Periods and L-values

Question

A famous theorem of Euler is that $\zeta(2n)$ is a rational number times $\pi^{2n}$. Work of Kummer, Herbrand, Ribet and others shows that the rational multiplier has number theoretic significance.

For more general L-functions attached to motives, the philosophy has emerged (Deligne, Beilinson, Bloch, Kato, etc.) that (in vague terms) their values at certain integers are algebraic multiples of transcendental numbers and that the particular algebraic number that's a multiple of the transcendental number contains information about the motive that the L-function is attached to.

But a (nonzero) algebraic multiple of a transcendental number is again transcendental, so an arbitrary real number does not have a well defined decomposition as a product of an algebraic number and a transcendental number.

Still, because of the theorems of Kummer, et. al. one suspects that powers of pi are (at least close to) the "right" "transcendental parts" of the L-function values to be looking at. Maybe one should really be looking a powers of $2\pi$? But it seems clear that one should not be looking at powers of $691\pi$ because otherwise the statement of Kummer's criterion for the regularity of a prime would have an exceptional clause involving the prime $691$.

Is there a conceptually motivated means of picking out the "right" "transcendental part" of a special value of an L-function?

Presumably the reason that Euler expressed his theorem in terms of $\pi$ is because $\pi$ was a commonly used symbol. (I've heard people argue that $2\pi$ is more conceptually primitive and that a label should have been made for the quantity $2\pi$ rather than for $\pi$ and am not sure what I think about this). In any case, there should be an a priori means to pin down the relevant transcendental number down "on the nose" (not up to a rational/algebraic multiple).

I have heard that Beilinson's conjecture give the transcendental number only up to a rational multiple and that the Bloch-Kato conjecture pins down the number. But I don't know enough to understand the statement of either conjecture and so am at present ill equipped to derive insight from reading the paper of Bloch and Kato. Are there more elementary considerations that give insight into how to pick out a particular transcendental number out of the set of all of its algebraic multiples?

Answer 1

The ingredient in the Beilinson and Bloch--Kato conjectures is a motive (over ${\mathbb Q}$, say). If we take the integral cohomology of this motive (mod torsion, say) we get an integral lattice. If we take some kind of Neron model, and take the algebraic de Rham cohomology of this, we get a second integral lattice. Now computing the determinant of the pairing of one of these on the other, we get a transcendental number, well defined up to a unit in ${\mathbb Z}^{\times}$, i.e. a sign.

This should give you an idea of how one can attach a canonical period to a motive, and is the basic idea underlying the construction of periods for motives. (Since one doesn't have Neron models in general, this idea is just heuristic as it stands, but I think it gives the right idea. If you apply it to ${\mathbb G}_m$, you should recover the period $2\pi i$.)

EDIT: I should point out that the above really is just a heuristic, explaining how there are two ways of getting integral structures in cohomology: in singular cohomology, one just takes integral cycles (i.e. "true" cycles on the motive, with no funny coefficients), and in de Rham cohomology, one takes algebraic differential forms that are defined over the integers, like the Neron differential $dx/2y$ on an elliptic curve with minimal Weierstrass equation $y^2 = f(x)$.

To actually get the correct periods for a given $L$-function, one has to do a little more manipulation than I indicated; e.g. for an elliptic curve over ${\mathbb Q}$, one will integrate the Neron differential over the a basis for the real integral cycles (i.e. the cycles that are fixed by the action of complex conjugation on $E({\mathbb C})$; these are rank one subgroup of the cohomoloy of $E({\mathbb C})$). But hopefully what I wrote above gives some intuition for what is going on.