Comments on: Distinguished Lecture Series II: Shou-wu Zhang, “Gross-Zagier formula and Birch and Swinnerton-Dyer conjecture ” http://terrytao.wordpress.com/2007/05/03/distinguished-lecture-series-ii-shou-wu-zhang-%e2%80%9cgross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-%e2%80%9d/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Thu, 07 Aug 2008 21:37:40 +0000 http://wordpress.org/?v=MU By: Terence Tao http://terrytao.wordpress.com/2007/05/03/distinguished-lecture-series-ii-shou-wu-zhang-%e2%80%9cgross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-%e2%80%9d/#comment-823 Terence Tao Sun, 06 May 2007 05:19:47 +0000 http://terrytao.wordpress.com/2007/05/03/distinguished-lecture-series-ii-shou-wu-zhang-%e2%80%9cgross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-%e2%80%9d/#comment-823 Hi Jordan! Thanks for your comments. As for the "philosophy", I know at least one expert in Diophantine equations who assigns your name to it, but I can believe it has been around for a while. I went back to my notes for Shou-wu's talk and tried to reconstruct his comments more precisely. I don't really understand what's going on, but it was something like this: just as moduli of elliptic curves give modular curves, moduli of abelian varieties give Shimura varieties (and this was only because $latex H^1(X)$ was abelian, apparently). Shou-wu made a "moral" at the end of the talk that a really good way to find rational points on a variety is to somehow get nontrivial maps into them from "richer" varieties such as modular varieties, in which the individual points on the variety themselves have algebraic structure. He did concede though that at the present time, this strategy has not really borne much fruit with regard to general Shimura varieties (there was some connection here between Shimura varieties involving a congruence subgroup in $latex Sp_{2g}({\Bbb Z})$ and a moduli space $latex {\mathcal M}_g$ of curves, or a moduli space of subgroups of $latex PSL_2({\Bbb R})$, but I can't reconstruct what the deal was from my notes). My notes also state "For most moduli spaces, there is no "concrete" embedding into projective space", which is presumably what you just said above. I'm not sure what "concrete" means here, but Shou-wu gave the exponential function $latex x \mapsto e^{2\pi i x}$ modeling the torus $latex {\Bbb R}/{\Bbb Z}$ in the complex plane as an analogy. Hi Jordan! Thanks for your comments. As for the “philosophy”, I know at least one expert in Diophantine equations who assigns your name to it, but I can believe it has been around for a while.

I went back to my notes for Shou-wu’s talk and tried to reconstruct his comments more precisely. I don’t really understand what’s going on, but it was something like this: just as moduli of elliptic curves give modular curves, moduli of abelian varieties give Shimura varieties (and this was only because H^1(X) was abelian, apparently). Shou-wu made a “moral” at the end of the talk that a really good way to find rational points on a variety is to somehow get nontrivial maps into them from “richer” varieties such as modular varieties, in which the individual points on the variety themselves have algebraic structure. He did concede though that at the present time, this strategy has not really borne much fruit with regard to general Shimura varieties (there was some connection here between Shimura varieties involving a congruence subgroup in Sp_{2g}({\Bbb Z}) and a moduli space {\mathcal M}_g of curves, or a moduli space of subgroups of PSL_2({\Bbb R}), but I can’t reconstruct what the deal was from my notes).

My notes also state “For most moduli spaces, there is no “concrete” embedding into projective space”, which is presumably what you just said above. I’m not sure what “concrete” means here, but Shou-wu gave the exponential function x \mapsto e^{2\pi i x} modeling the torus {\Bbb R}/{\Bbb Z} in the complex plane as an analogy.

]]> By: Jordan Ellenberg http://terrytao.wordpress.com/2007/05/03/distinguished-lecture-series-ii-shou-wu-zhang-%e2%80%9cgross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-%e2%80%9d/#comment-818 Jordan Ellenberg Sat, 05 May 2007 23:03:07 +0000 http://terrytao.wordpress.com/2007/05/03/distinguished-lecture-series-ii-shou-wu-zhang-%e2%80%9cgross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-%e2%80%9d/#comment-818 I agree with the philosophy ascribed to me above but I'm certain I don't deserve any primary credit for articulating it! <i>(This is apparently quite remarkable; most moduli spaces do not have concrete representations as algebraic varieties. I gather from Shou-wu that this phenomenon is ultimately due to the fact that elliptic curves are an abelian variety.)</i> I'm not sure I agree with this, though (but I may not be completely clear about what you're saying.) The moduli spaces that you meet in arithmetic geometry do have representations as algebraic varieties, or at least algebraic stacks; what is rare is to have, as with X_0(N), a reasonably nice way of writing down <i>explicit equations</i> for such a variety. I think that's pretty rare even when you are studying moduli spaces of abelian varieties, so I'm not quite sure of the meaning of Shou-Wu's comment here. But in general it usually turns out not to be so useful to have explicit equations! (That said, I did write a paper where it was quite important to have explicit equations for a Hilbert modular surface, and it was a huge pain. But it would have been a better paper if there were a way to get the result without writing the equations down!) It tends to be more important just to know that there <i>is</i> a moduli space which is an algebraic variety, and to know its properties (dimension, smoothness, properness, morphisms to other varieties of interest, etc.) I agree with the philosophy ascribed to me above but I’m certain I don’t deserve any primary credit for articulating it!

(This is apparently quite remarkable; most moduli spaces do not have concrete representations as algebraic varieties. I gather from Shou-wu that this phenomenon is ultimately due to the fact that elliptic curves are an abelian variety.)

I’m not sure I agree with this, though (but I may not be completely clear about what you’re saying.) The moduli spaces that you meet in arithmetic geometry do have representations as algebraic varieties, or at least algebraic stacks; what is rare is to have, as with X_0(N), a reasonably nice way of writing down explicit equations for such a variety. I think that’s pretty rare even when you are studying moduli spaces of abelian varieties, so I’m not quite sure of the meaning of Shou-Wu’s comment here.

But in general it usually turns out not to be so useful to have explicit equations! (That said, I did write a paper where it was quite important to have explicit equations for a Hilbert modular surface, and it was a huge pain. But it would have been a better paper if there were a way to get the result without writing the equations down!) It tends to be more important just to know that there is a moduli space which is an algebraic variety, and to know its properties (dimension, smoothness, properness, morphisms to other varieties of interest, etc.)

]]> By: Not Even Wrong » Blog Archive » All Sorts of Stuff http://terrytao.wordpress.com/2007/05/03/distinguished-lecture-series-ii-shou-wu-zhang-%e2%80%9cgross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-%e2%80%9d/#comment-815 Not Even Wrong » Blog Archive » All Sorts of Stuff Sat, 05 May 2007 20:28:33 +0000 http://terrytao.wordpress.com/2007/05/03/distinguished-lecture-series-ii-shou-wu-zhang-%e2%80%9cgross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-%e2%80%9d/#comment-815 [...] 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 rational [...] [...] 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 rational [...]

]]> By: Emmanuel Kowalski http://terrytao.wordpress.com/2007/05/03/distinguished-lecture-series-ii-shou-wu-zhang-%e2%80%9cgross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-%e2%80%9d/#comment-791 Emmanuel Kowalski Thu, 03 May 2007 20:39:28 +0000 http://terrytao.wordpress.com/2007/05/03/distinguished-lecture-series-ii-shou-wu-zhang-%e2%80%9cgross-zagier-formula-and-birch-and-swinnerton-dyer-conjecture-%e2%80%9d/#comment-791 The question of average rank of elliptic curves (or average order of vanishing) seems quite tricky, and there are really two schools of thought as to whether, ordering by discriminant, one should or not get 50% of rank 0 and rank 1, and a set with ``density 0'' of curves with rank > 1. There's a fascinating account of the most recent numerical evidence in: <a HREF="http://www.ams.org/bull/2007-44-02/S0273-0979-07-01138-X/home.html" rel="nofollow">Average ranks of elliptic curves: Tension between data and conjecture</a> Baur Bektemirov; Barry Mazur; William Stein; Mark Watkins Bull. Amer. Math. Soc. 44 (2007), 233-254. The Random Matrix models for L-functions tend to strongly suggest the 50/50 split between rank 0 and 1 (assuming the B-SD conjecture), with some function field evidence by Katz and Sarnak. In some other families of abelian varieties (obtained from modular curves in ways similar to elliptic curves, though involving modular forms with not-necessarily integral coefficients), the results are quite a bit more complete: there, it has been shown that at least 25% of the relevant abelian varieties have rank 0 (50% among those which should have even rank), and at least 43.75% have rank 1 (i.e., 87.5% among those which should have odd rank have rank 1). See e.g. http://www.math.u-bordeaux1.fr/~kowalski/high-derivatives.pdf (joint with P. Michel and J. Vanderkam). Incidentally, these results arise from some form of very precise approximate orthogonality for the Fourier coefficients of the relevant modular forms (the Petersson formula). If the Fourier coefficients of elliptic curves, averaged properly with the desired ordering, have the same type of approximate orthogonality, there would be a similar result. This is described a bit in the following survey : http://www.math.u-bordeaux1.fr/~kowalski/elliptic-curves-families.pdf The question of average rank of elliptic curves (or average order of vanishing) seems quite tricky, and there are really two schools of thought as to whether, ordering by discriminant, one should or not get 50% of rank 0 and rank 1, and a set with “density 0” of curves with rank > 1. There’s a fascinating account of the most recent numerical evidence in:

Average ranks of elliptic curves: Tension between data and conjecture
Baur Bektemirov; Barry Mazur; William Stein; Mark Watkins
Bull. Amer. Math. Soc. 44 (2007), 233-254.

The Random Matrix models for L-functions tend to strongly suggest the 50/50 split between rank 0 and 1 (assuming the B-SD conjecture), with some function field evidence by Katz and Sarnak.

In some other families of abelian varieties (obtained from modular curves in ways similar to elliptic curves, though involving modular forms with not-necessarily integral coefficients), the results are quite a bit more complete: there, it has been shown that at least 25% of the relevant abelian varieties have rank 0 (50% among those which should have even rank), and at least 43.75% have rank 1 (i.e., 87.5% among those which should have odd rank have rank 1). See e.g.

http://www.math.u-bordeaux1.fr/~kowalski/high-derivatives.pdf

(joint with P. Michel and J. Vanderkam).

Incidentally, these results arise from some form of very precise approximate orthogonality for the Fourier coefficients of the relevant modular forms (the Petersson formula). If the Fourier coefficients of elliptic curves, averaged properly with the desired ordering, have the same type of approximate orthogonality, there would be a similar result. This is described a bit in the following survey :

http://www.math.u-bordeaux1.fr/~kowalski/elliptic-curves-families.pdf

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