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

Δ(q) := q · &prod n=1,…,&infin (1 - qn)24.
We write &tau(n) for the coefficient of qn in Δ(q). The adjoint L-function of &Delta is defined by
L(ad Δ,s) := p (1 - &tau'(p)p-s + &tau'(p)p-2s - p-3s )-1
where
&tau'(p) := &tau(p)&frasl p11 - 1.
The class number formula (proved in this form by Goro Shimura and Haruzo Hida) in this case states that
L(ad &Delta,1) = 223&frasl11! · &pi13 · &intF |&Delta(e2πi(x+iy))|2y10 dx dy.
Here F is the standard fundamental domain for the action of SL2(Z) on the complex upper half plane:
F = { zC ; Im(z) > 0 and |z| ≥ 1 and |Re(z)| ≤ ½ }.

The regulator in this case is the product of π13 and the integral, while the nuisance factor is 223&frasl11!. 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.

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