Timeline for Average rank of elliptic curves, excluding those of low rank
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Nov 2, 2013 at 4:01 | comment | added | ThisNameForSale | The only place where Watkins uses RMT is the 3/8 exponent for log; he says the 19/24 can be derived more crudely. For RMT, model $L$-values via charpolys and apply a discretization process in an arithmetic analog. One has $Prob(P(1)\le t)\sim M^{3/8}\sqrt{t}$ as $t\sim 0$ for the charpoly $P$ of an orthog matrix, where $M$ is the matrix size. For ellcurv we guess the same for $Prob(L(E,1)\le t)$ with an arith factor that averages out. He matches $M\sim\log N$ as ellcurv conductor, mucks cond vs disc vs $1/\Omega^{1/12}$, discretizes as Sha is integral, and ends with the 3/8 he started with. | |
| Nov 2, 2013 at 3:43 | comment | added | ThisNameForSale | I think it is a different approach. The Watkins paper only references a to-come work with Granville, which seems to be the one cited in my answer. They work in the twist case there, but would basically heuristically bound the number of (integral?) points $(A,B,X,Y,Z)$ on $Y^2Z=X^3+AXZ^2+BZ^3$ in some ranges, and compare this to the number of such points (via ellipsoids) that a curve of rank $r$ with parameters of size $(A,B)$ would generate in said ranges. I think Lang-Vojta guessing is similar. Maybe I will try to write this as an answer, if I can work out how the non-twist case differs. | |
| Nov 2, 2013 at 2:21 | comment | added | Daniel Hast | Thanks for the answer; I figured it would be pretty speculative. Is the conjecture you mention based in a more refined version of the heuristic described in Will Sawin's answer, or is it a different approach? | |
| Nov 1, 2013 at 20:23 | comment | added | Will Sawin | So this would imply that the average value is $n$ for $n\leq 22$ and undetermined for $n >22$? | |
| Nov 1, 2013 at 20:09 | history | answered | Matt Young | CC BY-SA 3.0 |