## Swipe to navigate through the chapters of this book

Published in:

2016 | OriginalPaper | Chapter

# Some Results in the Theory of Low-Lying Zeros of Families of L-Functions

Authors : Blake Mackall, Steven J. Miller, Christina Rapti, Caroline Turnage-Butterbaugh, Karl Winsor

Published in:

Publisher:

## Abstract

While Random Matrix Theory has successfully modeled the limiting behavior of many quantities of families of L-functions, especially the distributions of zeros and values, the theory frequently cannot see the arithmetic of the family. In some situations this requires an extended theory that inserts arithmetic factors that depend on the family, while in other cases these arithmetic factors result in contributions which vanish in the limit, and are thus not detected. In this chapter we review the general theory associated with one of the most important statistics, the n-level density of zeros near the central point. According to the Katz–Sarnak density conjecture, to each family of L-functions there is a corresponding symmetry group (which is a subset of a classical compact group) such that the behavior of the zeros near the central point as the conductors tend to infinity agrees with the behavior of the eigenvalues near 1 as the matrix size tends to infinity. We show how these calculations are done, emphasizing the techniques, methods, and obstructions to improving the results, by considering in full detail the family of Dirichlet characters with square-free conductors. We then move on and describe how we may associate a symmetry constant with each family, and how to determine the symmetry group of a compound family in terms of the symmetries of the constituents. These calculations allow us to explain the remarkable universality of behavior, where the main terms are independent of the arithmetic, as we see that only the first two moments of the Satake parameters survive to contribute in the limit. Similar to the Central Limit Theorem, the higher moments are only felt in the rate of convergence to the universal behavior. We end by exploring the effect of lower order terms in families of elliptic curves. We present evidence supporting a conjecture that the average second moment in one-parameter families without complex multiplication has, when appropriately viewed, a negative bias, and end with a discussion of the consequences of this bias on the distribution of low-lying zeros, in particular relations between such a bias and the observed excess rank in families.
Appendix
Available only for authorised users
Footnotes
1
The derivation is by doing a contour integral of the logarithmic derivative of the completed L-function times the test function, using the Euler product and shifting contours; see [RudSa] for details.

2
It is worth noting that these formulas hold without assuming GRH. In that case, however, the zeros no longer lie on a common line and we lose the correspondence with eigenvalues of Hermitian matrices.

3
A similar absorbtion holds in other families, so long as the Satake parameters satisfy | α i (p) | ≤ Cp δ for some δ < 1∕6.

4
We comment on this in greater length when we consider the family of all characters with square-free modulus. Briefly, a constancy in the conductors allows us to pass certain sums through the test functions to the coefficients. This greatly simplifies the analysis of the 1-level density; unfortunately cross terms arise in the 2-level and higher cases, and the savings vanish (see [Mil1, Mil2]).

5
The definition of the 1-level density as a sum of a test function at scaled zeros is well defined even if GRH fails; however, in that case the zeros are no longer on a line and we thus lose the ability to talk about spacings between zeros. Thus in many of the arguments in the subject GRH is only used to interpret the quantities studied, though there are exceptions (in [ILS] the authors use GRH for Dirichlet L-functions to expand Kloosterman sums).

6
It is easy to handle the case where the conductors are monotone by rescaling the zeros by the average log-conductor; as remarked many times above the general case is more involved.

7
The Satake parameters | α π, i  | are bounded by p δ for some δ, and it is conjectured that we may take δ = 0. While this conjecture is open in general, for many forms there is significant progress towards these bounds with some δ < 1∕2. See, for example, recent work of Kim and Sarnak [Kim, KimSa]. For our purposes, we only need to be able to take δ < 1∕6, as such an estimate and trivial bounding suffices to show that the sum over all primes and all ν ≥ 3 is O(1∕logR).

8
There are some situations where arithmetic enters. The standard example is that in estimating moments of L-functions one has a product a k g k , where a k is an arithmetic factor coming from the arithmetic of the form and g k arises from random matrix theory. See, for example, [CFKRS, KeSn1, KeSn2].

9
These bounds cannot be improved, as Miller [Mil3] found a family where there is a term of size p 3∕2.

10
Following [ILS] we can remove the weights, but their presence facilitates the application of the Petersson formula.

Literature
[AAILMZ]
L. Alpoge, N. Amersi, G. Iyer, O. Lazarev, S.J. Miller, L. Zhang, Maass waveforms and low-lying zeros, in Analytic Number Theory: In Honor of Helmut Maier’s 60th birthday (Carl Pomerance, Michael Th. Rassias, editors), Springer-Verlag (2015)
[AFMY]
M. Asada, E. Fourakis, A. Kwon, S.J. Miller, K. Yang, Biases in the second moments of Fourier coefficients in families of L-functions. Preprint (2015)
[AMil]
L. Alpoge, S.J. Miller, Low lying zeros of Maass form L-functions. Int. Math. Res. Not. 24 pp. (2014). doi:​10.​1093/​imrn/​rnu012
[BFMT-B]
O. Barrett, F.W.K. Firk, S.J. Miller, C. Turnage-Butterbaugh, From quantum systems to L-functions: pair correlation statistics and beyond, in Open Problems in Mathematics, ed. by J. Nash Jr., M.T. Rassias, Springer-Verlag (2016)
[BhSh1]
M. Bhargava, A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves (2010). arXiv:1006.1002
[BhSh2]
M. Bhargava, A. Shankar, Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0 (2010). arXiv:1007.0052
[BS-D1]
B. Birch, H.P.F. Swinnerton-Dyer, Notes on elliptic curves. I. J. Reine Angew. Math. 212, 7–25 (1963)
[BS-D2]
B. Birch, H.P.F. Swinnerton-Dyer, Notes on elliptic curves. II. J. Reine Angew. Math. 218, 79–108 (1965)
[Br]
A. Brumer, The average rank of elliptic curves I. Invent. Math. 109, 445–472 (1992)
[CFKRS]
J.B. Conrey, D. Farmer, P. Keating, M. Rubinstein, N. Snaith, Integral moments of L-functions. Proc. Lond. Math. Soc. (3) 91 (1), 33–104 (2005)
[CFZ1]
J.B. Conrey, D.W. Farmer, M.R. Zirnbauer, Autocorrelation of ratios of L-functions. Commun. Number Theory Phys. 2 (3), 593–636 (2008)
[CFZ2]
J.B. Conrey, D.W. Farmer, M.R. Zirnbauer, Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the classical compact groups (2005). http://​arxiv.​org/​abs/​math-ph/​0511024
[Con]
J.B. Conrey, L-Functions and random matrices, in Mathematics Unlimited — 2001 and Beyond (Springer, Berlin, 2001), pp. 331–352 MATH
[ConSn]
J.B. Conrey, N.C. Snaith, Applications of the L-functions ratios conjecture. Proc. Lond. Math. Soc. 93 (3), 594–646 (2007)
[ConSn2]
J.B. Conrey, N.C. Snaith, Triple correlation of the Riemann zeros (2007). http://​arxiv.​org/​abs/​math/​0610495
[Da]
H. Davenport, Multiplicative Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 74 (Springer, New York, 1980). Revised by H. Montgomery
[DHKMS1]
E. Dueñez, D.K. Huynh, J.C. Keating, S.J. Miller, N. Snaith, The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI. J. Phys. A Math. Theor. 43, 405204 (27pp.) (2010)
[DHKMS2]
E. Dueñez, D.K. Huynh, J.C. Keating, S.J. Miller, N. Snaith, Models for zeros at the central point in families of elliptic curves (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith). J. Phys. A Math. Theor. 45, 115207 (32pp.) (2012)
[DuMil1]
E. Dueñez, S.J. Miller, The low lying zeros of a GL(4) and a GL(6) family of L-functions. Compos. Math. 142 (6), 1403–1425 (2006)
[DuMil2]
E. Dueñez, S.J. Miller, The effect of convolving families of L-functions on the underlying group symmetries. Proc. Lond. Math. Soc. (2009). doi: 10.​1112/​plms/​pdp018 MATH
[Ed]
H.M. Edwards, Riemann’s Zeta Function (Academic, New York, 1974) MATH
[ER-GR]
A. Entin, E. Roditty-Gershon, Z. Rudnick, Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and random matrix theory. Geom. Funct. Anal. 23 (4), 1230–1261 (2013)
[Fe1]
S. Fermigier, Zéros des fonctions L de courbes elliptiques. Exp. Math. 1 (2), 167–173 (1992)
[Fe2]
S. Fermigier, Étude expérimentale du rang de familles de courbes elliptiques sur $$\mathbb{Q}$$. Exp. Math. 5 (2), 119–130 (1996)
[FioMil]
D. Fiorilli, S.J. Miller, Surpassing the ratios conjecture in the 1-level density of Dirichlet L-functions. Algebra and Number Theory 9 (1), 13–52 (2015)
[FirMil]
F.W.K. Firk, S.J. Miller, Nuclei, primes and the random matrix connection. Symmetry 1, 64–105 (2009). doi:​10.​3390/​sym1010064
[FoIw]
E. Fouvry, H. Iwaniec, Low-lying zeros of dihedral L-functions. Duke Math. J. 116 (2), 189–217 (2003)
[FoPo]
E. Fouvrey, J. Pomykala, Rang des courbes elliptiques et sommes d’exponentelles. Monat. Math. 116, 111–125 (1993) CrossRef
[For]
P. Forrester, Log-Gases and Random Matrices. London Mathematical Society Monograph, vol. 34 (Princeton University Press, Princeton, NJ, 2010)
[FrMil]
J. Freeman, S.J. Miller, Determining Optimal Test Functions for Bounding the Average Rank in Families of L-Functions, in SCHOLAR – a Scientific Celebration Highlighting Open Lines of Arithmetic Research, Conference in Honour of M. Ram Murty’s Mathematical Legacy on his 60th Birthday (A. C. Cojocaru, C. David and F. Pappaardi, editors), Contemporary Mathematics, AMS and CRM, 655 (2015)
[Gao]
P. Gao, N-level density of the low-lying zeros of quadratic Dirichlet L-functions. Ph.D. thesis, University of Michigan (2005)
[GJMMNPP]
J. Goes, S. Jackson, S.J. Miller, D. Montague, K. Ninsuwan, R. Peckner, T. Pham, A unitary test of the ratios conjecture. J. Number Theory 130 (10), 2238–2258 (2010)
[GoHuKe]
S.M. Gonek, C.P. Hughes, J.P. Keating, A hybrid Euler-Hadamard product formula for the Riemann zeta function. Duke Math. J. 136, 507–549 (2007)
[GolKon]
D. Goldfeld, A. Kontorovich, On the GL(3) Kuznetsov formula with applications to symmetry types of families of L-functions. Automorphic representations and L-functions, 263–310, Tata Inst. Fundam. Res. Stud. Math., 22, Tata Inst. Fund. Res., Mumbai (2013)
[Gü]
A. Güloğlu, Low-lying zeros of symmetric power L-functions. Int. Math. Res. Not. 2005 (9), 517–550 (2005)
[Ha]
B. Hayes, The spectrum of Riemannium. Am. Sci. 91 (4), 296–300 (2003) CrossRef
[H-B]
R. Heath-Brown, The average analytic rank of elliptic curves. Duke Math. J. 122 (3), 591–623 (2004)
[HuMil]
C. Hughes, S.J. Miller, Low-lying zeros of L-functions with orthogonal symmetry. Duke Math. J. 136 (1), 115–172 (2007)
[HuRud]
C. Hughes, Z. Rudnick, Linear statistics of low-lying zeros of L-functions. Q. J. Math. Oxford 54, 309–333 (2003)
[HuyKeSn]
D.K. Huynh, J.P. Keating, N.C. Snaith, Lower order terms for the one-level density of elliptic curve L-functions. J. Number Theory 129 (12), 2883–2902 (2009)
[HuyMM]
D.K. Huynh, S.J. Miller, R. Morrison, An elliptic curve family test of the ratios conjecture. J. Number Theory 131, 1117–1147 (2011)
[ILS]
H. Iwaniec, W. Luo, P. Sarnak, Low lying zeros of families of L-functions. Inst. Hautes Études Sci. Publ. Math. 91, 55–131 (2000)
[Iw]
H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms. Biblioteca de la Revista Matemática Iberoamericana (1995)
[IwKo]
H. Iwaniec, E. Kowalski, Analytic Number Theory. AMS Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004)
[KaSa1]
N. Katz, P. Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy. AMS Colloquium Publications, vol. 45 (American Mathematical Society, Providence, RI, 1999)
[KaSa2]
N. Katz, P. Sarnak, Zeros of zeta functions and symmetries. Bull. Am. Math. Soc. 36, 1–26 (1999)
[KeSn1]
J.P. Keating, N.C. Snaith, Random matrix theory and ζ(1∕2 + it). Commun. Math. Phys. 214 (1), 57–89 (2000)
[KeSn2]
J.P. Keating, N.C. Snaith, Random matrix theory and L-functions at s = 1∕2. Commun. Math. Phys. 214 (1), 91–110 (2000)
[KeSn3]
J.P. Keating, N.C. Snaith, Random matrices and L-functions. Random matrix theory. J. Phys. A 36 (12), 2859–2881 (2003)
[Kim]
H. Kim, Functoriality for the exterior square of GL 2 and the symmetric fourth of GL 2. J. Am. Math. Soc. 16 (1), 139–183 (2003)
[KimSa]
H. Kim, P. Sarnak, Appendix: Refined estimates towards the Ramanujan and Selberg conjectures. Appendix to [Kim]
[LevMil]
J. Levinson, S.J. Miller, The n-level density of zeros of quadratic Dirichlet L-functions. Acta Arith. 161, 145–182 (2013)
[MaTe]
J. Matz, N. Templier, Sato-Tate equidistribution for families of Hecke-Maass forms on $$\mathrm{SL}(n, \mathbb{R})/\mathrm{SO}(n)$$. Preprint (2015), http://​arxiv.​org/​abs/​1505.​07285
[Meh]
M. Mehta, Random Matrices, 2nd edn. (Academic, Boston, 1991)
[Mic]
P. Michel, Rang moyen de familles de courbes elliptiques et lois de Sato-Tate. Monat. Math. 120, 127–136 (1995)
[Mil1]
S.J. Miller, 1- and 2-level densities for families of elliptic curves: evidence for the underlying group symmetries. Ph.D. thesis, Princeton University (2002)
[Mil2]
S.J. Miller, 1- and 2-level densities for families of elliptic curves: evidence for the underlying group symmetries. Compos. Math. 140, 952–992 (2004)
[Mil3]
S.J. Miller, Variation in the number of points on elliptic curves and applications to excess rank. C. R. Math. Rep. Acad. Sci. Canada 27 (4), 111–120 (2005)
[Mil5]
S.J. Miller, A symplectic test of the L-functions ratios conjecture. Int. Math. Res. Not. 2008, 36 pp. (2008). Article ID rnm146
[Mil6]
S.J. Miller, Lower order terms in the 1-level density for families of holomorphic cuspidal newforms. Acta Arith. 137, 51–98 (2009)
[Mil7]
S.J. Miller, An orthogonal test of the L-functions ratios conjecture. Proc. Lond. Math. Soc. (2009). doi:​10.​1112/​plms/​pdp009 MATH
[MilMo]
S.J. Miller, D. Montague, An orthogonal test of the L-functions ratios conjecture. II. Acta Arith. 146, 53–90 (2011)
[MilPe]
S.J. Miller, R. Peckner, Low-lying zeros of number field L-functions. J. Number Theory 132, 2866–2891 (2012)
[MMRW]
B. Mackall, S.J. Miller, C. Rapti, K. Winsor, Lower-order biases in elliptic curve fourier coefficients in families, in Frobenius Distributions: Lang-Trotter and Sato-Tate Conjectures (David Kohel and Igor Shparlinski, editors), Contemporary Mathematics 663, AMS, Providence, RI (2016)
[Mon]
H.L. Montgomery, The pair correlation of zeros of the zeta function, in Analytic Number Theory, Proceedings of Symposia in Pure Mathematics, vol. 24 (American Mathematical Society, Providence, RI, 1973), pp. 181–193
[MonVa]
H.L. Montgomery, R.C. Vaughan, The large sieve. Mathematika 20, 119–134 (1973)
[Od1]
A. Odlyzko, On the distribution of spacings between zeros of the zeta function. Math. Comput. 48 (177), 273–308 (1987)
[Od2]
A. Odlyzko, The 10 22-nd zero of the Riemann zeta function, Proceedings of Conference on Dynamical, Spectral and Arithmetic Zeta-Functions, ed. by M. van Frankenhuysen, M.L. Lapidus, Contemporary Mathematics Series (American Mathematical Society, Providence, RI, 2001). http://​www.​research.​att.​com/​~amo/​doc/​zeta.​html
[OS1]
A.E. Özlük, C. Snyder, Small zeros of quadratic L-functions. Bull. Aust. Math. Soc. 47 (2), 307–319 (1993)
[OS2]
A.E. Özlük, C. Snyder, On the distribution of the nontrivial zeros of quadratic L-functions close to the real axis. Acta Arith. 91 (3), 209–228 (1999)
[RiRo]
G. Ricotta, E. Royer, Statistics for low-lying zeros of symmetric power L-functions in the level aspect. Preprint, Forum Math. 23, 969–1028 (2011)
[Ro]
E. Royer, Petits zéros de fonctions L de formes modulaires. Acta Arith. 99 (2), 147–172 (2001)
[RoSi]
M. Rosen, J. Silverman, On the rank of an elliptic surface. Invent. Math. 133, 43–67 (1998)
[Rub]
M. Rubinstein, Low-lying zeros of L–functions and random matrix theory. Duke Math. J. 109, 147–181 (2001)
[RudSa]
Z. Rudnick, P. Sarnak, Zeros of principal L-functions and random matrix theory. Duke Math. J. 81, 269–322 (1996)
[SaShTe]
P. Sarnak, S.W. Shin, N. Templier, Families of L-functions and their symmetry, in Families of Automorphic Forms and the Trace Formula, ed. by W. Müller, S.W. Shin, N. Templier (Springer, New York, 2016)
[ShTe]
S.W. Shin, N. Templier, Sato-Tate Theorem for Families and low-lying zeros of automorphic L-functions. With Appendix 1 by R. Kottwitz and Appendix 2 by R. Cluckers, J. Gordon and I. Halupczok. Invent. Math. 203 (1), 1–177 (2016)
[Sil]
J. Silverman, The average rank of an algebraic family of elliptic curves. J. Reine Angew. Math. 504, 227–236 (1998)
[Ti]
E. Titchmarsh, D.R. Heath-Brown, The Theory of the Riemann Zeta-Function (Oxford University Press, Oxford, 1986)
[Wa]
M. Watkins, Rank distribution in a family of cubic twists (2007). http://​arxiv.​org/​pdf/​math.​NT/​0412427
[Ya]
A. Yang, Low-lying zeros of Dedekind zeta functions attached to cubic number fields. Preprint
[Yo1]
M. Young, Lower-order terms of the 1-level density of families of elliptic curves. Int. Math. Res. Not. 2005 (10), 587–633 (2005)
[Yo2]
M. Young, Low-lying zeros of families of elliptic curves. J. Am. Math. Soc. 19 (1), 205–250 (2006)
Title
Some Results in the Theory of Low-Lying Zeros of Families of L-Functions
Authors
Blake Mackall
Steven J. Miller
Christina Rapti
Caroline Turnage-Butterbaugh
Karl Winsor