To give another example of a
class number formula,
we introduce the
Ramanujan
*modular form*

Δ(We write &tau(q) :=q· &prod_{n=1,…,&infin}(1 -q^{n})^{24}.

whereL(ad Δ,s) := ∏_{p}(1 - &tau'(p)p^{-s}+ &tau'(p)p^{-2s}-p^{-3s })^{-1}

&tau'(The class number formula (proved in this form by Goro Shimura and Haruzo Hida) in this case states thatp) :=^{&tau(p)}&frasl_{ p11}- 1.

HereL(ad &Delta,1) =^{223}&frasl_{11!}· &pi^{13}· &int_{F}|&Delta(e^{2πi(x+iy)})|^{2}y^{10}dx dy.

F= {z∈C; Im(z) > 0 and |z| ≥ 1 and |Re(z)| ≤ ½ }.

The regulator in this case is the product of π^{13} and the
integral, while the nuisance factor is
^{223}&frasl_{11!}.
Thus the class number for the adjoint of
Δ should simply equal one.

The last step is to relate this class number to an interesting
arithmetic object. Conjectures of
Spencer Bloch and
Kazuya Kato
predict in this case that the class number should equal the size of a certain
group called the
*adjoint Selmer group* of &Delta. This fact (in
this and in more general situations) has been the subject of a great deal
of recent study, beginning with the pioneering work of
Matthias Flach and
Richard Taylor and
Andrew Wiles.
Unfortunately, we have not yet been able to compute this
Selmer group precisely. At this point we do
know that it is
finite (see *Algebraic
cycles, modular forms and Euler systems* and
*Geometric Euler systems for
locally isotropic motives*) and, using work of
Fred Diamond, Flach and Li Guo,
that its order is not divisible by any
prime *p* > 11 except possibly
691.