zeta(-101) \mod 5^3.

These can be phrased as a rigorous theorem.

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

S={sZ<0:s1mod(p1)}.S = \{ s \in \mathbb{Z}_{<0}: s \equiv -1 \mod (p-1) \}.

Then for any s1,s2Ss_1, s_2 \in S, we have

s1s2modpn    ζp(s1)ζp(s2)modpn+1.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 ζp(s)\zeta_p(s) is pp-adically continuous on the set SZS \subset \mathbb{Z}.

The connection to pp-adic LL-functions

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

ζp:SZp\zeta_p: S \to \mathbb{Z}_p

defined on a dense subset SS of Zp\mathbb{Z}_p. So we can ask: can we extend ζp\zeta_p to a continuous function to all of Zp\mathbb{Z}_p? That is, can we define a continuous function ZpZp\mathbb{Z}_p \to \mathbb{Z}_p whose restriction to SS agrees with ζp\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 ζp:ZpZp\zeta_p: \mathbb{Z}_p \to \mathbb{Z}_p which extends the function ζp\zeta_p defined on SS defined earlier. That is, for any sSs \in S, we have ζp(s)=(1ps)ζ(s)\zeta_p(s) = (1-p^{-s}) \zeta(s).

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

Analogy

What we have just done is a pp-adic analogue of the process of analytic continuation for complex LL-functions. In the complex story, we define an LL-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 pp-adically continuous function ζp\zeta_p on a set SS, and then we "analytically continue" ζp\zeta_p to all of Zp\Z_p. We then show that this extension is unique. We call this function the Kubota-Leopoldt pp-adic LL-function.

The General Definition of the Kubota-Leopoldt pp-adic zeta function

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

S={sZ<0:s1mod(p1)}.S = \{ s \in \mathbb{Z}_{<0}: s \equiv -1 \mod (p-1) \}.

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

Si={sZ<0:simod(p1)}.S_i = \{ s \in \mathbb{Z}_{<0}: s \equiv i \mod (p-1) \}.

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

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

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

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

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

Who cares?

So who cares about pp-adic zeta functions? A priori, they are just numerical curiosities. It turns out, however, that pp-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-adic zeta function}Iwasawa Main Conjecture{class groups of cyclotomic fields}\{ 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 pp-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 ζp(i)\zeta_p^{(i)}, the i-th branch of the Kubota-Leopoldt pp-adic zeta function, encodes information about the χi\chi^{i}-th eigenspaces of the class groups of cyclotomic fields, where χ\chi is the cyclotomic character. (This is why we need all the branches of the pp-adic LL-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.