Lowlying zeros of families of elliptic curves
Author:
Matthew P. Young
Journal:
J. Amer. Math. Soc. 19 (2006), 205250
MSC (2000):
Primary 11M41, 11F30, 11G05, 11G40, 11L20, 11L40
Published electronically:
September 7, 2005
MathSciNet review:
2169047
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Abstract: There is a growing body of work giving strong evidence that zeros of families of functions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the KatzSarnak philosophy. The random matrix model makes predictions for the average distribution of zeros near the central point for families of functions. We study these lowlying zeros for families of elliptic curve functions. For these functions there is special arithmetic interest in any zeros at the central point (by the conjecture of Birch and SwinnertonDyer and the impressive partial results towards resolving the conjecture). We calculate the density of the lowlying zeros for various families of elliptic curves. Our main foci are the family of all elliptic curves and a large family with positive rank. An important challenge has been to obtain results with test functions that are concentrated close to the origin since the central point is a location of great arithmetical interest. An application of our results is an improvement on the upper bound of the average rank of the family of all elliptic curves (conditional on the Generalized Riemann Hypothesis (GRH)). The upper bound obtained is less than , which shows that a positive proportion of curves in the family have algebraic rank equal to analytic rank and finite TateShafarevich group. We show that there is an extra contribution to the density of the lowlying zeros from the family with positive rank (presumably from the ``extra" zero at the central point).
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M. Young, Lowlying zeros of families of elliptic curves, http://arxiv.org/abs/math.NT/ 0406330.
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M. Young, Random matrix theory and families of elliptic curves, Ph.D. thesis, Rutgers University, 2004.
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 C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over : wild adic exercises, J. Amer. Math. Soc. 14(4), 843939 (2001). MR 1839918 (2002d:11058)
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 A. Brumer, The average rank of elliptic curves. I, Invent. Math. 109(3), 445472 (1992). MR 1176198 (93g:11057)
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 A. Brumer, J. Silverman, The number of elliptic curves over with conductor , Manuscripta Math. 91, 95102 (1996). MR 1404420 (97e:11062)
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 J. B. Conrey, Lfunctions and random matrices, Mathematics unlimited2001 and beyond, Springer, Berlin, 331352 (2001). MR 1852163 (2002g:11134)
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 W. Duke and E. Kowalski, A problem of Linnik for elliptic curves and meanvalue estimates for automorphic representations, with an appendix by Dinakar Ramakrishnan. Invent. Math. 139(1), 139 (2000). MR 1728875 (2001b:11034)
 [FNT]
 E. Fouvry, M. Nair, and G. Tenenbaum, L'ensemble exceptionnel dans la conjecture de Szpiro, Bull. Soc. Math. France 120(4), 485506 (1992).MR 1194273 (94a:11076)
 [GZ]
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 [HB]
 D. R. HeathBrown, The average analytic rank of elliptic curves, Duke Math J. 122 (2004), no. 3, 591623. MR2057019 (2004m:11084)
 [HBP]
 D.R. HeathBrown and S.J. Patterson, The distribution of Kummer sums at prime arguments, J. Reine und Angew. Math., 310, 111136 (1979). MR 0546667 (81e:10033)
 [He1]
 H. Helfgott, On the behavior of root numbers in families of elliptic curves, http://www. arxiv.org/abs/math.NT/0408141.
 [He2]
 H. Helfgott, The parity problem for reducible cubic forms, http://www.arxiv.org/abs/ math.NT/0408142.
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 H. Iwaniec and E. Kowalski, Analytic Number Theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004. MR 2061214 (2005h:11005)
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 H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of families of Lfunctions, Inst. Hautes Études Sci. Publ. Math. no. 91, 55131 (2001).MR 1828743 (2002h:11081)
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 N. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy. American Mathematical Society Colloquium Publications, 45. American Mathematical Society, Providence, RI, 1999. MR 1659828 (2000b:11070)
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 N. Katz and P. Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc., 36, 126 (1999). MR 1640151 (2000f:11114)
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 V. Kolyvagin, The MordellWeil and ShafarevichTate groups for Weil elliptic curves (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 11541180, 1327; translation in Math. USSRIzv. 33 (1989), no. 3, 473499. MR 0984214 (90f:11035)
 [KM1]
 E. Kowalski and P. Michel, The analytic rank of and zeros of automorphic functions, Duke Math. J. 100 (1999), no. 3, 503542. MR 1719730 (2001b:11060)
 [KM2]
 E. Kowalski and P. Michel, Explicit upper bound for the (analytic) rank of , Israel J. Math. 120 (2000), part A, 179204.MR 1815375 (2002e:11065)
 [Ku]
 D. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), no. 2, 193237. MR 0434947 (55:7910)
 [Mic]
 P. Michel, Rang moyen de familles de courbes elliptiques et lois de SatoTate, Monatsh. Math. 120(2), 127136 (1995).MR 1348365 (96j:11077)
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 S. J. Miller,  and level densities for families of elliptic curves: evidence for underlying group symmetries, Compositio Mathematics 104, 952992 (2004). MR 2059225 (2005c:11085)
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 K. Rubin and Alice Silverberg, Rank frequencies for quadratic twists of elliptic curves, Experiment. Math. 10(4), 559569 (2001).MR 1881757 (2002k:11081)
 [R]
 M. Rubinstein, Lowlying zeros of functions and random matrix theory, Duke Math. J. 109(1), 147181 (2001). MR 1844208 (2002f:11114)
 [Sch]
 W. Schmidt, Equations over Finite Fields, an Elementary Approach, SpringerVerlag, Berlin, 1976. MR 0429733 (55:2744)
 [Si1]
 J. Silverman, The Arithmetic of Elliptic Curves, SpringerVerlag, New York, 1986. MR 0817210 (87g:11070)
 [Si2]
 J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, SpringerVerlag, New York, 1994. MR 1312368 (96b:11074)
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 [ST]
 C. L. Stewart and J. Top, On ranks of twists of elliptic curves and powerfree values of binary forms, J. Amer. Math. Soc. 8(4), 943973 (1995).MR 1290234 (95m:11055)
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 [W]
 A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. Math (2) 141(3), 443551 (1995). MR 1333035 (96d:11071)
 [Y1]
 M. Young, Lowlying zeros of families of elliptic curves, http://arxiv.org/abs/math.NT/ 0406330.
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Additional Information
Matthew P. Young
Affiliation:
American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 943062244
Email:
myoung@aimath.org
DOI:
http://dx.doi.org/10.1090/S0894034705005035
PII:
S 08940347(05)005035
Received by editor(s):
April 6, 2005
Published electronically:
September 7, 2005
Article copyright:
© Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
