Here’s something enticing and strange: there are two “cut and paste” invariants of the same group which are equal to dual zeta values!

• the euler characteristic \( \chi(SL_2(\mathbb{Z})) = \zeta(-1), \) and
• the tamagawa measure of \( \mu(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}) = \zeta(2)\).

I repeat, these apriori unrelated invariants of our beloved example case \( SL_2(\mathbb{Z}), \) are dual zeta values! Is it a general pattern? Let’s define and explain these results to spiritually prepare ourselves for the general picture!

Euler Characteristics of Arithmetic Groups – star of the show: \( \chi(SL_2(\mathbb{Z})) = \zeta(-1) \)

An arithmetic group is the integer points of an algebraic group (i.e. a group object in abelian varieties). For example, \( SL_2(\mathbb{Z}) \) is the arithmetic group associated to the algebraic group \( \mathbb{SL}_2 \). In general, \( \chi(G) \) for arithmetic groups is related to L-functions and zeta functions.

The euler characteristic is defined for a torsion free group \( G’ \) as follows: \(\chi(G') = \sum_i (-1)^i H^i(G', \mathbb{Z}).\) This is supposed to generalize the notion of euler characteristic for surfaces of polyhedra where \( \chi(P) = V - F + E \), the analogy is to consider \( \dim H^0(P, \mathbb{Z}) = V \), \( \dim H^1(P, \mathbb{Z}) = E \), and \( \dim H^2(P, \mathbb{Z}) = F\).

For example:

• if we consider \( F_n \) the free group of rank \( n \geq 0 \) then \( BF_n \) has one vertex and n \( 1 \)-cells, therefore, \( \chi(F_n) = 1-n \)
• free abelian group of rank \( n \geq 1 \) then \(n\)-torus \( (S^1)^n \) has euler characteristic \( \chi(S^1), …, \chi(S^1)=0.\)

The euler characteristic is defined for any group \( G \), in terms of \( G’ \) a torsion free group, \(\chi(G) = [G: G']^{-1}\chi(G')\)

Note: This doesn’t depend on choice of \( G \), consider a two different torsion free subgroups \( \Gamma’ \) and \( \Gamma’’\), and define \( \Gamma_0 := \Gamma’ \cap \Gamma’’ \) , then

\[\frac{\chi(\Gamma')}{[\Gamma: \Gamma']} = \frac{\chi(\Gamma_0)/[\Gamma': \Gamma]}{[\Gamma: \Gamma']} = \frac{\chi(\Gamma_0)}{[\Gamma : \Gamma_0]} = \frac{\chi(\Gamma'')}{[\Gamma:\Gamma'']}\]

It’s time to stare at the Gauss Bonnet Thm: Let \( Y \) be closed Riemannian manfiold, then there’s a canonical measure \( \mu \) in \( Y \) such that \(\chi(Y) = \mu(Y).\)

There’s a version for orbifolds, \( Y = BG\) and let \( X \) be its universal cover, there’s a canonical measure such that

\[\chi(Y)= \mu(Y)\]

Further, \( \chi(BG) = \chi(G). \)

Let’s look at \( SL_2(\mathbb{Z}) \). The free group \( F_2 \) is a torsion free group living inside of \( SL_2(\mathbb{Z}) \) via the ping pong lemma. It has index 12, not sure why tbh.

Thus, \(\chi(SL_2(\mathbb{Z})) = [F_2: SL_2(\mathbb{Z})]^{-1} \chi(F_2) = \frac{1}{12} (1-2) = -\frac{1}{12} = \zeta(-1).\)

The Tamagawa Number of Arithmetic Groups – star of the show: \( \mu_\infty(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{Q})) = \zeta(2) \)

Given an arithmetic group \( G \), it comes equipped with a canonical measure on its adelic points \( G(\mathbb{A}) \). All other measures are a constant times this measure. It is defined by considering the measure associated to a differential form.

Given an algebraic group \( G \), pick any \( \omega \) which is a top degree invariant differential form of \( G \) defined over \( \mathbb{Q}. \) Then, define the measure \( \mu \) place-wise (a la the adelic perspective of Tate) for each \( G (K_v) \) by \(\mu_v := |\omega|_v \) on \((\mathbb{Q}, +)\). We then renormalize such that:

• for finite places \(v \), \( \mu_v(\mathbb{Z}_v)=1 \),
• and at infinite places \(\mu_\infty([0,1]) = 1.\)

You might be like – we chose \( \omega \) there, seems noncanonical to me. Well it is noncanonical, if we consider it individually over each place, but all together, no matter which choice of \( \omega \) we take, we will get the same answer because for any number \( f \), the product of its valuation over global fields will always be one: \( \prod |f|_v = 1. \)

Non-archimedian case (family friendly) \( \prod_p \mu_p(SL_2(\mathbb{Z}_p)) = \zeta(2)^{-1} \)

In this section we wish to show that \(\prod_p \mu_p(SL_2(\mathbb{Z}\_p)) = \prod_{p} (1-p^{-2})^{-1} = \zeta(2)^{-1}.\)

We may compute the Tamagawa measure for \( SL_2(\mathbb{R}) \) using Iwasawa decomposition. We get that the Haar measure on \( SL_2(\mathbb{Q}_p) \) is \( |x^{-1}|_pdxdydz, \) where \( |x^{-1}|_p = p^{-v_p(x)} \). A nice reference on this is the notes of Paul Garrett.

Now consider the reduction map:

\[f: SL_2(\mathbb{Z}) \to SL_2(\mathbb{F}_p)\]

\(\mu\_p(SL_2(\mathbb{Z}\_p))= \# SL_2(\mathbb{F}\_p)\mu\_p(\text{ker } f)\) and \(\mu_p(\text{ker } f) = \int_{x,y,z} \|x^{-1}\|_p dxdydz = p^{-3}.\)

Lemma: \( \# GL_2(\mathbb{F}_p) = (p^2-1)(p^2-p), \),

Proof: This because when forming a matrix in \( GL_2 \), the first column \( r_1 \) of the matrix can be anything but the 0 vector (giving us \( p^2 - 1\) choices), and second column of the matrix can be anything other than a multiple of the first vector (it can’t be \( r_2 = cr_1 \) ).

Lemma: \( \# SL_2(\mathbb{F}_p) = 1-p^2, \)

Thus, we have proven: \( \mu(SL_2(\mathbb{Z}_p)) = \frac{1}{1-p^{-2}}, \) and we may conclude:

\[\prod_{p} \mu_p(SL_2(\mathbb{Z}\_p)) = \prod\_{p} (1-p^{-2})^{-1} = \zeta(2)^{-1}.\]

Let’s get Archimedian \( \mu_\infty(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{Q})) = \zeta(2) \) (to be continued…)