John Baez
L-functions

Preface

These are notes of ongoing work by John Baez and James Dolan. Here we would like to tackle the question “what sort of thing has an $L$-function?” The ideas here are very preliminary, and may build on the somewhat more developed section on zeta functions. One well-known answer is:

• A Galois representation, meaning perhaps a continuous finite-dimensional representation of $\mathrm{Gal}(\overline{\mathbb{Q}}\mid \mathbb{Q})$ with its usual topology as a profinite group.

Galois actions

Let $G=\mathrm{Gal}(\overline{\mathbb{Q}}\mid \mathbb{Q})$ regarded as a profinite group.

Claim: the category of continous transitive actions of $G$ is equivalent to the category of finite-dimensional extensions of $\mathbb{Q}$.

A commutative semisimple algebra over $\mathbb{Q}$ is a finite cartesian product of fields that are finite extensions of $\mathbb{Q}$.

Claim: the category of continuous actions of $G$ on finite sets is equivalent to the category of finite-dimensional commutative semisimple algebras over $\mathbb{Q}$. This category is a topos. Moreover, each object in this category gives a Dirichlet series, namely its zeta function.

Claim: the above map from finite-dimensional commutative semsimple algebras over $\mathbb{Q}$ to Dirichlet series factors through the topos $[\mathrm{finite}\mathrm{semsimple}\mathrm{commutative}\mathrm{rings}{]}^{\mathrm{op}}$.

(Here we are using some process of getting Dirichlet series from finite semsimple commutative rings.)

Conjecture: the topos $[\mathrm{finite}\mathrm{semsimple}\mathrm{commutative}\mathrm{rings}{]}^{\mathrm{op}}$ is equivalent to the topos of continuous finite $H$-sets where $H$ is some groupoid. What is $H$ like?

There is a topos of functors from the category of finite fields to $\mathrm{Set}$.

There is a topos of functors from the groupoid of finite fields to $\mathrm{Set}$.

a continuous transitive action of $G$ is an action of $G$ on a finite set, and this in turn must be (or come from??) the action of $G$ on some finite field. If we drop the ‘transitivity’ requirement and keep the finiteness, we get actions of $G$ on finite commutative semsimple algebras.

“structure types carried by algebraic completions of the rationals” versus “structure types carried by finite fields”….

Counting Points

Number of points in the Gaussian and Eisenstein elliptic curve over $F_p$ for various low primes:

p = 2 gives 3, 3

p = 3 gives 4, 4

p = 5 gives 8, 6

p = 7 gives 8, 4

p = 11 gives 12, 12

p = 13 gives 8, 12

p = 17 gives 16, 18

p = 19 gives 20, 28

p = 23 gives 24, 24

p = 29 gives 40, 30

p = 31 gives 32, 28

p = 37 gives 40, 48

p = 41 gives 32, 42

p = 43 gives 44, 52

p = 47 gives 48, 48

p = 53 gives 40, 54

Primes of the form $p=4n-1$ seem to give $p+1$ points on the Gaussian elliptic curve. Primes of the form $4n+1$ seem to be very erratic.

Primes of the form $p=6n-1$ seem to give $p+1$ points on the Eisenstein curve.

These are the primes that are inert for the Gaussian and Eisenstein integers!

Miscellaneous Junk

Other Stuff Types from Number Theory

It is well-known that starting from Riemann zeta function we can construct Dirichlet series of other multiplicative arithmetic functions. Many of the constructions live at the level of species. Here are some examples.

Starting from the Riemann species $Z$ we can form a new species $Z{\cdot }_{D}Z$. By definition, this species assigns to any finite set the collection of ways of writing that set as a product of two sets and then making each of those two sets into a product of finite fields.

Proposition: If $d(n)$ denotes the number of positive divisors of the natural number $n$, then

$$\overline{Z{\cdot }_{D}Z}(s)=\sum _{n>1}d(n)n^{-s}$$\overline{Z \cdot_D Z}(s) = \sum_{n \gt 1} d(n) n^{-s}

Proof: this follows from the well-known result

$$\zeta (s{)}^2=\sum _{n>1}d(n)n^{-s},$$\zeta(s)^2 = \sum_{n \gt 1} d(n) n^{-s} \, ,

which is also easy to see from scratch.

We can also construct species whose Dirichlet series are translated versions of the Riemann zeta function:

Definition: Let ${\sigma }_k(n)$ be the sum of the $k$th powers of the positive divisors of $n$.

Proposition:

$$\overline{Z{\cdot }_D{Z}_{{\mathbb{Z}}^k}}(s)=\sum _{n\ge 1}{\sigma }_k{n}^{-s}$$\overline{Z \cdot_D Z_{\mathbb{Z}^k}}(s) = \sum_{n \ge 1} \sigma_k n^{-s}

Other Junk

More generally, any species over the Riemann species has a zeta function, and any linear species over the Riemann species has an $L$-function.

More generally, we can repeat the above replacing ‘commutative ring’ by ‘commutative $k$-algebra’ for some commutative ring $k$, and we claim that we get the zeta function of $k$, at least when $k$ is a Dedekind domain or something.

Note something interesting: for each finite semsimple commutative ring $R$ and each prime $p$, we get a ‘Frobenius automorphism’ $F_p:R\to R$. When $R$ is a field of characteristic $p$, this is given by $x\mapsto x^p$. If it’s a field of characteristic $q\ne p$, it acts trivially. In general it’s a product of fields and we define the Frobenius factor by factor.

So, for each prime we get a natural automorphism of the identity functor

$$1:\mathrm{core}(FinComSepAlg)\to \mathrm{core}(FinComSepAlg)$$1 : core(\Fin\Com\Sep\Alg) \to core(\Fin\Com\Sep\Alg)

Is it true that for each prime we get a quandle structure on the set $S$ of finite commutative semsimple algebra structures on the $n$-element set? Each element of $S$ gives a Frobenius $F_p:n\to n$ which then acts on $S$.