Comments on: Distinguished Lecture Series I: Shou-wu Zhang, “Overview of rational points on curves” http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Thu, 07 Aug 2008 21:41:40 +0000 http://wordpress.org/?v=MU By: Jose A. Martinez http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-29874 Jose A. Martinez Sun, 18 May 2008 14:20:57 +0000 http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-29874 Hello all, I got for some material an effective nonlinear susceptibility in the form $latex \chi(E; E^{1/2}, \frac{\partial E}{\partial t^{loc}}, E^2, E^3,\ldots)$, so my question is how can I obtain a wave equation for such a medium. Some refereces? In advance, many thanks! J.A. M. Pd. I'm enrolled in a BS degree in photonics. Hello all,

I got for some material an effective nonlinear susceptibility in the form
\chi(E; E^{1/2}, \frac{\partial E}{\partial t^{loc}}, E^2, E^3,\ldots), so my question is how can I obtain a wave equation for such a medium. Some refereces?

In advance, many thanks!

J.A. M.
Pd. I’m enrolled in a BS degree in photonics.

]]> By: Anonymous http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-29867 Anonymous Sun, 18 May 2008 04:04:55 +0000 http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-29867 Very deep lecture. $latex \alpha+\beta=\gamma$ Very deep lecture.

\alpha+\beta=\gamma

]]> By: Dyadic models « What’s new http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-5158 Dyadic models « What’s new Fri, 27 Jul 2007 15:34:47 +0000 http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-5158 [...] factorisation of large numbers) are surprisingly easy to solve in the dyadic world (see e.g. Zhang’s talk for the dyadic version of FLT), but nobody knows how to convert the dyadic arguments to the [...] [...] factorisation of large numbers) are surprisingly easy to solve in the dyadic world (see e.g. Zhang’s talk for the dyadic version of FLT), but nobody knows how to convert the dyadic arguments to the [...]

]]> By: Ars Mathematica » Blog Archive » Tao on Zhang http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-1004 Ars Mathematica » Blog Archive » Tao on Zhang Sat, 19 May 2007 03:49:29 +0000 http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-1004 [...] Overview of rational points on curves [...] [...] Overview of rational points on curves [...]

]]> By: Not Even Wrong » Blog Archive » All Sorts of Stuff http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-814 Not Even Wrong » Blog Archive » All Sorts of Stuff Sat, 05 May 2007 20:23:47 +0000 http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-814 [...] continues to come up with amazingly good blog entries. His latest is a series of three postings (here, here and here), reporting on my colleague Shouwu Zhang’s lectures at UCLA on the topic of [...] [...] continues to come up with amazingly good blog entries. His latest is a series of three postings (here, here and here), reporting on my colleague Shouwu Zhang’s lectures at UCLA on the topic of [...]

]]> By: Distinguished Lecture Series III: Shou-wu Zhang, “Triple L-series and effective Mordell conjecture” « What’s new http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-796 Distinguished Lecture Series III: Shou-wu Zhang, “Triple L-series and effective Mordell conjecture” « What’s new Fri, 04 May 2007 16:52:20 +0000 http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-796 [...] discussed in the first lecture, one of the landmark achievements in the higher genus theory is Faltings’ theorem (proving [...] [...] discussed in the first lecture, one of the landmark achievements in the higher genus theory is Faltings’ theorem (proving [...]

]]> By: Distinguished Lecture Series II: Shou-wu Zhang, “Gross-Zagier formula and Birch and Swinnerton-Dyer conjecture ” « What’s new http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-786 Distinguished Lecture Series II: Shou-wu Zhang, “Gross-Zagier formula and Birch and Swinnerton-Dyer conjecture ” « What’s new Thu, 03 May 2007 17:24:12 +0000 http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-786 [...] In the previous lecture, Shou-wu defined an elliptic curve E as a genus 1 curve with a marked point P (which will eventually become the group identity element). It is traditional to move this point P to the point at infinity and view the curve affinely, in which case the curve can be placed via an affine transformation in the normal form [...] [...] In the previous lecture, Shou-wu defined an elliptic curve E as a genus 1 curve with a marked point P (which will eventually become the group identity element). It is traditional to move this point P to the point at infinity and view the curve affinely, in which case the curve can be placed via an affine transformation in the normal form [...]

]]> By: Terence Tao http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-780 Terence Tao Thu, 03 May 2007 01:45:05 +0000 http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-780 Dear Emmanuel and bb, Thanks very much for the corrections; they are especially helpful for me as I am trying to learn this subject properly. (I sort of absorbed bits and pieces of this through osmosis while a graduate student at Princeton - I was there when Wiles proved his theorem, in particular - but it's only now that I think I'm getting a bit of a handle on things.) Dear Emmanuel and bb,

Thanks very much for the corrections; they are especially helpful for me as I am trying to learn this subject properly. (I sort of absorbed bits and pieces of this through osmosis while a graduate student at Princeton - I was there when Wiles proved his theorem, in particular - but it’s only now that I think I’m getting a bit of a handle on things.)

]]> By: bb http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-779 bb Thu, 03 May 2007 01:08:26 +0000 http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-779 I'd just like to point out a tiny mistake in the definition of the Tate-Shafarevich group (Sha, for short). What you've defined (the set of genus 1 curves over a field k with jacobian E, upto isomorphism) is actually the cohomology group H^1(k,E) (this can also be defined as the set of curves over k with a free and transitive E-action defined over k). In order to get Sha, we must account for local obstructions as we're trying to capture the failure of the Hasse principle. So over a number field k, we define Sha(E,k) to be the subgroup of H^1(k,E) consisting of curves which have points over all completions of k. This is the group that's conjectured to be finite -- the H^1(k,E) could very well be infinite. Also, as perhaps an interesting aside, I believe that in the function field case the truth of BSD conjecture is now equivalent to the finiteness of Sha which, in turn, follows from the 2-dimensional case of the remarkable conjecture of Artin that any smooth projective variety over a finite field (in fact, any proper scheme over Spec(Z) works!) has a finite Brauer group (Morita equivalence classes of central simple algebras). I’d just like to point out a tiny mistake in the definition of the Tate-Shafarevich group (Sha, for short). What you’ve defined (the set of genus 1 curves over a field k with jacobian E, upto isomorphism) is actually the cohomology group H^1(k,E) (this can also be defined as the set of curves over k with a free and transitive E-action defined over k). In order to get Sha, we must account for local obstructions as we’re trying to capture the failure of the Hasse principle. So over a number field k, we define Sha(E,k) to be the subgroup of H^1(k,E) consisting of curves which have points over all completions of k. This is the group that’s conjectured to be finite — the H^1(k,E) could very well be infinite.

Also, as perhaps an interesting aside, I believe that in the function field case the truth of BSD conjecture is now equivalent to the finiteness of Sha which, in turn, follows from the 2-dimensional case of the remarkable conjecture of Artin that any smooth projective variety over a finite field (in fact, any proper scheme over Spec(Z) works!) has a finite Brauer group (Morita equivalence classes of central simple algebras).

]]> By: Emmanuel Kowalski http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-776 Emmanuel Kowalski Wed, 02 May 2007 21:52:11 +0000 http://terrytao.wordpress.com/2007/05/02/distinguished-lecture-series-i-shou-wu-zhang-overview-of-rational-points-on-curves/#comment-776 The ``Lefschetz principle'' is presumably a reference to the Grothendieck-Lefschetz trace formula, which states that the number of points on an algebraic variety $latex U$ defined over a finite field $latex \mathbf{F}_q$ with $latex q$ elements is the alternating sum of traces of the ``global'' Frobenius automorphism acting on some finite-dimensional vector spaces, the index $latex j$ ranging from 0 to $latex 2d$, where $latex d$ is the dimension. For a smooth irreducible projective curve, $latex d=1$, the first trace ($latex j=0$) is $latex 1$ (trivial action on a one-dimensional space), the last trace ($latex j=2$) is $latex q$ (multiplication by $latex q$ on a one-dimensional space) and the middle (interesting) trace ($latex j=1$) is the trace on a vector space of dimension $latex 2g$ (analogue of the classical first (co)homology group of a Riemann surface), written as sum of eigenvalues $latex \lambda_{i}$. The “Lefschetz principle” is presumably a reference to the Grothendieck-Lefschetz
trace formula, which states that the number of points on an algebraic variety U
defined over a finite field \mathbf{F}_q with q elements is the alternating
sum of traces of the “global” Frobenius automorphism acting on some finite-dimensional
vector spaces, the index j ranging from 0 to 2d, where d is the dimension.
For a smooth irreducible projective curve, d=1, the first trace (j=0) is 1 (trivial action on a one-dimensional space), the last trace (j=2) is q (multiplication by q on a
one-dimensional space) and the middle (interesting) trace (j=1) is the trace on
a vector space of dimension 2g (analogue of the classical first (co)homology group of a Riemann surface), written as sum of eigenvalues \lambda_{i}.

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