Geodesic
On the Average Number of 2-Selmer Elements of Elliptic Curves over $\mathbb F_q(X)$ with Two Marked Points
Documenta mathematica, Tome 24 (2019), pp. 1179-1223.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We consider elliptic curves over global fields of positive characteristic with two distinct marked non-trivial rational points. Restricting to a certain subfamily of the universal one, we show that the average size of the 2-Selmer groups of these curves exists, in a natural sense, and equals 12. Along the way, we consider a map from these 2-Selmer groups to the moduli space of $G$-torsors over an algebraic curve, where $G$ is isogenous to $\mathrm{SL}_2^4$, and show that the images of 2-Selmer elements under this map become equidistributed in the limit.
Classification : 11G05, 14H60
Keywords: elliptic curves, rational points, Selmer groups 
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     author = {Thorne, Jack A.},
     title = {On the {Average} {Number} of {2-Selmer} {Elements} of {Elliptic} {Curves} over \(\mathbb {F_q(X)\)} with {Two} {Marked} {Points}},
     journal = {Documenta mathematica},
     pages = {1179--1223},
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     volume = {24},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a33/}
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Thorne, Jack A. On the Average Number of 2-Selmer Elements of Elliptic Curves over \(\mathbb F_q(X)\) with Two Marked Points. Documenta mathematica, Tome 24 (2019), pp. 1179-1223. http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a33/