## On the Average Number of 2-Selmer Elements of Elliptic Curves over $\mathbb F_q(X)$ with Two Marked Points

##### Doc. Math. 24, 1179-1223 (2019)
DOI: 10.25537/dm.2019v24.1179-1223

### Summary

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.

11G05, 14H60

### Keywords/Phrases

elliptic curves, rational points, Selmer groups

### References

• [BD09]. Kai Behrend and Ajneet Dhillon. Connected components of moduli stacks of torsors via Tamagawa numbers. Canad. J. Math., 61(1):3-28, 2009. DOI 10.4153/CJM-2009-001-5; zbl 1219.14030; MR2488447; arxiv math/0503383.
• [BH]. Manjul Bhargava and Wei Ho. On average sizes of Selmer groups and ranks in families of elliptic curves having marked points. Available at http://www-personal.umich.edu/~weiho/papers/Selmer-averages-families.pdf.
• [BH04]. Indranil Biswas and Yogish I. Holla. Harder-Narasimhan reduction of a principal bundle. Nagoya Math. J., 174:201-223, 2004. DOI 10.1017/S0027763000008850; zbl 1056.14046; MR2066109.
• [BLR90]. Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud. Néron models, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1990. zbl 0705.14001; MR1045822.
• [BPGN97]. L. Brambila-Paz, I. Grzegorczyk, and P. E. Newstead. Geography of Brill-Noether loci for small slopes. J. Algebraic Geom., 6(4):645-669, 1997. zbl 0937.14018; MR1487229; arxiv alg-geom/9511003.
• [BS]. Lior Bary-Soroker. Letter to Bjorn Poonen. September 16, 2013. Available at http://www.math.tau.ac.il/~barylior/files/let_poonen.pdf.
• [BS15]. Manjul Bhargava and Arul Shankar. Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. Ann. of Math. (2), 181(1):191-242, 2015. DOI 10.4007/annals.2015.181.1.3; zbl 1307.11071; MR3272925; arxiv 1006.1002.
• [CFS10]. John E. Cremona, Tom A. Fisher, and Michael Stoll. Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves. Algebra Number Theory, 4(6):763-820, 2010. DOI 10.2140/ant.2010.4.763; zbl 1222.11073; MR2728489; arxiv 0908.1741.
• [Cha97]. Nick Chavdarov. The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy. Duke Math. J., 87(1):151-180, 1997. DOI 10.1215/S0012-7094-97-08707-X; zbl 0941.14006; MR1440067.
• [dJ02]. A. J. de Jong. Counting elliptic surfaces over finite fields. Mosc. Math. J., 2(2):281-311, 2002. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. zbl 1031.11033; MR1944508.
• [HLHN14]. Q. P. Ho, V. B. Lê Hung, and B. C. Ngô. Average size of 2-Selmer groups of elliptic curves over function fields. Math. Res. Lett., 21(6):1305-1339, 2014. DOI 10.4310/MRL.2014.v21.n6.a6; zbl 1316.14055; MR3335849; arxiv 1310.7963.
• [Hum95]. James E. Humphreys. Conjugacy classes in semisimple algebraic groups, volume 43 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1995. zbl 0834.20048; MR1343976.
• [IMP03]. S. Ilangovan, V. B. Mehta, and A. J. Parameswaran. Semistability and semisimplicity in representations of low height in positive characteristic. In A tribute to C. S. Seshadri (Chennai, 2002), Trends Math., pages 271-282. Birkhäuser, Basel, 2003. zbl 1067.20061; MR2017588.
• [KR71]. B. Kostant and S. Rallis. Orbits and representations associated with symmetric spaces. Amer. J. Math., 93:753-809, 1971. DOI 10.2307/2373470; zbl 0224.22013; MR0311837.
• [Lev07]. Paul Levy. Involutions of reductive Lie algebras in positive characteristic. Adv. Math., 210(2):505-559, 2007. DOI 10.1016/j.aim.2006.07.002; zbl 1173.17019; MR2303231; arxiv math/0501334.
• [Liu02]. Qing Liu. Algebraic geometry and arithmetic curves, volume 6 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné, Oxford Science Publications. zbl 0996.14005; MR1917232.
• [Mir81]. Rick Miranda. The moduli of Weierstrass fibrations over $\mathbb P^1$. Math. Ann., 255(3):379-394, 1981. DOI 10.1007/BF01450711; zbl 0438.14023; MR0615858.
• [Poo03]. Bjorn Poonen. Squarefree values of multivariable polynomials. Duke Math. J., 118(2):353-373, 2003. DOI 10.1215/S0012-7094-03-11826-8; zbl 1047.11021; MR1980998; arxiv math/0203292.
• [PR12]. Bjorn Poonen and Eric Rains. Random maximal isotropic subspaces and Selmer groups. J. Amer. Math. Soc., 25(1):245-269, 2012. DOI 10.1090/S0894-0347-2011-00710-8; zbl 1294.11097; MR2833483; arxiv 1009.0287.
• [RR84]. S. Ramanan and A. Ramanathan. Some remarks on the instability flag. Tohoku Math. J. (2), 36(2):269-291, 1984. DOI 10.2748/tmj/1178228852; zbl 0567.14027; MR0742599;.
• [Sad12]. Mohammad Sadek. Minimal genus one curves. Funct. Approx. Comment. Math., 46(part 1):117-131, 2012. DOI 10.7169/facm/2012.46.1.9; zbl 1286.11098; MR2951733; arxiv 1002.0451.
• [Sch15]. Simon Schieder. The Harder-Narasimhan stratification of the moduli stack of $G$-bundles via Drinfeld's compactifications. Selecta Math. (N.S.), 21(3):763-831, 2015. DOI 10.1007/s00029-014-0161-y; zbl 1341.14006; MR3366920; arxiv 1212.6814.
• [Shi91]. Tetsuji Shioda. Theory of Mordell-Weil lattices. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 473-489. Math. Soc. Japan, Tokyo, 1991. zbl 0746.14009; MR1159235.
• [Slo80]. Peter Slodowy. Simple singularities and simple algebraic groups, volume 815 of Lecture Notes in Mathematics. Springer, Berlin, 1980. zbl 0441.14002; MR0584445.
• [SS70]. T. A. Springer and R. Steinberg. Conjugacy classes. In Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Lecture Notes in Mathematics, Vol. 131, pages 167-266. Springer, Berlin, 1970. zbl 0249.20024; MR0268192.
• [Tat75]. J. Tate. Algorithm for determining the type of a singular fiber in an elliptic pencil. Lecture Notes in Math., Vol. 476, pages 33-52, 1975. DOI 10.1007/BFb0097582; zbl 1214.14020; MR0393039.
• [{Tho}13]. Jack A. Thorne. Vinberg's representations and arithmetic invariant theory. Algebra Number Theory, 7(9):2331-2368, 2013. DOI 10.2140/ant.2013.7.2331; zbl 1321.11045; MR3152016.
• [Tho15]. Jack A. Thorne. $E_6$ and the arithmetic of a family of non-hyperelliptic curves of genus 3. Forum Math. Pi, 3:e1, 41, 2015. DOI 10.1017/fmp.2014.2; zbl 1346.14058; MR3298319.
• [Wei95]. André Weil. Adèles et groupes algébriques. In Séminaire Bourbaki, Vol. 5, pages Exp. No. 186, 249-257. Soc. Math. France, Paris, 1995. zbl 0118.15801; MR1603471.

### Affiliation

Thorne, Jack A.
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom