A heuristic for boundedness of ranks of elliptic curves. (English) Zbl 1469.11173

Park, Jennifer; Poonen, Bjorn; Voight, John; Wood, Melanie Matchett

J. Eur. Math. Soc. (JEMS) 21, No. 9, 2859-2903 (2019).

Summary: We present a heuristic that suggests that ranks of elliptic curves \(E\) over \(\mathbb{Q}\) are bounded. In fact, it suggests that there are only finitely many \(E\) of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.

MSC:

11G05	Elliptic curves over global fields
11G40	\(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11P21	Lattice points in specified regions
14G25	Global ground fields in algebraic geometry

Keywords:

elliptic curve; rank; Shafarevich-Tate group

Software:

Magma; ecdata

References:

[1]	Batyrev, V. V., Manin, Yu. I.: Sur le nombre des points rationnels de hauteur borné des variétés algébriques. Math. Ann. 286, 27-43 (1990)Zbl 0679.14008 MR 1032922 · Zbl 0679.14008
[2]	Bektemirov, B., Mazur, B., Stein, W., Watkins, M.: Average ranks of elliptic curves: tension between data and conjecture. Bull. Amer. Math. Soc. (N.S.) 44, 233-254 (2007)Zbl 1190.11032 MR 2291676 · Zbl 1190.11032
[3]	Bhargava, M., Kane, D. M., Lenstra, H. W., Jr., Poonen, B., Rains, E.: Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves. Cambridge J. Math. 3, 275-321 (2015)Zbl 1329.14071 MR 3393023 · Zbl 1329.14071
[4]	Bhargava, M., Skinner, C., Zhang, W.: A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture.arXiv:1407.1826v2(2014)
[5]	Brumer, A.: The average rank of elliptic curves. I. Invent. Math. 109, 445-472 (1992) Zbl 0783.14019 MR 1176198 · Zbl 0783.14019
[6]	Brumer, A., McGuinness, O.: The behavior of the Mordell-Weil group of elliptic curves. Bull. Amer. Math. Soc. (N.S.) 23, 375-382 (1990)Zbl 0741.14010 MR 1044170 · Zbl 0741.14010
[7]	Cassels, J. W. S.: Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung. J. Reine Angew. Math. 211, 95-112 (1962)Zbl 106.03706 MR 0163915 · Zbl 0106.03706
[8]	Cassels, J. W. S.: Diophantine equations with special reference to elliptic curves. J. London Math. Soc. 41, 193-291 (1966)Zbl 0138.27002 MR 0199150 · Zbl 0138.27002
[9]	Cohen, H., Lenstra, H. W., Jr.: Heuristics on class groups of number fields. In: Number Theory, Noordwijkerhout 1983, Lecture Notes in Math. 1068, Springer, Berlin, 33-62 (1984)Zbl 0558.12002 MR 756082 · Zbl 0558.12002
[10]	Cohen, H., Martinet, J.: Étude heuristique des groupes de classes des corps de nombres. J. Reine Angew. Math. 404, 39-76 (1990)Zbl 0699.12016 MR 1037430 · Zbl 0699.12016
[11]	Conrey, J. B., Ghosh, A.: Remarks on the generalized Lindelöf hypothesis. Funct. Approx. Comment. Math. 36, 71-78 (2006)Zbl 1196.11121 MR 2296639 · Zbl 1196.11121
[12]	Conrey, J. B., Keating, J. P., Rubinstein, M. O., Snaith, N. C.: On the frequency of vanishing of quadratic twists of modular L-functions. In: Number Theory for the Millennium, I (Urbana, IL, 2000), A K Peters, Natick, MA, 301-315 (2002) Zbl 1044.11035 MR 1956231 · Zbl 1044.11035
[13]	Conrey, J. B., Rubinstein, M. O., Snaith, N. C., Watkins, M.: Discretisation for odd quadratic twists. In: Ranks of Elliptic Curves and Random Matrix Theory, London Math. Soc. Lecture Note Ser. 341, Cambridge Univ. Press, Cambridge, 201-214 (2007)Zbl 1193.11064 MR 2322346 A heuristic for boundedness of ranks of elliptic curves2899 · Zbl 1193.11064
[14]	Cornut, C.: Mazur’s conjecture on higher Heegner points. Invent. Math. 148, 495-523 (2002)Zbl 111.11029 MR 1908058 · Zbl 1111.11029
[15]	Cremona, J. E.: Algorithms for Modular Elliptic Curves. 2nd ed., Cambridge Univ. Press, Cambridge (1997)Zbl 0872.14041 MR 1628193 · Zbl 0872.14041
[16]	D ˛abrowski, A., J˛edrzejak, T., Szymaszkiewicz, L.: Behaviour of the order of Tate- Shafarevich groups for the quadratic twists of (X0)(49). In: Elliptic Curves, Modular Forms and Iwasawa Theory, Springer Proc. Math. Statist. 188, Springer, Cham, 125- 159 (2016)Zbl 1398.11087 MR 3629650 · Zbl 1398.11087
[17]	Delaunay, C.: Heuristics on Tate-Shafarevitch groups of elliptic curves defined over Q. Experiment. Math. 10, 191-196 (2001)Zbl 045.11038 MR 1837670 · Zbl 1045.11038
[18]	Delaunay, C.: Moments of the orders of Tate-Shafarevich groups. Int. J. Number Theory 1, 243-264 (2005)Zbl 1082.11042 MR 2173383 · Zbl 1082.11042
[19]	Delaunay, C.: Heuristics on class groups and on Tate-Shafarevich groups: the magic of the Cohen-Lenstra heuristics. In: Ranks of Elliptic Curves and Random Matrix Theory, London Math. Soc. Lecture Note Ser. 341, Cambridge Univ. Press, Cambridge, 323-340 (2007)Zbl 1231.11129 MR 2322355 · Zbl 1231.11129
[20]	Delaunay, C.,Duquesne, S.: Numerical investigations related to the derivatives of the L-series of certain elliptic curves. Experiment. Math. 12, 311-317 (2003) Zbl 1083.11041 MR 2034395 · Zbl 1083.11041
[21]	Delaunay, C., Jouhet, F.: p‘-torsion points in finite abelian groups and combinatorial identities. Adv. Math. 258, 13-45 (2014)Zbl 1286.11087 MR 3190422 · Zbl 1286.11087
[22]	Delaunay, C., Roblot, X.-F.: Regulators of rank one quadratic twists. J. Théor. Nombres Bordeaux 20, 601-624 (2008)Zbl 1201.11057 MR 2523310 · Zbl 1201.11057
[23]	Delaunay, C.,Watkins, M.: The powers of logarithm for quadratic twists. In: Ranks of Elliptic Curves and Random Matrix Theory, London Math. Soc. Lecture Note Ser. 341, Cambridge Univ. Press, Cambridge, 189-193 (2007)Zbl 1213.11117 MR 2322344 · Zbl 1213.11117
[24]	Deninger, C.: The Hilbert-Pólya strategy and height pairings. In: Casimir Force, Casimir Operators and the Riemann Hypothesis, de Gruyter, Berlin, 275-283 (2010) Zbl 225.14018 MR 2777722 · Zbl 1225.14018
[25]	Dujella, A.: Infinite families of elliptic curves with high rank and prescribed torsion. http://web.math.pmf.unizg.hr/∼duje/tors/generic.html(2017)
[26]	Duke, W., Rudnick, Z., Sarnak, P.: Density of integer points on affine homogeneous varieties. Duke Math. J. 71, 143-179 (1993)Zbl 0798.11024 MR 1230289 · Zbl 0798.11024
[27]	Durrett, R.: Probability: Theory and Examples. 4th ed. Cambridge Ser. Statist. Probab. Math., Cambridge Univ. Press, Cambridge (2010)Zbl 1202.60001 MR 2722836 · Zbl 1202.60001
[28]	Elkies, N. D.: Curves Dy2=x3−xof odd analytic rank. In: Algorithmic Number Theory (Sydney, 2002), Lecture Notes in Computer Sci. 2369, Springer, Berlin, 244- 251 (2002)Zbl 1058.11034 MR 2041088 · Zbl 1058.11034
[29]	Elkies, N. D.: Z28in E(Q), etc. Email to [e-mail] (2006)
[30]	Eskin, A., Katznelson, Y. R.: Singular symmetric matrices. Duke Math. J. 79, 515- 547 (1995)Zbl 0832.11036 MR 1344769 · Zbl 0832.11036
[31]	Farmer, D. W., Gonek, S. M., Hughes, C. P.: The maximum size of L-functions. J. Reine Angew. Math. 609, 215-236 (2007)Zbl 1234.11109 MR 2350784 · Zbl 1234.11109
[32]	Fouvry, É., Klüners, J.: On the 4-rank of class groups of quadratic number fields. Invent. Math. 167, 455-513 (2007)Zbl 1126.11062 MR 2276261 · Zbl 1126.11062
[33]	Friedman, E., Washington, L. C.: On the distribution of divisor class groups of curves over a finite field. In: Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 227-239 (1989)Zbl 0693.12013 MR 1024565 2900Jennifer Park et al. · Zbl 0693.12013
[34]	Furstenberg, H., Katznelson, Y.: A density version of the Hales-Jewett theorem. J. Anal. Math. 57, 64-119 (1991)Zbl 0770.05097 MR 1191743 · Zbl 0770.05097
[35]	Gerth, F., III: Extension of conjectures of Cohen and Lenstra. Exposition. Math. 5, 181-184 (1987)Zbl 0613.12003 MR 887792 · Zbl 0613.12003
[36]	Goldfeld, D.: Conjectures on elliptic curves over quadratic fields. In: Number Theory, Carbondale 1979, Lecture Notes in Math. 751, Springer, Berlin, 108-118 (1979) Zbl 0417.14031 MR 564926 · Zbl 0417.14031
[37]	Goldfeld, D., Szpiro, L.: Bounds for the order of the Tate-Shafarevich group. Compos. Math. 97, 71-87 (1995)Zbl 860.11032 MR 1355118 · Zbl 0860.11032
[38]	Granville, A.: Harald Cramér and the distribution of prime numbers. Scand. Actuar. J. 1995, 12-28Zbl 0833.01018 MR 1349149 · Zbl 0833.01018
[39]	Granville, A.: ABC allows us to count squarefrees. Int. Math. Res. Notices 1998, 991-1009Zbl 0924.11018 MR 1654759 · Zbl 0924.11018
[40]	Gross, B. H.: Heights and the special values of L-series. In: Number Theory (Montreal, Que., 1985), CMS Conf. Proc. 7, Amer. Math. Soc., Providence, RI, 115- 187 (1987)Zbl 0623.10019 MR 894322 · Zbl 0623.10019
[41]	Hall, P.: A partition formula connected with Abelian groups. Comment. Math. Helv. 11, 126-129 (1938)Zbl 0019.39705 MR 1509594 · JFM 64.0060.04
[42]	Harron, R., Snowden, A.: Counting elliptic curves with prescribed torsion. J. Reine Angew. Math. 729, 151-170 (2017)Zbl 1378.11067 MR 3680373 · Zbl 1378.11067
[43]	Hindry, M.: Why is it difficult to compute the Mordell-Weil group? In: Diophantine Geometry, CRM Ser. 4, Ed. Normale, Pisa, 197-219 (2007)Zbl 1219.11099 MR 2349656 · Zbl 1219.11099
[44]	Hindry, M., Pacheco, A.: An analogue of the Brauer-Siegel theorem for abelian varieties in positive characteristic. Moscow Math. J. 16, 45-93 (2016)Zbl 1382.11041 MR 3470576 · Zbl 1382.11041
[45]	Hindry, M., Silverman, J. H.: The canonical height and integral points on elliptic curves. Invent. Math. 93, 419-450 (1988)Zbl 0657.14018 MR 948108 · Zbl 0657.14018
[46]	Honda, T.: Isogenies, rational points and section points of group varieties. Japan. J. Math. 30, 84-101 (1960)Zbl 0109.39602 MR 0155828 · Zbl 0109.39602
[47]	Hortsch, R.: Counting elliptic curves of bounded Faltings height.arXiv:1505.05112v1 (2015) · Zbl 1355.11068
[48]	Iwaniec, H.,Sarnak, P.: Perspectives on the analytic theory of L-functions. In: GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal., Special Volume, 705-741 (2000) Zbl 0996.11036 MR 1826269 · Zbl 0996.11036
[49]	Katz, N. M.,Sarnak, P.: Random Matrices, Frobenius Eigenvalues, and Monodromy. Amer. Math. Soc. Colloq. Publ. 45, Amer. Math. Soc., Providence, RI (1999) Zbl 0958.11004 MR 1659828 · Zbl 0958.11004
[50]	Katz, N. M., Sarnak, P.: Zeroes of zeta functions and symmetry. Bull. Amer. Math. Soc. (N.S.) 36, 1-26 (1999)Zbl 0921.11047 MR 1640151 · Zbl 0921.11047
[51]	Keating, J. P., Snaith, N. C.: Random matrix theory and L-functions at s = 1/2. Comm. Math. Phys. 214, 91-110 (2000)Zbl 1051.11047 MR 1794267 · Zbl 1051.11047
[52]	Kohnen, W., Zagier, D.: Values of L-series of modular forms at the center of the critical strip. Invent. Math. 64, 175-198 (1981)Zbl 0468.10015 MR 629468 · Zbl 0468.10015
[53]	Kurchanov, P. F.: The zeta-function of elliptic curves over certain abelian extensions of imaginary quadratic fields. Mat. Sb. (N.S.) 102 (144), 56-70, 151 (1977) (in Russian)Zbl 0443.12004MR 0498391 · Zbl 0443.12004
[54]	Lang, S.: Conjectured Diophantine estimates on elliptic curves. In: Arithmetic and Geometry, Vol. I, Progr. Math. 35, Birkhäuser Boston, Boston, MA, 155-171 (1983) Zbl 0529.14017 MR 717593 A heuristic for boundedness of ranks of elliptic curves2901 · Zbl 0529.14017
[55]	Lang, S.: Old and new conjectured Diophantine inequalities. Bull. Amer. Math. Soc. (N.S.) 23, 37-75 (1990)Zbl 0714.11034 MR 1005184 · Zbl 0714.11034
[56]	Lang, S.: Algebraic Number Theory. 2nd ed. Grad. Texts in Math. 110, Springer, New York (1994)Zbl 0811.11001 MR 1282723 · Zbl 0811.11001
[57]	Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235-265 (1997)Zbl 0898.68039 MR 1484478 · Zbl 0898.68039
[58]	Mazur, B.: Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. 47, 33-186 (1978) (1977)Zbl 0394.14008 MR 488287 · Zbl 0394.14008
[59]	Mazur, B.: Modular curves and arithmetic. In: Proc. International Congress of Mathematicians (Warszawa, 1983), Vol. 1, PWN, Warszawa, 185-211 (1984) Zbl 0597.14023 MR 804682 · Zbl 0597.14023
[60]	Merel, L.: Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124, 437-449 (1996)Zbl 0936.11037 MR 1369424 · Zbl 0936.11037
[61]	Mestre, J.-F.: Construction d’une courbe elliptique de rang ≥ 12. C. R. Acad. Sci. Paris Sér. I Math. 295, 643-644 (1982)Zbl 0541.14027 MR 688896 · Zbl 0541.14027
[62]	Mestre, J.-F.: Formules explicites et minorations de conducteurs de variétés algébriques. Compos. Math. 58, 209-232 (1986)Zbl 0607.14012 MR 844410 · Zbl 0607.14012
[63]	Mordell, L. J.: On the rational solutions of the indeterminate equations of the third and fourth degrees. Proc. Cambridge Philos. Soc. 21, 179-192 (1922) · JFM 48.0140.03
[64]	Néron, A.: Problèmes arithmétiques et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps. Bull. Soc. Math. France 80, 101-166 (1952) Zbl 0049.30803 MR 0056951 · Zbl 0049.30803
[65]	Néron, A.: Propriétés arithmétiques de certaines familles de courbes algébriques. In: Proc. International Congress of Mathematicians (Amsterdam, 1954), Vol. III, Noordhoff, Groningen, and North-Holland, Amsterdam, 481-488 (1956)Zbl 0074.15901 MR 0087210 · Zbl 0074.15901
[66]	Ono, K., Skinner, C.: Non-vanishing of quadratic twists of modular L-functions. Invent. Math. 134, 651-660 (1998)Zbl 0937.11017 MR 1660945 · Zbl 0937.11017
[67]	Poincaré, H.: Sur les propriétés arithmétiques des courbes algébriques. J. Math. Pures Appl. (5) 7, 161-234 (1901)JFM 32.0564.06 · JFM 32.0564.06
[68]	Poincaré, H.: Œuvres d’Henri Poincaré, Volume 5. Gauthier-Villars, Paris (1950) Zbl 0041.37403
[69]	Polymath, D. H. J.: A new proof of the density Hales-Jewett theorem. Ann. of Math. (2) 175, 1283-1327 (2012)Zbl 1267.11010 MR 2912706 · Zbl 1267.11010
[70]	Poonen, B.: Squarefree values of multivariable polynomials. Duke Math. J. 118, 353-373 (2003)Zbl 1047.11021 MR 1980998 · Zbl 1047.11021
[71]	Poonen, B., Rains, E.: Random maximal isotropic subspaces and Selmer groups. J. Amer. Math. Soc. 25, 245-269 (2012)Zbl 1294.11097 MR 2833483 · Zbl 1294.11097
[72]	Poonen, B., Stoll, M.: The Cassels-Tate pairing on polarized abelian varieties. Ann. of Math. (2) 150, 1109-1149 (1999)Zbl 1024.11040 MR 1740984 · Zbl 1024.11040
[73]	Rubin, K., Silverberg, A.: Ranks of elliptic curves in families of quadratic twists. Experiment. Math. 9, 583-590 (2000)Zbl 0959.11023 MR 1806293 · Zbl 0959.11023
[74]	Rubin, K., Silverberg, A.: Rank frequencies for quadratic twists of elliptic curves. Experiment. Math. 10, 559-569 (2001)Zbl 1035.11025 MR 1881757 · Zbl 1035.11025
[75]	Rubin, K., Silverberg, A.: Ranks of elliptic curves. Bull. Amer. Math. Soc. (N.S.) 39, 455-474 (2002)Zbl 1052.11039 MR 1920278 · Zbl 1052.11039
[76]	Sarnak, P.: Some Applications of Modular Forms. Cambridge Tracts in Math. 99, Cambridge Univ. Press, Cambridge (1990)Zbl 1141.11023 MR 1102679 2902Jennifer Park et al. · Zbl 0721.11015
[77]	Schmidt, W. M.: Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height. Duke Math. J. 35, 327-339 (1968)Zbl 0172.06304 MR 0224562 · Zbl 0172.06304
[78]	Serre, J.-P.: Lectures on the Mordell-Weil Theorem. 3rd ed., Aspects of Math., Vieweg, Braunschweig (1997)Zbl 0863.14013 MR 1757192 · Zbl 0863.14013
[79]	Shioda, T.: Some remarks on elliptic curves over function fields. In: Journées Arithmétiques (Genève, 1991), Astérisque 209, 12, 99-114 (1992)Zbl 0820.14016 MR 1211006 · Zbl 0820.14016
[80]	Silverman, J. H.: Lower bound for the canonical height on elliptic curves. Duke Math. J. 48, 633-648 (1981)Zbl 0475.14033 MR 630588 · Zbl 0475.14033
[81]	Silverman, J. H.: Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math. 342, 197-211 (1983)Zbl 0505.14035 MR 703488 · Zbl 0505.14035
[82]	Silverman, J. H.: Heights and elliptic curves. In: Arithmetic Geometry (Storrs, CN, 1984), Springer, New York, 253-265 (1986)Zbl 0603.14020 MR 861979 · Zbl 0603.14020
[83]	Silverman, J. H.: The Arithmetic of Elliptic Curves. Grad. Texts in Math. 106, Springer, New York (1992)MR 1329092
[84]	Silverman, J. H.: The Arithmetic of Elliptic Curves. 2nd ed. Grad. Texts in Math. 106, Springer, Dordrecht (2009)Zbl 1194.11005 MR 2514094 · Zbl 1194.11005
[85]	Stein, W., Wuthrich, C.: Algorithms for the arithmetic of elliptic curves using Iwasawa theory. Math. Comp. 82, 1757-1792 (2013)Zbl 1336.11047 MR 3042584 · Zbl 1336.11047
[86]	Stewart, C. L., Top, J.: On ranks of twists of elliptic curves and power-free values of binary forms. J. Amer. Math. Soc. 8, 943-973 (1995)Zbl 0857.11026 MR 1290234 · Zbl 0857.11026
[87]	Tate, J. T.: The arithmetic of elliptic curves. Invent. Math. 23, 179-206 (1974) Zbl 0296.14018 MR 0419359 · Zbl 0296.14018
[88]	Tate, J.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. In: Séminaire Bourbaki, Vol. 9, exp. 306, Soc. Math. France, Paris, 415-440 (1995) Zbl 0199.55604 MR 1610977
[89]	Tate, J., Shafarevich, I. R.: The rank of elliptic curves. Dokl. Akad. Nauk SSSR 175, 770-773 (1967) (in Russian)Zbl 0168.42201 MR 0237508 · Zbl 0168.42201
[90]	Ulmer, D.: Elliptic curves with large rank over function fields. Ann. of Math. (2) 155, 295-315 (2002)Zbl 1109.11314 MR 1888802 · Zbl 1109.11314
[91]	Vatsal, V.: Special values of anticyclotomic L-functions. Duke Math. J. 116, 219-261 (2003)Zbl 1065.11048 MR 1953292 · Zbl 1065.11048
[92]	Venkatesh, A., Ellenberg, J. S.: Statistics of number fields and function fields. In: Proc. International Congress of Mathematicians (Allahabad, 2010), Vol. II, Hindustan Book Agency, New Delhi, 383-402 (2010)Zbl 1259.11106 MR 2827801 · Zbl 1259.11106
[93]	Waldspurger, J.-L.: Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl. (9) 60, 375-484 (1981)Zbl 0431.10015 MR 646366 · Zbl 0431.10015
[94]	Wang, Y., Stanley, R. P.: The Smith normal form distribution of a random integer matrix. SIAM J. Discrete Math. 31, 2247-2268 (2017)Zbl 1372.05006 MR 3706911 · Zbl 1372.05006
[95]	Watkins, M.: Some heuristics about elliptic curves. Experiment. Math. 17, 105-125 (2008)Zbl 1151.14025 MR 2410120 · Zbl 1151.14025
[96]	Watkins, M.: On elliptic curves and random matrix theory. J. Théor. Nombres Bordeaux 20, 829-845 (2008)Zbl 1193.11058 MR 2523320 · Zbl 1193.11058
[97]	Watkins, M.: A discursus on 21 as a bound for ranks of elliptic curves over Q, and sundry related topics.http://magma.maths.usyd.edu.au/∼watkins/papers/ DISCURSUS.pdf(2015) A heuristic for boundedness of ranks of elliptic curves2903 [Wat+14]Watkins, M., Donnelly, S., Elkies, N. D., Fisher, T., Granville, A., Rogers, N. F.: Ranks of quadratic twists of elliptic curves. Publ. Math. Besançon 2014/2, 63-98 (2014)Zbl 1365.11070 MR 3381037 · Zbl 1365.11070
[98]	de Weger, B. M. M.: A + B = C and big X’s. Quart. J. Math. Oxford Ser. (2) 49, 105-128 (1998)Zbl 0917.11020 MR 1617347 · Zbl 0917.11020
[99]	Wiles, A.: The Birch and Swinnerton-Dyer conjecture. In: The Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, 31-41 (2006)Zbl 1194.11006 MR 2238272 · Zbl 1194.11006
[100]	Wong, S.: On the density of elliptic curves. Compos. Math. 127, 23-54 (2001) Zbl 1003.11023 MR 1832985 · Zbl 1003.11023
[101]	Wood, M. M.: The distribution of sandpile groups of random graphs. J. Amer. Math. Soc. 30, 915-958 (2017)Zbl 1366.05098 MR 3671933 · Zbl 1366.05098
[102]	Wood, M. M.: Random integral matrices and the Cohen-Lenstra heuristics. Amer. J. Math. 141, 383-398 (2019)Zbl 07057016 MR 3928040 · Zbl 1446.11170
[103]	Young, M. P.: On the non-vanishing of elliptic curve L-functions at the central point. Proc. London Math. Soc. (3) 93, 1-42 (2006)Zbl 1174.11052 MR 2235480 · Zbl 1174.11052