Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T04:56:19.678Z Has data issue: false hasContentIssue false
  • English
  • Français

Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one

Part of: Number theory Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 November 2011

Robert L. Miller
Robert L. Miller*
Affiliation:
Warwick Mathematics Institute Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom The Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley CA 94720-5070, USA

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we rigorously prove the formula for 16714 of the 16725 such curves of conductor less than 5000.

MSC classification

Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture) 11-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 327 - 350
Copyright
Copyright © London Mathematical Society 2011

References

[1]Agashe, A., Ribet, K. and Stein, W. A., ‘The Manin constant’, Pure Appl. Math. Q. 2 (2006) no. 2, 617–636 part 2; MR 2251484(2007c:11076).CrossRefGoogle Scholar
[2]Agashe, A. and Stein, W., ‘Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero’, Math. Comp. 74 (2005) no. 249, 455–484.CrossRefGoogle Scholar
[3]Belabas, K., Cohen, H.et al., PARI/GP, The Bordeaux Group, http://pari.math.u-bordeaux.fr/.Google Scholar
[4]Birch, B. J. and Swinnerton-Dyer, H. P. F., ‘Notes on elliptic curves. I’, J. Reine Angew. Math. 212 (1963) 7–25.CrossRefGoogle Scholar
[5]Birch, B. J. and Swinnerton-Dyer, H. P. F., ‘Notes on elliptic curves. II’, J. Reine Angew. Math. 218 (1965) 79–108.CrossRefGoogle Scholar
[6]Breuil, C., Conrad, B., Diamond, F. and Taylor, R., ‘On the modularity of elliptic curves over ℚ: wild 3-adic exercises’, J. Amer. Math. Soc. 14 (2001) no. 4, 843–939.CrossRefGoogle Scholar
[7]Buhler, J. P., Gross, B. H. and Zagier, D. B., ‘On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3’, Math. Comp. 44 (1985) no. 170, 473–481; MR 777279(86g:11037).CrossRefGoogle Scholar
[8]Bump, D., Friedberg, S. and Hoffstein, J., ‘Non-vanishing theorems for L-functions of modular forms and their derivatives’, Invent. Math. 102 (1990) no. 3, 543–618.CrossRefGoogle Scholar
[9]Cannon, J., Steele, A.et al., MAGMA Computational Algebra System, The University of Sydney, http://magma.maths.usyd.edu.au/magma/.Google Scholar
[10]Cassels, J. W. S., ‘Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer’, J. Reine Angew. Math. 217 (1965) 180–199.CrossRefGoogle Scholar
[11]Cha, B., ‘Vanishing of some cohomology groups and bounds for the Shafarevich–Tate groups of elliptic curves’, PhD Thesis, Johns Hopkins University, 2003.Google Scholar
[12]Cha, B., ‘Vanishing of some cohomology goups and bounds for the Shafarevich–Tate groups of elliptic curves’, J. Number Theory 111 (2005) 154–178.CrossRefGoogle Scholar
[13]Connell, I., Elliptic curve handbook, http://www.math.mcgill.ca/connell/public/ECH1, 1999.Google Scholar
[14]Cremona, J. E., Algorithms for modular elliptic curves, 2nd edn (Cambridge University Press, Cambridge, UK, 1997).Google Scholar
[15]Cremona, J., ‘The elliptic curve database for conductors to 130000’, Algorithmic number theory, Lecture Notes in Computer Science 4076 (Springer, Berlin, 2006) 11–29; MR 2282912(2007k:11087).CrossRefGoogle Scholar
[16]Cremona, J. E. and Fisher, T. A., ‘On the equivalence of binary quartics’, J. Symbolic Comput. 44 (2009) no. 6, 673–682.CrossRefGoogle Scholar
[17]Cremona, J. E. and Mazur, B., ‘Visualizing elements in the Shafarevich–Tate group’, Experiment. Math. 9 (2000) no. 1, 13–28.CrossRefGoogle Scholar
[18]Cremona, J. E., Prickett, M. and Siksek, S., ‘Height difference bounds for elliptic curves over number fields’, J. Number Theory 116 (2006) no. 1, 42–68.CrossRefGoogle Scholar
[19]Cremona, J. E. and Stoll, M., ‘Minimal models for 2-coverings of elliptic curves’, LMS J. Comput. Math. 5 (2002) 220–243 (electronic).Google Scholar
[20]Darmon, H., Rational points on modular elliptic curves, CBMS Regional Conference Series in Mathematics 101 (Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004).Google Scholar
[21]Edixhoven, B., ‘On the Manin constants of modular elliptic curves’, Arithmetic algebraic geometry (Texel, 1989) (Birkhäuser, Boston, MA, 1991) 25–39.CrossRefGoogle Scholar
[22]Fisher, T., ‘On 5 and 7 descents for elliptic curves’, PhD Thesis, University of Cambridge, 2000.Google Scholar
[23]Fisher, T., ‘Finding rational points on elliptic curves using 6-descent and 12-descent’, J. Algebra 320 (2008) no. 2, 853–884.CrossRefGoogle Scholar
[24]Flynn, E. V., Leprévost, F., Schaefer, E. F., Stein, W. A., Stoll, M. and Wetherell, J. L., ‘Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves’, Math. Comp. 70 (2001) no. 236, 1675–1697; MR 1836926(2002d:11072)(electronic).CrossRefGoogle Scholar
[25]Greenberg, R. and Vatsal, V., ‘On the Iwasawa invariants of elliptic curves’, Invent. Math. 142 (2000) no. 1, 17–63; MR 1784796(2001g:11169).CrossRefGoogle Scholar
[26]Grigorov, G., ‘Kato’s Euler system and the main conjecture’, PhD Thesis, Harvard University, 2005.Google Scholar
[27]Grigorov, G., Jorza, A., Patrikis, S., Stein, W. and Tarniţǎ, C., ‘Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves’, Math. Comp. 78 (2009) 2397–2425.CrossRefGoogle Scholar
[28]Gross, B. H., ‘Kolyvagin’s work on modular elliptic curves’, L-functions and arithmetic (Durham, 1989), London Mathematical Society Lecture Note Series 153 (Cambridge University Press, Cambridge, UK, 1991) 235–256.CrossRefGoogle Scholar
[29]Gross, B. and Zagier, D., ‘Heegner points and derivatives of L-series’, Invent. Math. 84 (1986) no. 2, 225–320.CrossRefGoogle Scholar
[30]Hochschild, G. and Serre, J.-P., ‘Cohomology of group extensions’, Trans. Amer. Math. Soc. 74 (1953) 110–134.CrossRefGoogle Scholar
[31]Jetchev, D., ‘Global divisibility of Heegner points and Tamagawa numbers’, Compos. Math. 144 (2008) no. 4, 811–826.CrossRefGoogle Scholar
[32]Jones, J. W., ‘Iwasawa L-functions for multiplicative abelian varieties’, Duke Math. J. 59 (1989) no. 2, 399–420; MR 1016896(90m:11094).CrossRefGoogle Scholar
[33]Kato, K., ‘p-adic Hodge theory and values of zeta functions of modular forms’, Astérisque 295 (2004) 117–290 ix.Google Scholar
[34]Kolyvagin, V. A., ‘Euler systems’, The Grothendieck festschrift, vol. II, Progress in Mathematics 87 (Birkhäuser, Boston, MA, 1990) 435–483.Google Scholar
[35]Lang, S., Number theory. III, vol. 60 (Springer, 1991).CrossRefGoogle Scholar
[36]Matsuno, K., ‘Finite Λ-submodules of Selmer groups of abelian varieties over cyclotomic ℤp-extensions’, J. Number Theory 99 (2003) no. 2, 415–443; MR 1969183(2004c:11098).CrossRefGoogle Scholar
[37]Mazur, B., Tate, J. and Teitelbaum, J., ‘On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer’, Invent. Math. 84 (1986) no. 1, 1–48; MR 830037(87e:11076).CrossRefGoogle Scholar
[38]Merriman, J. R., Siksek, S. and Smart, N. P., ‘Explicit 4-descents on an elliptic curve’, Acta Arith. 77 (1996) no. 4, 385–404.CrossRefGoogle Scholar
[39]Miller, R. L. and Stoll, M., Explicit isogeny descent on elliptic curves, http://arxiv.org/abs/1010.3334, 2010.Google Scholar
[40]Milne, J. S., Arithmetic duality theorems, second edn (BookSurge, Charleston, SC, 2006).Google Scholar
[41]Murty, M. R. and Murty, V. K., ‘Mean values of derivatives of modular L-series’, Ann. of Math. (2) 133 (1991) no. 3, 447–475.CrossRefGoogle Scholar
[42]Razar, M. J., ‘The non-vanishing of L(1) for certain elliptic curves with no first descents’, Amer. J. Math. 96 (1974) 104–126; MR 0360596(50#13044a).CrossRefGoogle Scholar
[43]Razar, M. J., ‘A relation between the two-component of the Tate–Šafarevič group and L(1) for certain elliptic curves’, Amer. J. Math. 96 (1974) 127–144; MR 0360597(50#13044b).CrossRefGoogle Scholar
[44]Rubin, K., ‘Congruences for special values of L-functions of elliptic curves with complex multiplication’, Invent. Math. 71 (1983) no. 2, 339–364.CrossRefGoogle Scholar
[45]Rubin, K., ‘The main conjectures of Iwasawa theory for imaginary quadratic fields’, Invent. Math. 103 (1991) no. 1, 25–68.CrossRefGoogle Scholar
[46]Schaefer, E. F. and Stoll, M., ‘How to do a p-descent on an elliptic curve’, Trans. Amer. Math. Soc. 356 (2004) 1209–1231.CrossRefGoogle Scholar
[47]Serf, P., ‘The rank of elliptic curves over real quadratic number fields of class number 1’, PhD Thesis, Universität des Saarlandes, 1995.Google Scholar
[48]Siksek, S., ‘Descents on curves of genus 1’, PhD Thesis, University of Exeter, 1995.Google Scholar
[49]Silverman, J. H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106 (Springer, New York, 1992).Google Scholar
[50]Silverman, J. H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151 (Springer, New York, 1994).CrossRefGoogle Scholar
[51]Skinner, E. and Urban, C., ‘The Iwasawa main conjectures for GL2’,http://www.math.columbia.edu/∼urban/eurp/MC.pdf.Google Scholar
[52]Stamminger, S., ‘Explicit 8-descent on elliptic curves’, PhD Thesis, International University Bremen, 2005.Google Scholar
[53]Stein, W., ‘Explicit approaches to modular abelian varieties’, PhD Thesis, University of California at Berkeley, 2000.Google Scholar
[54]Stein, W. and Wuthrich, C., ‘Algorithms for the arithmetic of elliptic curves using Iwasawa theory’, http://wstein.org/papers/shark, 2011.Google Scholar
[55]Stein, W.et al., Sage: Open Source Mathematical Software, The Sage Group,http://www.sagemath.org, 2010.Google Scholar
[56]Tate, J., On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki 9 (Société Mathématique de France, Paris, 1995) 415–440. Exp. No. 306.Google Scholar
[57]Waldspurger, J.-L., ‘Sur les valuers de certaines fonctions L automorphes en leur centre de symétrie’, Compositio Math. 54 (1985) no. 2, 173–242.Google Scholar
[58]Werner, A., ‘Local heights on abelian varieties with split multiplicative reduction’, Compositio Math. 107 (1997) no. 3, 289–317; MR 1458753(98c:14039).CrossRefGoogle Scholar
[59]Wiles, A. J., ‘Modular elliptic curves and Fermat’s last theorem’, Ann. of Math. (2) 2 (1995) no. 3, 443–551.CrossRefGoogle Scholar
[60]Womack, T., ‘Explicit descent on elliptic curves’, PhD Thesis, University of Nottingham, 2003.Google Scholar
[61]Woo, J., ‘Arithmetic of elliptic curves and surfaces: descents and quadratic sections’, PhD Thesis, Harvard University, 2010.Google Scholar
[62]Zhang, S.-W., ‘Gross–Zagier formula for GL(2) II’, Heegner points and Rankin L-series, Mathematical Sciences Research Institute Publications 49 (Cambridge University Press, Cambridge, 2004) 191–214.CrossRefGoogle Scholar