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Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields

About this Title

Lisa Berger, Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA, Chris Hall, Department of Mathematics, Western University, London, Ontario, N6A 5B7, Canada, René Pannekoek, Department of Mathematics, Imperial College London SW7 2AZ, United Kingdom, Jennifer Park, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, Rachel Pries, Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA, Shahed Sharif, Department of Mathematics, CSU San Marcos, San Marcos, CA 92096, USA, Alice Silverberg, Department of Mathematics, UC Irvine, Irvine, CA 92697, USA and Douglas Ulmer, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

Publication: Memoirs of the American Mathematical Society
Publication Year: 2020; Volume 266, Number 1295
ISBNs: 978-1-4704-4219-4 (print); 978-1-4704-6253-6 (online)
DOI: https://doi.org/10.1090/memo/1295
Published electronically: July 22, 2022
Keywords: Curve, function field, Jacobian, abelian variety, finite field, Mordell-Weil group, torsion, rank, $L$-function, Birch and Swinnerton-Dyer conjecture, Tate-Shafarevich group, Tamagawa number, endomorphism algebra, descent, height, NĂ©ron model, Kodaira-Spencer map, monodromy
MSC: Primary 11G10, 11G30; Secondary 11G40, 14G05, 14G25, 14K15

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Table of Contents

Chapters

  • Introduction
  • 1. The curve, explicit divisors, and relations
  • 2. Descent calculations
  • 3. Minimal regular model, local invariants, and domination by a product of curves
  • 4. Heights and the visible subgroup
  • 5. The $L$-function and the BSD conjecture
  • 6. Analysis of $J[p]$ and $\operatorname {NS}(\mathcal {X}_d)_{\mathrm {tor}}$
  • 7. Index of the visible subgroup and the Tate-Shafarevich group
  • 8. Monodromy of $\ell$-torsion and decomposition of the Jacobian
  • A. An additional hyperelliptic family

Abstract

We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $\mathbb {F}_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, we compute the $L$-function of $J$ over $\mathbb {F}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb {F}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^\nu +1$, and $K_d := \mathbb {F}_p(\mu _d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the Tate-Shafarevich group of $J$ over $K_d$ is $[J(K_d):V]^2$. When $r>2$, we prove that the “new” part of $J$ is isogenous over $\bar {\mathbb {F}_p(t)}$ to the square of a simple abelian variety of dimension $\phi (r)/2$ with endomorphism algebra $\mathbb {Z} [\mu _r]^+$. For a prime $\ell$ with $\ell \nmid pr$, we prove that $J[\ell ](L)=\{0\}$ for any abelian extension $L$ of $\bar {\mathbb {F}}_p(t)$.

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References
  • Ahmed Abbes, RĂ©duction semi-stable des courbes d’aprĂšs Artin, Deligne, Grothendieck, Mumford, Saito, Winters, $\ldots$, Courbes semi-stables et groupe fondamental en gĂ©omĂ©trie algĂ©brique (Luminy, 1998) Progr. Math., vol. 187, BirkhĂ€user, Basel, 2000, pp. 59–110 (French). MR 1768094
  • M. Artin, Coverings of the rational double points in characteristic $p$, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 11–22. MR 0450263
  • Lucian Bădescu, Algebraic surfaces, Universitext, Springer-Verlag, New York, 2001. Translated from the 1981 Romanian original by Vladimir MaƟek and revised by the author. MR 1805816
  • Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004.
  • Lisa Berger, Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields, J. Number Theory 128 (2008), no. 12, 3013–3030. MR 2464851, DOI 10.1016/j.jnt.2008.03.009
  • Siegfried Bosch, Werner LĂŒtkebohmert, and Michel Raynaud, NĂ©ron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822
  • Irene I. Bouw and Stefan Wewers, Computing $L$-functions and semistable reduction of superelliptic curves, Glasg. Math. J. 59 (2017), no. 1, 77–108. MR 3576328, DOI 10.1017/S0017089516000057
  • Nils Bruin, Bjorn Poonen, and Michael Stoll, Generalized explicit descent and its application to curves of genus 3, Forum Math. Sigma 4 (2016), Paper No. e6, 80. MR 3482281, DOI 10.1017/fms.2016.1
  • Henri Cohen, Number theory. Vol. I. Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, Springer, New York, 2007. MR 2312337
  • Ricardo P. Conceição, Chris Hall, and Douglas Ulmer, Explicit points on the Legendre curve II, Math. Res. Lett. 21 (2014), no. 2, 261–280. MR 3247055, DOI 10.4310/MRL.2014.v21.n2.a5
  • Brian Conrad, Chow’s $K/k$-image and $K/k$-trace, and the Lang-NĂ©ron theorem, Enseign. Math. (2) 52 (2006), no. 1-2, 37–108. MR 2255529
  • SchĂ©mas en groupes. I: PropriĂ©tĂ©s gĂ©nĂ©rales des schĂ©mas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois Marie 1962/64 (SGA 3); DirigĂ© par M. Demazure et A. Grothendieck. MR 0274458
  • W. J. Gordon, Linking the conjectures of Artin-Tate and Birch-Swinnerton-Dyer, Compositio Math. 38 (1979), no. 2, 163–199. MR 528839
  • Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
  • A. Grothendieck, ÉlĂ©ments de gĂ©omĂ©trie algĂ©brique. I. Le langage des schĂ©mas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
  • Alexander Grothendieck, GĂ©omĂ©trie formelle et gĂ©omĂ©trie algĂ©brique, SĂ©minaire Bourbaki, Vol. 5, Soc. Math. France, Paris, 1995, pp. Exp. No. 182, 193–220, errata p. 390 (French). MR 1603467
  • RevĂȘtements Ă©tales et groupe fondamental (SGA 1), Documents MathĂ©matiques (Paris) [Mathematical Documents (Paris)], vol. 3, SociĂ©tĂ© MathĂ©matique de France, Paris, 2003 (French). SĂ©minaire de gĂ©omĂ©trie algĂ©brique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61]; Directed by A. Grothendieck; With two papers by M. Raynaud; Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. MR 2017446
  • Chris Hall, Monodromy of some superelliptic curves, in preparation.
  • Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558
  • Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR 499562
  • Kazuya Kato and Fabien Trihan, On the conjectures of Birch and Swinnerton-Dyer in characteristic $p>0$, Invent. Math. 153 (2003), no. 3, 537–592. MR 2000469, DOI 10.1007/s00222-003-0299-2
  • Steven L. Kleiman, Relative duality for quasicoherent sheaves, Compositio Math. 41 (1980), no. 1, 39–60. MR 578050
  • Steven L. Kleiman, The Picard scheme, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 235–321. MR 2223410, DOI 10.1016/j.aim.2005.06.006
  • Anthony W. Knapp, Advanced algebra, Cornerstones, BirkhĂ€user Boston, Inc., Boston, MA, 2007. Along with a companion volume Basic algebra. MR 2360434
  • Kunihiko Kodaira, Complex manifolds and deformation of complex structures, Reprint of the 1986 English edition, Classics in Mathematics, Springer-Verlag, Berlin, 2005. Translated from the 1981 Japanese original by Kazuo Akao. MR 2109686
  • Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie ErnĂ©; Oxford Science Publications. MR 1917232
  • Qing Liu, Dino Lorenzini, and Michel Raynaud, NĂ©ron models, Lie algebras, and reduction of curves of genus one, Invent. Math. 157 (2004), no. 3, 455–518. [See corrigendum in: 3858404]. MR 2092767, DOI 10.1007/s00222-004-0342-y
  • J. S. Milne, On the arithmetic of abelian varieties, Invent. Math. 17 (1972), 177–190. MR 330174, DOI 10.1007/BF01425446
  • James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 167–212. MR 861976
  • Carl Pomerance and Douglas Ulmer, On balanced subgroups of the multiplicative group, Number theory and related fields, Springer Proc. Math. Stat., vol. 43, Springer, New York, 2013, pp. 253–270. MR 3081046, DOI 10.1007/978-1-4614-6642-0_{1}4
  • Bjorn Poonen and Edward F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math. 488 (1997), 141–188. MR 1465369
  • Rachel Pries and Douglas Ulmer, Arithmetic of abelian varieties in Artin-Schreier extensions, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8553–8595. MR 3551581, DOI 10.1090/S0002-9947-2016-06641-6
  • Kenneth A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804. MR 457455, DOI 10.2307/2373815
  • Takeshi Saito, Vanishing cycles and geometry of curves over a discrete valuation ring, Amer. J. Math. 109 (1987), no. 6, 1043–1085. MR 919003, DOI 10.2307/2374585
  • Jean-Pierre Serre, Sur la topologie des variĂ©tĂ©s algĂ©briques en caractĂ©ristique $p$, Symposium internacional de topologĂ­a algebraica International symposium on algebraic topology, Universidad Nacional AutĂłnoma de MĂ©xico and UNESCO, Mexico City, 1958, pp. 24–53 (French). MR 0098097
  • Jean-Pierre Serre, Facteurs locaux des fonctions zĂȘta des varietĂ©s algĂ©briques (dĂ©finitions et conjectures), SĂ©minaire Delange-Pisot-Poitou. 11e annĂ©e: 1969/70. ThĂ©orie des nombres. Fasc. 1: ExposĂ©s 1 Ă  15; Fasc. 2: ExposĂ©s 16 Ă  24, SecrĂ©tariat Math., Paris, 1970, pp. 15 (French). MR 3618526
  • Jean-Pierre Serre, Groupes algĂ©briques et corps de classes, Hermann, Paris, 1975 (French). DeuxiĂšme Ă©dition; Publication de l’Institut de MathĂ©matique de l’UniversitĂ© de Nancago, No. VII; ActualitĂ©s Scientifiques et Industrielles, No. 1264. MR 0466151
  • Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
  • Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
  • Tetsuji Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math. 108 (1986), no. 2, 415–432. MR 833362, DOI 10.2307/2374678
  • Tetsuji Shioda, Mordell-Weil lattices for higher genus fibration over a curve, New trends in algebraic geometry (Warwick, 1996) London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 359–373. MR 1714831, DOI 10.1017/CBO9780511721540.015
  • Tetsuji Shioda and Toshiyuki Katsura, On Fermat varieties, Tohoku Math. J. (2) 31 (1979), no. 1, 97–115. MR 526513, DOI 10.2748/tmj/1178229881
  • John Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, SĂ©minaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 306, 415–440. MR 1610977
  • D. T. TÚĭt and I. R. Ć afarevič, The rank of elliptic curves, Dokl. Akad. Nauk SSSR 175 (1967), 770–773 (Russian). MR 0237508
  • Douglas Ulmer, Elliptic curves over function fields, Arithmetic of $L$-functions, IAS/Park City Math. Ser., vol. 18, Amer. Math. Soc., Providence, RI, 2011, pp. 211–280. MR 2882692, DOI 10.1090/pcms/018/09
  • Douglas Ulmer, Elliptic curves with large rank over function fields, Ann. of Math. (2) 155 (2002), no. 1, 295–315. MR 1888802, DOI 10.2307/3062158
  • Douglas Ulmer, $L$-functions with large analytic rank and abelian varieties with large algebraic rank over function fields, Invent. Math. 167 (2007), no. 2, 379–408. MR 2270458, DOI 10.1007/s00222-006-0018-x
  • Douglas Ulmer, On Mordell-Weil groups of Jacobians over function fields, J. Inst. Math. Jussieu 12 (2013), no. 1, 1–29. MR 3001733, DOI 10.1017/S1474748012000618
  • Douglas Ulmer, Curves and Jacobians over function fields, Arithmetic geometry over global function fields, Adv. Courses Math. CRM Barcelona, BirkhĂ€user/Springer, Basel, 2014, pp. 283–337. MR 3586808
  • Douglas Ulmer, Explicit points on the Legendre curve, J. Number Theory 136 (2014), 165–194. MR 3145329, DOI 10.1016/j.jnt.2013.09.010
  • Douglas Ulmer, Explicit points on the Legendre curve III, Algebra Number Theory 8 (2014), no. 10, 2471–2522. MR 3298546, DOI 10.2140/ant.2014.8.2471
  • Douglas Ulmer and JosĂ© Felipe Voloch, On the number of rational points on special families of curves over function fields, New Zealand J. Math. 47 (2017), 1–7. MR 3668802
  • Douglas Ulmer and Yuri G. Zarhin, Ranks of Jacobians in towers of function fields, Math. Res. Lett. 17 (2010), no. 4, 637–645. MR 2661169, DOI 10.4310/MRL.2010.v17.n4.a5
  • Claire Voisin, Hodge theory and complex algebraic geometry. I, Reprint of the 2002 English edition, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2007. Translated from the French by Leila Schneps. MR 2451566
  • JosĂ© Felipe Voloch, Diophantine approximation on abelian varieties in characteristic $p$, Amer. J. Math. 117 (1995), no. 4, 1089–1095. MR 1342843, DOI 10.2307/2374961
  • Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575
  • AndrĂ© Weil, Adeles and algebraic groups, Progress in Mathematics, vol. 23, BirkhĂ€user, Boston, Mass., 1982. With appendices by M. Demazure and Takashi Ono. MR 670072
  • Robert A. Wilson, The finite simple groups, Graduate Texts in Mathematics, vol. 251, Springer-Verlag London, Ltd., London, 2009. MR 2562037