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I would like to learn more about the background of this talk, but found no text on that theme. Do you know more? Edit: An interesting talk by Miranda Cheng (slides).

Edit: A talk today on the theme, has anyone a text or slides?: http://www.mpim-bonn.mpg.de/de/node/4590

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I can say a little about the work of Brown and Schretz, since Brown gave a talk at BIRS last month.

If you take a graph with $N$ edges and some restriction on valence (called a Feynman graph), there is a certain integral on an $N-1$-dimensional domain with boundary that in small cases appeared to yield linear combinations of multiple zeta values. This was a discovery of Broadhurst and Kreimer, by numerical brute force.

This raised the question of which graphs yield combinations of multiple zeta values. This is interesting in part because when multiple zeta values show up in an integral, it is a sign that mixed Tate motives are lurking somewhere. In fact, Brown showed last year that $MT(\mathbb{Z}) = \pi_1^{mot}(\mathbb{P}^1 \setminus \{0,1,\infty \})$, i.e., every mixed Tate motive over $\mathbb{Z}$ has periods in $\mathbb{Q}[(2 i \pi)^{-1}, MZV]$. The connection between graphs and motives comes through work of Bloch, Esnault and Kreimer, where they defined motives of graphs $G$ by making a variety $X_G$ by some blow-up construction and taking $H^{N-1}$. You can ask if these motives are mixed Tate, but this is apparently way too hard. An easier question is computing cohomology, and easier still is counting points over finite fields.

Kontsevich conjectured that the function that takes a prime power $q$ and returns the number of $\mathbb{F}_q$-points in $X_G$ is a polynomial, and it was true for small $G$, but this recent work gives a construction of counterexamples. Some of the functions are (or appear to be) $q$-expansions of modular forms of small level and weight, but that seems to be a transition zone between polynomials and wilder functions that no one has been able to identify.

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  • $\begingroup$ Do you have a reference for the construction of $X_G$? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jun 20, 2011 at 19:49
  • $\begingroup$ I added a link under the authors' names. $\endgroup$
    – S. Carnahan ♦
    Commented Jun 20, 2011 at 20:33
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    $\begingroup$ Nice answer! If I'm not mistaken (according to a recent talk of Marcolli) the invalidity of Kontsevich's conjecture was first found by <a href="arxiv.org/abs/math/0012198">Belkale and Brosnan</a>, who found the corresponding motives could be "arbitrarily far from mixed Tate". $\endgroup$
    – David Ben-Zvi
    Commented Jun 21, 2011 at 0:40
  • $\begingroup$ Oh, I should have mentioned earlier work. Brown mentioned those results (and some others) during his talk, and said that his contribution to the knowledge of counterexamples was an explicit computationally tractable construction that used almost no technology. $\endgroup$
    – S. Carnahan ♦
    Commented Jun 21, 2011 at 4:05
  • $\begingroup$ "a transition zone ... that no one has been able to identify" - sounds like a case for junq.info $\endgroup$
    – Thomas Riepe
    Commented Jun 22, 2011 at 9:01
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You may have a look here : http://people.math.jussieu.fr/~brown/K3inphi4.pdf

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