Riemann hypothesis via absolute geometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:08:54Z http://mathoverflow.net/feeds/question/69389 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69389/riemann-hypothesis-via-absolute-geometry Riemann hypothesis via absolute geometry Zoran Škoda 2011-07-03T10:00:54Z 2011-07-05T16:26:52Z <p>Several leading mathematicians (e.g. <a href="http://ncatlab.org/nlab/show/Yuri+Manin" rel="nofollow">Yuri Manin</a>) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute algebraic geometry over the field of one element; some like Mochizuki and <a href="http://ncatlab.org/nlab/show/Nikolai+Durov" rel="nofollow">Durov</a> are thinking of a possible application of $\mathbf{F}_1$-geometry to an even stronger <em>abc</em> conjecture. It seems that this is one of the driving forces for studying algebraic geometry over $\mathbf{F}_1$ and that the main obstacle to materializing this proof is that the geometry over $\mathbf{F}_1$ (cf. MO <a href="http://mathoverflow.net/questions/2300/what-is-the-field-with-one-element" rel="nofollow">what is the field with one element</a>, <a href="http://mathoverflow.net/questions/23394/applications-of-algebraic-geometry-over-a-field-with-one-element" rel="nofollow">applications of algebaric geometry over a field with one element</a>) is still <em>not</em> satisfactorily developed. Even a longer-term attacker of the Riemann hypothesis from outside the algebraic geometry community, <a href="http://ncatlab.org/nlab/show/Alain+Connes" rel="nofollow">Alain Connes</a>, has concentrated recently in his collaboration with Katia Consani on the development of a version of geometry over $\mathbf{F}_1$.</p> <p>Could somebody outline for us the ideas in the folklore sketch of the proof of the Riemann hypothesis via absolute geometry ? Is the proof analogous to the <a href="http://ncatlab.org/nlab/show/Pierre+Deligne" rel="nofollow">Deligne</a>'s proof (<a href="http://www.numdam.org/item?id=PMIHES_1974__43__273_0" rel="nofollow">article</a>) of the Riemann-Weil conjecture (see <a href="http://en.wikipedia.org/wiki/Weil_conjectures" rel="nofollow">wikipedia</a> and MathOverflow question <a href="http://mathoverflow.net/questions/215/equivalent-statements-of-riemann-hypothesis-in-the-weil-conjectures" rel="nofollow">equivalent-statements-of-riemann-hypothesis-in-the-weil-conjectures</a>) ? </p> <p><a href="http://ncatlab.org/nlab/show/Alexander+Grothendieck" rel="nofollow">Grothendieck</a> was not happy with Deligne's proof, since he expected that the proof would/should be based on substantial progress on <a href="http://ncatlab.org/nlab/show/motive" rel="nofollow">motives</a> and the standard conjectures on algebraic cycles. Is there any envisioned progress in the motivic picture based on $\mathbf{F}_1$-geometry, or even envisioned extensions of the motivic picture ? </p> http://mathoverflow.net/questions/69389/riemann-hypothesis-via-absolute-geometry/69394#69394 Answer by David Speyer for Riemann hypothesis via absolute geometry David Speyer 2011-07-03T13:07:45Z 2011-07-05T16:26:52Z <p>Warning: I am not an expert here but I'll give this a shot.</p> <p>In the analogy between number fields and function field, Riemann's zeta funnction is the $\zeta$ function for <code>$\mathrm{Spec} \ \mathbb{Z}$</code>. Note that <code>$\mathrm{Spec} \ \mathbb{Z}$</code> is one dimensional. So proving the Riemann hypothesis should be like proving the Weil conjectures for a curve, which was done by Weil. Deligne's achievement was to prove the Weil conjectures for higher dimensional varieties which, according to this analogy, should be less relevant.</p> <p>I <a href="http://sbseminar.wordpress.com/2010/04/19/the-weil-conjectures-curves/" rel="nofollow">wrote a blog post</a> about one of the standard ways to prove the Riemann hypothesis for a curve $X$ (over $\mathbb{F}_p$). Note that a central role is played by the surface $X \times X$. I believe the $\mathbb{F}_1$ approach is to invent some object which can be called <code>$(\mathrm{Spec} \ \mathbb{Z}) \times_{\mathbb{F}_1} (\mathrm{Spec} \ \mathbb{Z})$</code>.</p> http://mathoverflow.net/questions/69389/riemann-hypothesis-via-absolute-geometry/69446#69446 Answer by S. Carnahan for Riemann hypothesis via absolute geometry S. Carnahan 2011-07-04T04:42:40Z 2011-07-04T04:42:40Z <p>Last fall, there was <a href="http://www-math.mit.edu/~kedlaya/conference2010/" rel="nofollow">a conference in Nagoya</a> about precisely this question (oddly enough, funded by a "Riemann Hypothesis" DARPA grant). Since I was attending a different conference at the same university at the same time, I didn't get to see all of the talks. However, <a href="http://math.mit.edu/~kedlaya/papers/nagoya2010.pdf" rel="nofollow">Kedlaya's overview talk</a>, which is listed among others on the <a href="http://www-math.mit.edu/~kedlaya/conference2010/schedule.html" rel="nofollow">schedule page</a>, is rather informative.</p> <p>Essentially, one hopes to get the completed $L$-function of an $\mathbb{F}_1$-scheme $X$ by cohomological means, by choosing a holomorphic family of operators (analogous to $1-q^{-s}\text{Frob}_q$ in the function field setting), and taking the determinant of the action on the cohomology of $X$ (which is expected to be infinite dimensional). This is basically a generalization of the Grothendieck-Lefschetz trace formula to a cohomology theory that is not yet known. There is some algebraic evidence that some form of the de Rham-Witt complex with a suitable alteration at infinity is such a cohomology theory, but I don't know what the appropriate family of operators ought to be. I am told that there are promising hints coming from the world of dynamical systems and foliated spaces, and this is where non-commutative geometry seems to enter the picture.</p>