Rational points on symmetric squares of constant algebraic curves over function fields

Berg, Jennifer; Voloch, José Felipe

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 467-480

We consider smooth projective curves $C/𝔽$ over a finite field and their symmetric squares $C^{\left(2\right)}$. For a global function field $K/𝔽$, we study the $K$-rational points of $C^{\left(2\right)}$. We describe the adelic points of $C^{\left(2\right)}$ surviving Frobenius descent and how the $K$-rational points fit there. Our methods also lead to an explicit bound on the number of $K$-rational points of $C^{\left(2\right)}$ satisfying an additional condition. Some of our results apply to arbitrary constant subvarieties of abelian varieties, however we produce examples which show that not all of our stronger conclusions extend.

On considère des courbes projectives lisses $C/𝔽$ sur un corps fini et leurs carrés symétriques $C^{\left(2\right)}$. Pour un corps de fonctions global $K/𝔽$, nous étudions les points $K$-rationnels de $C^{\left(2\right)}$. Nous décrivons les points adéliques de $C^{\left(2\right)}$ survivant à la descente de Frobenius et décrivons comment les points $K$-rationnels y sont situés. Nos méthodes conduisent également à une borne explicite pour le nombre de points $K$-rationnels de $C^{\left(2\right)}$ satisfaisant une condition supplémentaire. Certains de nos résultats s’appliquent à des sous-variétés constantes arbitraires de variétés abéliennes, mais nous produisons des exemples qui montrent que certaines des nos conclusions les plus fortes ne s’étendent pas.

Classification : 11G35, 14G05
Keywords: Rational points, Descent obstruction, Global fields

[1] Abramovich, Dan Subvarieties of semiabelian varieties, Compos. Math., Volume 90 (1994) no. 1, pp. 37-52 | Zbl | Numdam | MR

[2] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip A.; Harris, Joseph Geometry of algebraic curves. Volume I, Grundlehren der Mathematischen Wissenschaften, 267, Springer, 1985, xvi+386 pages | DOI

[3] Artin, Michael; Milne, James S. Duality in the flat cohomology of curves, Invent. Math., Volume 35 (1976), pp. 111-129 | Zbl | DOI | MR

[4] Creutz, Brendan There are no transcendental Brauer-Manin obstructions on abelian varieties, Int. Math. Res. Not. (2020) no. 9, pp. 2684-2697 | DOI | MR | Zbl

[5] Creutz, Brendan; Viray, Bianca; Voloch, José Felipe The $d$-primary Brauer-Manin obstruction for curves, Res. Number Theory, Volume 4 (2018) no. 2, 26, 16 pages | MR | Zbl

[6] Creutz, Brendan; Voloch, José Felipe The Brauer-Manin obstruction for constant curves over global function fields, Ann. Inst. Fourier, Volume 72 (2022) no. 1, pp. 43-58 | DOI | MR | Zbl

[7] Griffiths, Phillip A.; Harris, Joseph Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, 1994, xiv+813 pages (Reprint of the 1978 original) | DOI

[8] Gunther, Joseph Random hypersurfaces and embedding curves in surfaces over finite fields, J. Pure Appl. Algebra, Volume 221 (2017) no. 1, pp. 89-97 | MR | Zbl | DOI

[9] Kempf, George On the Geometry of a Theorem of Riemann, Ann. Math., Volume 98 (1973) no. 1, pp. 178-185 | DOI | MR | Zbl

[10] Resende de Macedo, A. Differential fppf descent obstructions, Ph. D. Thesis, University of Texas at Austin (2017)

[11] Martens, Henrik H. On the varieties of special divisors on a curve, J. Reine Angew. Math., Volume 227 (1967), pp. 111-120 | MR | Zbl

[12] Poonen, Bjorn; Voloch, José Felipe The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields, Ann. Math., Volume 171 (2010) no. 1, pp. 511-532 | Zbl | DOI | MR

[13] Pries, Rachel A short guide to $p$-torsion of abelian varieties in characteristic $p$., Computational arithmetic geometry (Contemporary Mathematics), Volume 463, American Mathematical Society, 2008, pp. 121-129 | DOI | MR | Zbl

[14] Stoll, Michael Finite descent obstructions and rational points on curves, Algebra Number Theory, Volume 1 (2007) no. 4, pp. 349-391 | MR | Zbl | DOI

[15] Voloch, José Felipe Differential descent obstructions over function fields, Proc. Am. Math. Soc., Volume 142 (2014) no. 10, pp. 3421-3424 | DOI | MR | Zbl

[16] Wutz, Franziska Bertini theorems for smooth hypersurface sections containing a subscheme over finite fields (2016) | arXiv