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De Jong showed in 2002 if the finite field $\mathbb{F}_q$ has characteristic not equal to 3, then the limsup of the average of 3-Selmer rank is bounded above, where the average is taken over the family of all elliptic curves over $\mathbb{F}_q(t)$, ordering by height.

Bhargava and Shankar has already computed the average size of $n$-Selmer groups for $n=2,3,4,5$ over the family of elliptic curves over $\mathbb{Q}$, and they have formulated the conjectural average for general $n$.

Question: Is there any work so far that calculates the precise average of $n$-Selmer group of elliptic curves over $\mathbb{F}_q(t)$? Do we expect same result for the function field case, and does the geometry of number technique carry over to the function field case as well?

Thanks!

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  • $\begingroup$ One expects that, for all $n$, over both number fields and function fields, the average rank of $n$-Selmer is the sum of all the divisors of $n$. I have heard that the same arguments as in the number field case work for $n\leq 5$, with the counting lattice points in nonconvex bodies replaced by somewhat simpler counting arguments. $\endgroup$ – Will Sawin Feb 19 '14 at 3:26
  • $\begingroup$ @Will Sawin, thanks! So am I right to say that we expect the lattice point counting problem in the function field case is simpler in general? If possible, can you say in more detail what is simpler? $\endgroup$ – user31415 Feb 19 '14 at 3:33
  • $\begingroup$ I can't say more with high confidence but I believe that, in the original proof, one considers elliptic curves with bounded invariants of certain types, and in certain fundamental domains, and these give certain polynomial inequalities that lattice points must satisfy. In the function field case, the notion of size is replaced with one of degree, and having a certain invariant have bounded degree is usually not a complicated curvy shape but instead a pretty simple all-or-none cutoff. So one might hope that counting gets a bit simpler. $\endgroup$ – Will Sawin Feb 19 '14 at 3:36
  • $\begingroup$ @WillSawin, cool! I was looking for something like "having a certain invariant have bounded degree is usually not a complicated curvy shape but instead a pretty simple all-or-none cutoff" - I will take this for granted for now. $\endgroup$ – user31415 Feb 19 '14 at 3:42
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    $\begingroup$ I believe that Ngo, Ho, and Le Hung have recently carried out a cohomological approach to the function field version of the techniques initiated by Bhargava and Shankar, making good use of the additional geometric structure present in equicharacteristic. Look on the arxiv. $\endgroup$ – user76758 Feb 19 '14 at 5:46

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