The Brauer–Manin obstruction for constant curves over global function fields
Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 43-58.

Let 𝔽 be a finite field and C,D smooth, geometrically irreducible, proper curves over 𝔽 and set K=𝔽(D). We consider Brauer–Manin and abelian descent obstructions to the existence of rational points and to weak approximation for the curve C 𝔽 K. In particular, we show that Brauer–Manin is the only obstruction to weak approximation and the Hasse principle in the case that the genus of D is less than that of C. We also show that we can identify the points corresponding to non-constant maps DC using Frobenius descents.

Soit 𝔽 un corps fini et C, D courbes lisses, géométriquement irréductibles, propres sur 𝔽 et soit K=𝔽(D). Nous considérons les obstructions de Brauer–Manin et de descente abélienne à l’existence de points rationnels et d’approximation faible pour la courbe C 𝔽 K. En particulier, nous montrons que l’obstruction de Brauer–Manin est la seule obstruction à l’approximation faible et le principe de Hasse dans le cas où le genre de D est inférieur à celui de C. On montre aussi que l’on peut identifier les points correspondant aux morphismes non constants DC en utilisant la descente de Frobenius.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3473
Classification: 11G20,  11G30,  14H25
Keywords: Rational points, Brauer–Manin obstruction, Global fields
Brendan, Creutz 1; José Felipe, Voloch 1

1 School of Mathematics and Statistics University of Canterbury Private Bag 4800 Christchurch 8140 (New Zealand)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2022__72_1_43_0,
     author = {Brendan, Creutz and Jos\'e Felipe, Voloch},
     title = {The {Brauer{\textendash}Manin} obstruction for constant curves over global function fields},
     journal = {Annales de l'Institut Fourier},
     pages = {43--58},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {1},
     year = {2022},
     doi = {10.5802/aif.3473},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3473/}
}
TY  - JOUR
TI  - The Brauer–Manin obstruction for constant curves over global function fields
JO  - Annales de l'Institut Fourier
PY  - 2022
DA  - 2022///
SP  - 43
EP  - 58
VL  - 72
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3473/
UR  - https://doi.org/10.5802/aif.3473
DO  - 10.5802/aif.3473
LA  - en
ID  - AIF_2022__72_1_43_0
ER  - 
%0 Journal Article
%T The Brauer–Manin obstruction for constant curves over global function fields
%J Annales de l'Institut Fourier
%D 2022
%P 43-58
%V 72
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3473
%R 10.5802/aif.3473
%G en
%F AIF_2022__72_1_43_0
Brendan, Creutz; José Felipe, Voloch. The Brauer–Manin obstruction for constant curves over global function fields. Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 43-58. doi : 10.5802/aif.3473. https://aif.centre-mersenne.org/articles/10.5802/aif.3473/

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

[2] Bogomolov, Fedor; Korotiaev, Mikhail; Tschinkel, Yuri A Torelli theorem for curves over finite fields, Pure Appl. Math. Q., Volume 6 (2010) no. 1, pp. 245-294 (Special Issue: In honor of John Tate.) | DOI | MR | Zbl

[3] Bruin, Nils; Stoll, Michael Deciding existence of rational points on curves: an experiment, Exp. Math., Volume 17 (2008) no. 2, pp. 181-189 | DOI | MR | Zbl

[4] Buium, Alexandru; Voloch, José Felipe Reduction of the Manin map modulo p, J. Reine Angew. Math., Volume 460 (1995), pp. 117-126 | 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] Demarche, Cyril; Harari, David Artin–Mazur–Milne duality for fppf cohomology, Algebra Number Theory, Volume 13 (2019) no. 10, pp. 2323-2357 | DOI | MR | Zbl

[7] González-Avilés, Cristian D.; Tan, Ki-Seng A generalization of the Cassels–Tate dual exact sequence, Math. Res. Lett., Volume 14 (2007) no. 2, pp. 295-302 | DOI | MR | Zbl

[8] González-Avilés, Cristian D.; Tan, Ki-Seng On the Hasse principle for finite group schemes over global function fields, Math. Res. Lett., Volume 19 (2012) no. 2, pp. 453-460 | DOI | MR | Zbl

[9] Görtz, Ulrich; Wedhorn, Torsten Algebraic geometry I. Schemes with examples and exercises, Advanced Lectures in Mathematics, Vieweg+Teubner, 2010, viii+615 pages | DOI

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

[11] Milne, James S. The Tate–Šafarevič group of a constant abelian variety, Invent. Math., Volume 6 (1968), pp. 91-105 | DOI | Zbl

[12] Milne, James S. Arithmetic duality theorems, Perspectives in Mathematics, 1, Academic Press Inc., 1986, x+421 pages

[13] Milne, James S. Algebraic groups. The theory of group schemes of finite type over a field, Cambridge Studies in Advanced Mathematics, 170, Cambridge University Press, 2017, xvi+644 pages | DOI

[14] Poonen, Bjorn Heuristics for the Brauer–Manin obstruction for curves, Exp. Math., Volume 15 (2006) no. 4, pp. 415-420 | DOI | MR | Zbl

[15] Poonen, Bjorn Rational points on varieties, Graduate Studies in Mathematics, 186, American Mathematical Society, 2017, xv+337 pages | DOI

[16] 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 | DOI | MR | Zbl

[17] Rössler, Damian Le groupe de Selmer des isogénies de hauteur un (preprint)

[18] Rössler, Damian On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic II, Algebra Number Theory, Volume 14 (2020) no. 5, pp. 1123-1173 | DOI | MR | Zbl

[19] Scharaschkin, Victor Local-global problems and the Brauer–Manin obstruction, Ph. D. Thesis, University of Michigan (1999), 59 pages | MR

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

[21] Tate, John Endomorphisms of abelian varieties over finite fields, Invent. Math., Volume 2 (1966), pp. 134-144 | DOI | MR | Zbl

[22] Waterhouse, William C.; Milne, James S. Abelian varieties over finite fields, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), American Mathematical Society, 1971, pp. 53-64 | Zbl

[23] Zilber, Boris A curve and its abstract Jacobian, Int. Math. Res. Not., Volume 2014 (2014) no. 5, pp. 1425-1439 | DOI | MR | Zbl

[24] Zilber, Boris A curve and its abstract Jacobian (2017) (Corrected version of [23], preprint, http://people.maths.ox.ac.uk/zilber/Jacobian.pdf)

Cited by Sources: