Low-lying zeros of families of elliptic curves

Author:
Matthew P. Young

Journal:
J. Amer. Math. Soc. **19** (2006), 205-250

MSC (2000):
Primary 11M41, 11F30, 11G05, 11G40, 11L20, 11L40

Published electronically:
September 7, 2005

MathSciNet review:
2169047

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 Katz-Sarnak philosophy. The random matrix model makes predictions for the average distribution of zeros near the central point for families of -functions. We study these low-lying 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 Swinnerton-Dyer and the impressive partial results towards resolving the conjecture).

We calculate the density of the low-lying 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 Tate-Shafarevich group. We show that there is an extra contribution to the density of the low-lying zeros from the family with positive rank (presumably from the ``extra" zero at the central point).

**[BCDT]**Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor,*On the modularity of elliptic curves over 𝐐: wild 3-adic exercises*, J. Amer. Math. Soc.**14**(2001), no. 4, 843–939 (electronic). MR**1839918**, 10.1090/S0894-0347-01-00370-8**[B]**Armand Brumer,*The average rank of elliptic curves. I*, Invent. Math.**109**(1992), no. 3, 445–472. MR**1176198**, 10.1007/BF01232033**[BS]**Armand Brumer and Joseph H. Silverman,*The number of elliptic curves over 𝐐 with conductor 𝐍*, Manuscripta Math.**91**(1996), no. 1, 95–102. MR**1404420**, 10.1007/BF02567942**[C]**J. Brian Conrey,*𝐿-functions and random matrices*, Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 331–352. MR**1852163****[DK]**W. Duke and E. Kowalski,*A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations*, Invent. Math.**139**(2000), no. 1, 1–39. With an appendix by Dinakar Ramakrishnan. MR**1728875**, 10.1007/s002229900017**[FNT]**É. Fouvry, M. Nair, and G. Tenenbaum,*L’ensemble exceptionnel dans la conjecture de Szpiro*, Bull. Soc. Math. France**120**(1992), no. 4, 485–506 (French, with English and French summaries). MR**1194273****[GZ]**Benedict H. Gross and Don B. Zagier,*Heegner points and derivatives of 𝐿-series*, Invent. Math.**84**(1986), no. 2, 225–320. MR**833192**, 10.1007/BF01388809**[H-B]**D. R. Heath-Brown,*The average analytic rank of elliptic curves*, Duke Math J. 122 (2004), no. 3, 591-623. MR2057019 (2004m:11084)**[H-BP]**D. R. Heath-Brown and S. J. Patterson,*The distribution of Kummer sums at prime arguments*, J. Reine Angew. Math.**310**(1979), 111–130. MR**546667****[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.**[IK]**Henryk Iwaniec and Emmanuel Kowalski,*Analytic number theory*, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR**2061214****[ILS]**Henryk Iwaniec, Wenzhi Luo, and Peter Sarnak,*Low lying zeros of families of 𝐿-functions*, Inst. Hautes Études Sci. Publ. Math.**91**(2000), 55–131 (2001). MR**1828743****[KS1]**Nicholas M. Katz and Peter Sarnak,*Random matrices, Frobenius eigenvalues, and monodromy*, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR**1659828****[KS2]**Nicholas M. Katz and Peter Sarnak,*Zeroes of zeta functions and symmetry*, Bull. Amer. Math. Soc. (N.S.)**36**(1999), no. 1, 1–26. MR**1640151**, 10.1090/S0273-0979-99-00766-1**[Ko]**V. A. Kolyvagin,*The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves*, Izv. Akad. Nauk SSSR Ser. Mat.**52**(1988), no. 6, 1154–1180, 1327 (Russian); English transl., Math. USSR-Izv.**33**(1989), no. 3, 473–499. MR**984214****[KM1]**E. Kowalski and P. Michel,*The analytic rank of 𝐽₀(𝑞) and zeros of automorphic 𝐿-functions*, Duke Math. J.**100**(1999), no. 3, 503–542. MR**1719730**, 10.1215/S0012-7094-99-10017-2**[KM2]**E. Kowalski and P. Michel,*Explicit upper bound for the (analytic) rank of 𝐽₀(𝑞)*. part A, Israel J. Math.**120**(2000), no. part A, 179–204. MR**1815375**, 10.1007/s11856-000-1276-8**[Ku]**Daniel Sion Kubert,*Universal bounds on the torsion of elliptic curves*, Proc. London Math. Soc. (3)**33**(1976), no. 2, 193–237. MR**0434947****[Mic]**Philippe Michel,*Rang moyen de familles de courbes elliptiques et lois de Sato-Tate*, Monatsh. Math.**120**(1995), no. 2, 127–136 (French, with English summary). MR**1348365**, 10.1007/BF01585913**[Mil]**Steven J. Miller,*One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries*, Compos. Math.**140**(2004), no. 4, 952–992. MR**2059225**, 10.1112/S0010437X04000582**[RS]**Karl Rubin and Alice Silverberg,*Rank frequencies for quadratic twists of elliptic curves*, Experiment. Math.**10**(2001), no. 4, 559–569. MR**1881757****[R]**Michael Rubinstein,*Low-lying zeros of 𝐿-functions and random matrix theory*, Duke Math. J.**109**(2001), no. 1, 147–181. MR**1844208**, 10.1215/S0012-7094-01-10916-2**[Sch]**Wolfgang M. Schmidt,*Equations over finite fields. An elementary approach*, Lecture Notes in Mathematics, Vol. 536, Springer-Verlag, Berlin-New York, 1976. MR**0429733****[Si1]**Joseph H. Silverman,*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210****[Si2]**Joseph H. Silverman,*Advanced topics in the arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR**1312368****[Si3]**Joseph H. Silverman,*The average rank of an algebraic family of elliptic curves*, J. Reine Angew. Math.**504**(1998), 227–236. MR**1656771**, 10.1515/crll.1998.109**[So]**Christopher D. Sogge,*Fourier integrals in classical analysis*, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. MR**1205579****[ST]**C. L. Stewart and J. Top,*On ranks of twists of elliptic curves and power-free values of binary forms*, J. Amer. Math. Soc.**8**(1995), no. 4, 943–973. MR**1290234**, 10.1090/S0894-0347-1995-1290234-5**[TW]**Richard Taylor and Andrew Wiles,*Ring-theoretic properties of certain Hecke algebras*, Ann. of Math. (2)**141**(1995), no. 3, 553–572. MR**1333036**, 10.2307/2118560**[W]**Andrew Wiles,*Modular elliptic curves and Fermat’s last theorem*, Ann. of Math. (2)**141**(1995), no. 3, 443–551. MR**1333035**, 10.2307/2118559**[Y1]**M. Young,*Low-lying zeros of families of elliptic curves*, http://arxiv.org/abs/math.NT/ 0406330.**[Y2]**M. Young,*Random matrix theory and families of elliptic curves*, Ph.D. thesis, Rutgers University, 2004.

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
11M41,
11F30,
11G05,
11G40,
11L20,
11L40

Retrieve articles in all journals with MSC (2000): 11M41, 11F30, 11G05, 11G40, 11L20, 11L40

Additional Information

**Matthew P. Young**

Affiliation:
American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94306-2244

Email:
myoung@aimath.org

DOI:
http://dx.doi.org/10.1090/S0894-0347-05-00503-5

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.