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If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles. For example, when $M=\mathbb{Q}$ is the trivial motive, $L(\mathbb{Q},s)$ is the Riemann Zeta function $\zeta(s)$.

There is a famous concjecture saying that all values of $L(M,s)$ at integers $n$ (which are not zero or poles, say; otherwise replace value by principal value) are periods -- that is to say have real and imaginary parts that can be expressed as (according to Wikipedia's definition) differences of volumes of region in of Euclideans pace given by polynomial inequalities with rational coefficients.

This conjecture is due (if I am not mistaken) to Deligne in the case where $n$ is a so-called critical value of $L(M,s)$ and to Beilinson in general. I won't recall the definition of critical here, but I can say that for $\zeta(s)$ the critical values are $n=2,4,6,8,\dots$ and $n=-1,-3,-5,\dots$. Of course, the conjecture of Deligne in this case was known to Euler who proved that $\zeta(2n)$ is a rational times the volume of the unit ball in $\mathbb{R}^{4n}$, and $\zeta(1-2n)$ is rational, for $n \geq 1$ (the later made completely rigourous by Riemann).

Now my question:

In which case (if any) where $n$ is not a critical value is it known that L(M,n) is a period?

Here is a second question, related to the first:

Do you know a good survey on the progresses oon Beilinson's conjecture?

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Nekovar has a 20-year-old survey (people.math.jussieu.fr/~nekovar/pu/mot.pdf), and there don't seem to be any that are much more recent (according to a quick search of mathscinet). Section 8 discusses what was known (proven for zeta functions of number fields and Dirichlet L-functions, some partial results for modular forms and elliptic curves, etc). There doesn't seem to have been any breakthroughs since then (searching "anywhere" with mathscinet for Beilinson Conjecture only pointed me to a couple of other partial results). –  B R Feb 3 '11 at 21:54

1 Answer 1

up vote 18 down vote accepted

In the case $M$ is the spectrum of a number field (so that $L(M,s)$ is the Dedekind zeta function associated to the number field), it is known thanks to Borel's theorem that all non-critical values $L(M,n)$ are indeed periods. EDIT : I should add that it is very easy to prove that $\zeta(n)$ is a period for every $n \geq 2$, by the following computation :

\begin{equation*} \zeta(n)=\sum_{k=1}^{\infty} \frac{1}{k^n} = \sum_{k=1}^{\infty} \int_0^1 \cdots \int_0^1 (x_1 \cdots x_n)^{k-1} dx_1 \cdots dx_n = \int_0^1 \cdots \int_0^1 \frac{dx_1 \cdots dx_n}{1-x_1 \cdots x_n}. \end{equation*} There is a pole at $(1,\ldots,1)$ in the last expression (one can further regularize), but since the integral is absolutely convergent, this is sufficient to prove that $\zeta(n)$ is a period.

In the case $M$ is the motive associated to a (classical) newform $f$ of weight $k \geq 2$, the non critical values are $L(f,m)$ with $m \geq k$. Beilinson's theorem states that each of these values is given, up to a standard factor, by the determinant of a regulator matrix (you can think about it as an analogue of the class number formula if you wish). In fact, in this case, the regulator matrix has size 1, so we just have a number. Unravelling the definition of Beilinson's regulator map, this implies that $L(f,m)$ is indeed a period (if one considers $L'(f,k-m)$ instead, maybe one has to invert a power of $\pi$). Note however that in general this expression as a period is far from explicit.

In more complicated cases like the symmetric powers of the motive associated to a modular form, then (to my knowledge) almost nothing is known, except in the case of CM elliptic curves, for which there is a general theorem by Deninger.

It is quite difficult to give an exhaustive list of all results in this area, and I may have forgotten to mention important results. In any case, it would be indeed nice to have such a list. So please don't hesitate to complete this answer.

For a good survey on Beilinson's conjectures, you may want to look at Nekovar's article "Beilinson conjectures". The link with periods is well-explained in Kontsevich-Zagier's article "Periods". Another reference I have in mind is Flach's article "The Equivariant Tamagawa Number Conjecture : A survey".

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