## Algebra & Number Theory

### On the Brauer–Siegel ratio for abelian varieties over function fields

Douglas Ulmer

#### Abstract

Hindry has proposed an analog of the classical Brauer–Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell–Weil group and the order of the Tate–Shafarevich group should have size comparable to the exponential differential height. Hindry–Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate–Shafarevich group and the regulator. We recover the results of Hindry–Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.

#### Article information

Source
Algebra Number Theory, Volume 13, Number 5 (2019), 1069-1120.

Dates
Revised: 27 February 2019
Accepted: 2 April 2019
First available in Project Euclid: 17 July 2019

https://projecteuclid.org/euclid.ant/1563328822

Digital Object Identifier
doi:10.2140/ant.2019.13.1069

Mathematical Reviews number (MathSciNet)
MR3981314

Zentralblatt MATH identifier
07083102

#### Citation

Ulmer, Douglas. On the Brauer–Siegel ratio for abelian varieties over function fields. Algebra Number Theory 13 (2019), no. 5, 1069--1120. doi:10.2140/ant.2019.13.1069. https://projecteuclid.org/euclid.ant/1563328822

#### References

• M. Artin, “Supersingular $K3$ surfaces”, Ann. Sci. École Norm. Sup. $(4)$ 7 (1974), 543–567.
• L. Berger, “Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields”, J. Number Theory 128:12 (2008), 3013–3030.
• L. Berger, C. Hall, R. Pannekoek, J. Park, R. Pries, S. Sharif, A. Silverberg, and D. Ulmer, “Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields”, 2015.
• S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 21, Springer, 1990.
• R. Brauer, “On the zeta-functions of algebraic number fields, II”, Amer. J. Math. 72 (1950), 739–746.
• R. P. Conceição, C. Hall, and D. Ulmer, “Explicit points on the Legendre curve II”, Math. Res. Lett. 21:2 (2014), 261–280.
• B. Conrad, “Chow's $K/k$-image and $K/k$-trace, and the Lang–Néron theorem”, Enseign. Math. $(2)$ 52:1-2 (2006), 37–108.
• F. R. Cossec and I. V. Dolgachev, Enriques surfaces, I, Progress in Mathematics 76, Birkhäuser, Boston, 1989.
• C. Davis and T. Occhipinti, “Explicit points on $y^2+xy-t^dy=x^3$ and related character sums”, J. Number Theory 168 (2016), 13–38.
• N. Dummigan, “The determinants of certain Mordell–Weil lattices”, Amer. J. Math. 117:6 (1995), 1409–1429.
• R. Griffon, “Analogue of the Brauer–Siegel theorem for some families of elliptic curves over function fields”, poster, 2015. Presented at the Silvermania conference at Brown University.
• R. Griffon, Analogues du théorème de Brauer–Siegel pour quelques familles de courbes elliptiques, Ph.D. thesis, Université Paris Diderot, 2016, http://math.richardgriffon.me/thesis/Griffon_thesis.pdf.
• R. Griffon, “A Brauer–Siegel theorem for Fermat surfaces over finite fields”, J. Lond. Math. Soc. $(2)$ 97:3 (2018), 523–549.
• A. Grothendieck, “Le groupe de Brauer, III: Exemples et compléments”, pp. 88–188 in Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam, 1968.
• L. H. Halle and J. Nicaise, “The Néron component series of an abelian variety”, Math. Ann. 348:3 (2010), 749–778.
• M. Hindry, “Why is it difficult to compute the Mordell–Weil group?”, pp. 197–219 in Diophantine geometry, edited by U. Zannier, CRM Series 4, Ed. Norm., Pisa, 2007.
• M. Hindry and A. Pacheco, “An analogue of the Brauer–Siegel theorem for abelian varieties in positive characteristic”, Mosc. Math. J. 16:1 (2016), 45–93.
• M. Hindry and J. H. Silverman, Diophantine geometry, Graduate Texts in Mathematics 201, Springer, 2000.
• K. Kato and F. Trihan, “On the conjectures of Birch and Swinnerton-Dyer in characteristic $p>0$”, Invent. Math. 153:3 (2003), 537–592.
• N. M. Katz, “On the ubiquity of “pathology” in products”, pp. 139–153 in Arithmetic and geometry, I, edited by M. Artin and J. Tate, Progr. Math. 35, Birkhäuser, Boston, 1983.
• N. Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, 2nd ed., Graduate Texts in Mathematics 58, Springer, 1984.
• S. Lang and A. Néron, “Rational points of abelian varieties over function fields”, Amer. J. Math. 81 (1959), 95–118.
• J. S. Milne, “On a conjecture of Artin and Tate”, Ann. of Math. $(2)$ 102:3 (1975), 517–533.
• J. S. Milne, Étale cohomology, Princeton Mathematical Series 33, Princeton University Press, 1980.
• L. Moret-Bailly, “Pinceaux de variétés abéliennes”, Astérisque 129 (1985), 266.
• A. Néron, “Quasi-fonctions et hauteurs sur les variétés abéliennes”, Ann. of Math. $(2)$ 82 (1965), 249–331.
• A. P. Ogg, “Elliptic curves and wild ramification”, Amer. J. Math. 89 (1967), 1–21.
• T. Saito, “Conductor, discriminant, and the Noether formula of arithmetic surfaces”, Duke Math. J. 57:1 (1988), 151–173.
• J.-P. Serre, Algebraic groups and class fields, Graduate Texts in Mathematics 117, Springer, 1988.
• T. Shioda, “An explicit algorithm for computing the Picard number of certain algebraic surfaces”, Amer. J. Math. 108:2 (1986), 415–432.
• T. Shioda and T. Katsura, “On Fermat varieties”, Tôhoku Math. J. $(2)$ 31:1 (1979), 97–115.
• J. Tate, “Endomorphisms of abelian varieties over finite fields”, Invent. Math. 2 (1966), 134–144.
• D. L. Ulmer, “$p$-descent in characteristic $p$”, Duke Math. J. 62:2 (1991), 237–265.
• D. Ulmer, “Elliptic curves with large rank over function fields”, Ann. of Math. $(2)$ 155:1 (2002), 295–315.
• D. Ulmer, “$L$-functions with large analytic rank and abelian varieties with large algebraic rank over function fields”, Invent. Math. 167:2 (2007), 379–408.
• D. Ulmer, “Elliptic curves over function fields”, pp. 211–280 in Arithmetic of $L$-functions, edited by C. Popescu et al., IAS/Park City Math. Ser. 18, Amer. Math. Soc., Providence, RI, 2011.
• D. Ulmer, “On Mordell–Weil groups of Jacobians over function fields”, J. Inst. Math. Jussieu 12:1 (2013), 1–29.
• D. Ulmer, “Curves and Jacobians over function fields”, pp. 283–337 in Arithmetic geometry over global function fields, edited by F. Bars et al., Springer, 2014.
• D. Ulmer, “Explicit points on the Legendre curve”, J. Number Theory 136 (2014), 165–194.
• D. Ulmer, “Explicit points on the Legendre curve III”, Algebra Number Theory 8:10 (2014), 2471–2522.
• A. Weil, “Remarques sur un mémoire d'Hermite”, Arch. Math. $($Basel$)$ 5 (1954), 197–202.
• A. Weil, Adeles and algebraic groups, Progress in Mathematics 23, Birkhäuser, Boston, 1982.