Zeros of quadratic Dirichlet 𝐿-functions in the hyperelliptic ensemble

Bui, H.; Florea, Alexandra

doi:10.1090/tran/7317

Zeros of quadratic Dirichlet $L$-functions in the hyperelliptic ensemble
HTML articles powered by AMS MathViewer

by H. M. Bui and Alexandra Florea PDF

Trans. Amer. Math. Soc. 370 (2018), 8013-8045 Request permission

Abstract:

We study the $1$-level density and the pair correlation of zeros of quadratic Dirichlet $L$-functions in function fields, as we average over the ensemble $\mathcal {H}_{2g+1}$ of monic, square-free polynomials with coefficients in $\mathbb {F}_q[x]$. In the case of the $1$-level density, when the Fourier transform of the test function is supported in the restricted interval $(\frac {1}{3},1)$, we compute a secondary term of size $q^{-\frac {4g}{3}}/g$, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For example, if the Fourier transform is supported in $(\frac {1}{3}, \frac {1}{2})$, we identify another lower order term of size $q^{-\frac {8g}{5}}/g$. We also compute the pair correlation, and as for the $1$-level density, we detect lower order terms under certain restrictions; for example, we see a term of size $q^{-g}/g^2$ when the Fourier transform is supported in $(\frac {1}{4},\frac {1}{2})$. The $1$-level density and the pair correlation allow us to obtain non-vanishing results for $L(\frac 12,\chi _D)$, as well as lower bounds for the proportion of simple zeros of this family of $L$-functions.

References

J. C. Andrade and J. P. Keating, Conjectures for the integral moments and ratios of $L$-functions over function fields, J. Number Theory 142 (2014), 102–148. MR 3208396, DOI 10.1016/j.jnt.2014.02.019
E. Bogomolny and J.P. Keating, Gutzwiller’s trace formula and spectral statistics: Beyond the diagonal approximation, Phys. Rev. Lett. 77 (1996), no. 8, 1472–1475.
J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Integral moments of $L$-functions, Proc. London Math. Soc. (3) 91 (2005), no. 1, 33–104. MR 2149530, DOI 10.1112/S0024611504015175
Brian Conrey, David W. Farmer, and Martin R. Zirnbauer, Autocorrelation of ratios of $L$-functions, Commun. Number Theory Phys. 2 (2008), no. 3, 593–636. MR 2482944, DOI 10.4310/CNTP.2008.v2.n3.a4
J. B. Conrey and N. C. Snaith, Applications of the $L$-functions ratios conjectures, Proc. Lond. Math. Soc. (3) 94 (2007), no. 3, 594–646. MR 2325314, DOI 10.1112/plms/pdl021
John Brian Conrey and Nina Claire Snaith, Correlations of eigenvalues and Riemann zeros, Commun. Number Theory Phys. 2 (2008), no. 3, 477–536. MR 2482941, DOI 10.4310/CNTP.2008.v2.n3.a1
Adrian Diaconu, Dorian Goldfeld, and Jeffrey Hoffstein, Multiple Dirichlet series and moments of zeta and $L$-functions, Compositio Math. 139 (2003), no. 3, 297–360. MR 2041614, DOI 10.1023/B:COMP.0000018137.38458.68
Alexei Entin, Edva Roditty-Gershon, and Zeév Rudnick, Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and random matrix theory, Geom. Funct. Anal. 23 (2013), no. 4, 1230–1261. MR 3077912, DOI 10.1007/s00039-013-0241-8
Daniel Fiorilli and Steven J. Miller, Surpassing the ratios conjecture in the 1-level density of Dirichlet $L$-functions, Algebra Number Theory 9 (2015), no. 1, 13–52. MR 3317760, DOI 10.2140/ant.2015.9.13
Daniel Fiorilli, James Parks, and Anders Södergren, Low-lying zeros of quadratic Dirichlet $L$-functions: lower order terms for extended support, Compos. Math. 153 (2017), no. 6, 1196–1216. MR 3705254, DOI 10.1112/S0010437X17007059
Alexandra Florea, The fourth moment of quadratic Dirichlet $L$-functions over function fields, Geom. Funct. Anal. 27 (2017), no. 3, 541–595. MR 3655956, DOI 10.1007/s00039-017-0409-8
Alexandra M. Florea, Improving the error term in the mean value of $L(\frac 12,\chi )$ in the hyperelliptic ensemble, Int. Math. Res. Not. IMRN 20 (2017), 6119–6148. MR 3712193, DOI 10.1093/imrn/rnv387
Alexandra Florea, The second and third moment of $L(1/2,\chi )$ in the hyperelliptic ensemble, Forum Math. 29 (2017), no. 4, 873–892. MR 3669007, DOI 10.1515/forum-2015-0152
Peng Gao, n-level density of the low-lying zeros of quadratic Dirichlet L-functions, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of Michigan. MR 2708011
D. R. Hayes, The expression of a polynomial as a sum of three irreducibles, Acta Arith. 11 (1966), 461–488. MR 201422, DOI 10.4064/aa-11-4-461-488
C. P. Hughes and Steven J. Miller, Low-lying zeros of $L$-functions with orthogonal symmetry, Duke Math. J. 136 (2007), no. 1, 115–172. MR 2271297, DOI 10.1215/S0012-7094-07-13614-7
C. P. Hughes and Z. Rudnick, Linear statistics of low-lying zeros of $L$-functions, Q. J. Math. 54 (2003), no. 3, 309–333. MR 2013141, DOI 10.1093/qjmath/54.3.309
D. K. Huynh, J. P. Keating, and N. C. Snaith, Lower order terms for the one-level density of elliptic curve $L$-functions, J. Number Theory 129 (2009), no. 12, 2883–2902. MR 2560841, DOI 10.1016/j.jnt.2008.12.008
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
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, DOI 10.1090/coll/045
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, DOI 10.1090/S0273-0979-99-00766-1
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, DOI 10.1112/S0010437X04000582
Steven J. Miller, A symplectic test of the $L$-functions ratios conjecture, Int. Math. Res. Not. IMRN 3 (2008), Art. ID rnm146, 36. MR 2416994, DOI 10.1093/imrn/rnm146
Steven J. Miller, An orthogonal test of the $L$-functions ratios conjecture, Proc. Lond. Math. Soc. (3) 99 (2009), no. 2, 484–520. MR 2533673, DOI 10.1112/plms/pdp009
Steven J. Miller and David Montague, An orthogonal test of the $L$-functions ratios conjecture, II, Acta Arith. 146 (2011), no. 1, 53–90. MR 2741191, DOI 10.4064/aa146-1-5
H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 181–193. MR 0337821
Hugh L. Montgomery, Distribution of the zeros of the Riemann zeta function, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 379–381. MR 0419378
A. E. Özlük and C. Snyder, On the distribution of the nontrivial zeros of quadratic $L$-functions close to the real axis, Acta Arith. 91 (1999), no. 3, 209–228. MR 1735673, DOI 10.4064/aa-91-3-209-228
Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. MR 1876657, DOI 10.1007/978-1-4757-6046-0
Michael Rubinstein, Low-lying zeros of $L$-functions and random matrix theory, Duke Math. J. 109 (2001), no. 1, 147–181. MR 1844208, DOI 10.1215/S0012-7094-01-10916-2
Zeév Rudnick, Traces of high powers of the Frobenius class in the hyperelliptic ensemble, Acta Arith. 143 (2010), no. 1, 81–99. MR 2640060, DOI 10.4064/aa143-1-5
Zeév Rudnick and Peter Sarnak, The $n$-level correlations of zeros of the zeta function, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 10, 1027–1032 (English, with English and French summaries). MR 1305671
Zeév Rudnick and Peter Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J. 81 (1996), no. 2, 269–322. A celebration of John F. Nash, Jr. MR 1395406, DOI 10.1215/S0012-7094-96-08115-6
K. Soundararajan, Nonvanishing of quadratic Dirichlet $L$-functions at $s=\frac 12$, Ann. of Math. (2) 152 (2000), no. 2, 447–488. MR 1804529, DOI 10.2307/2661390
André Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Publ. Inst. Math. Univ. Strasbourg, vol. 7, Hermann & Cie, Paris, 1948 (French). MR 0027151
Matthew P. Young, Lower-order terms of the 1-level density of families of elliptic curves, Int. Math. Res. Not. 10 (2005), 587–633. MR 2147004, DOI 10.1155/IMRN.2005.587
Matthew P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (2006), no. 1, 205–250. MR 2169047, DOI 10.1090/S0894-0347-05-00503-5