Non-vanishing of class group \(L\)-functions for number fields with a small regulator. (English) Zbl 1473.11210
Author’s abstract: “Let \(E/\mathbb{Q}\) be a number field of degree \(n\). We show that if \(\operatorname{Reg}(E)\ll_n |\!\operatorname{Disc}(E)|^{1/4}\) then the fraction of class group characters for which the Hecke \(L\)-function does not vanish at the central point is \(\gg_{n,\varepsilon} |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon}\). The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in \(\mathbf{PGL}_n(\mathbb{Z})\backslash \mathbf{PGL}_n(\mathbb{R})\) associated to the maximal order of \(E\), and the escape of mass of the torus orbit associated to the trivial ideal class.”
Reviewer’s remarks: The precise formulation of the main result of the paper is follows:
Theorem 1.1. Let \(E/\mathbb{Q}\) be a number field of degree \(n\). Denote by \(D\) its discriminant, by \(R\) the regulator of its ring of integers and by \(h\) the class number. For every class group character \(\chi\in\widehat{\text{Cl}(E)}\) let \(L(s,\chi)\) be the associated Hecke \(L\)-function.
Fix a real number \(1/2\le s<1\). There are effectively computable constants \(A,B>0\) that depend only on \(s\), \(n\) such that for every \(1/2>\varepsilon>0\), \[h^{-1}\#\{\chi\in \widehat{\text{Cl}(E)}\mid L(s,\chi)\ne 0\}\ge|D|^{-(1-s+\varepsilon)/2}\left(A-B\frac{R}{|D|^{s/2}}\right)\varepsilon^n.\] In the Introduction the author sets out his aims and compares his results to related earlier investigations of others. Then, in the subsection 1.2 he describes his working out of the paper. In Section 2 properties of the Epstein zeta function are dealed with and sharpened, whereas other means like Weil chambers and Fourier series are applied, in order to achieve his main results. In the proof of Theorems 2.6 one encounters the notion (and notation) of \(erfc\sqrt{\pi t}\), perhaps known to specialists, but perhaps not so well-known in general. It is only a little inconvenience. The whole paper is a good essay in fact, to be studied carefully, and very well suited for seminars also.
Nice stuff, the reviewer enjoined it; a very welcome to modern number theory.
Reviewer’s remarks: The precise formulation of the main result of the paper is follows:
Theorem 1.1. Let \(E/\mathbb{Q}\) be a number field of degree \(n\). Denote by \(D\) its discriminant, by \(R\) the regulator of its ring of integers and by \(h\) the class number. For every class group character \(\chi\in\widehat{\text{Cl}(E)}\) let \(L(s,\chi)\) be the associated Hecke \(L\)-function.
Fix a real number \(1/2\le s<1\). There are effectively computable constants \(A,B>0\) that depend only on \(s\), \(n\) such that for every \(1/2>\varepsilon>0\), \[h^{-1}\#\{\chi\in \widehat{\text{Cl}(E)}\mid L(s,\chi)\ne 0\}\ge|D|^{-(1-s+\varepsilon)/2}\left(A-B\frac{R}{|D|^{s/2}}\right)\varepsilon^n.\] In the Introduction the author sets out his aims and compares his results to related earlier investigations of others. Then, in the subsection 1.2 he describes his working out of the paper. In Section 2 properties of the Epstein zeta function are dealed with and sharpened, whereas other means like Weil chambers and Fourier series are applied, in order to achieve his main results. In the proof of Theorems 2.6 one encounters the notion (and notation) of \(erfc\sqrt{\pi t}\), perhaps known to specialists, but perhaps not so well-known in general. It is only a little inconvenience. The whole paper is a good essay in fact, to be studied carefully, and very well suited for seminars also.
Nice stuff, the reviewer enjoined it; a very welcome to modern number theory.
Reviewer: Robert W. van der Waall (Huizen)
MSC:
| 11R42 | Zeta functions and \(L\)-functions of number fields |
| 11M20 | Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\) |
| 11R04 | Algebraic numbers; rings of algebraic integers |
| 11M41 | Other Dirichlet series and zeta functions |
| 11E45 | Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
Keywords:
\(L\) function; class group; non-vanishing; escape of mass; equidistribution; peridic torus orbit; Epstein zeta function; Phragmén-Lindelöf principle; volume unit ballReferences:
| [1] | Balasubramanian, R. and Kumar Murty, V., Zeros of Dirichlet \(L\)-functions, Ann. Sci. Éc. Norm. Supér. (4)25 (1992), 567-615. · Zbl 0771.11033 |
| [2] | Blanksby, P. E. and Montgomery, H. L., Algebraic integers near the unit circle, Acta Arith. 18 (1971), 355-369. · Zbl 0221.12003 |
| [3] | Blomer, V., Non-vanishing of class group \(L\)-functions at the central point, Ann. Inst. Fourier (Grenoble)54 (2004), 831-847. · Zbl 1063.11040 |
| [4] | Blomer, V., Harcos, G. and Michel, P., Bounds for modular \(L\)-functions in the level aspect, Ann. Sci. Éc. Norm. Supér. (4)40 (2007), 697-740. · Zbl 1185.11034 |
| [5] | Bombieri, E. and Gubler, W., Heights in diophantine geometry, , vol. 4 (Cambridge University Press, Cambridge, 2006). · Zbl 1115.11034 |
| [6] | Burgess, D. A., On character sums and \(L\)-series. II, Proc. Lond. Math. Soc. (3)13 (1963), 524-536. · Zbl 0123.04404 |
| [7] | Dimitrov, V., A proof of the Schinzel-Zassenhaus conjecture on polynomials, Preprint (2019), arXiv:1912.12545. |
| [8] | Dobrowolski, E., On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), 391-401. · Zbl 0416.12001 |
| [9] | Duke, W., Number fields with large class group, in Number theory, , vol. 36 (American Mathematical Society, Providence, RI, 2004), 117-126. · Zbl 1104.11050 |
| [10] | Duke, W., Friedlander, J. B. and Iwaniec, H., The subconvexity problem for Artin \(L\)-functions, Invent. Math. 149 (2002), 489-577. · Zbl 1056.11072 |
| [11] | Einsiedler, M., Lindenstrauss, E., Michel, P. and Venkatesh, A., Distribution of periodic torus orbits on homogeneous spaces, Duke Math. J. 148 (2009), 119-174. · Zbl 1172.37003 |
| [12] | Einsiedler, M., Lindenstrauss, E., Michel, P. and Venkatesh, A., Distribution of periodic torus orbits and Duke’s theorem for cubic fields, Ann. of Math. (2)173 (2011), 815-885. · Zbl 1248.37009 |
| [13] | Epstein, P., Zur Theorie allgemeiner Zetafunktionen. II, Math. Ann. 63 (1906), 205-216. · JFM 37.0433.02 |
| [14] | Fröhlich, A., Artin root numbers and normal integral bases for quaternion fields, Invent. Math. 17 (1972), 143-166. · Zbl 0261.12008 |
| [15] | Hecke, E., Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 6 (1920), 11-51. · JFM 47.0152.01 |
| [16] | Iwaniec, H. and Sarnak, P., The non-vanishing of central values of automorphic \(L\)-functions and Landau-Siegel zeros, Israel J. Math. 120 (2000), 155-177. · Zbl 0992.11037 |
| [17] | Katz, N. M. and Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy, , vol. 45 (American Mathematical Society, Providence, RI, 1999). · Zbl 0958.11004 |
| [18] | Luo, W., Rudnick, Z. and Sarnak, P., On Selberg’s eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), 387-401. · Zbl 0844.11038 |
| [19] | Luo, W., Rudnick, Z. and Sarnak, P., On the generalized Ramanujan conjecture for \({\rm GL}(n)\), in Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), , vol. 66 (American Mathematical Society, Providence, RI, 1999), 301-310. · Zbl 0965.11023 |
| [20] | Michel, P. and Venkatesh, A., Heegner points and non-vanishing of Rankin/Selberg \(L\)-functions, in Analytic number theory, , vol. 7 (American Mathematical Society, Providence, RI, 2007), 169-183. · Zbl 1214.11063 |
| [21] | Rademacher, H., On the Phragmén-Lindelöf theorem and some applications, Math. Z. 72 (1959-1960), 192-204. · Zbl 0092.27703 |
| [22] | Sarnak, P., Shin, S. W. and Templier, N., Families of \(L\)-functions and their symmetry, in Families of automorphic forms and the trace formula, (Springer, Cham, 2016), 531-578. · Zbl 1417.11078 |
| [23] | Shankar, A., Södergren, A. and Templier, N., Sato-Tate equidistribution of certain families of Artin \(L\)-functions, Forum Math. Sigma7 (2019), e23. · Zbl 1469.11443 |
| [24] | Soundararajan, K., Nonvanishing of quadratic Dirichlet \(L\)-functions at \(s=\frac{1}{2} \), Ann. of Math. (2)152 (2000), 447-488. · Zbl 0964.11034 |
| [25] | Soundararajan, K., Weak subconvexity for central values of \(L\)-functions, Ann. of Math. (2)172 (2010), 1469-1498. · Zbl 1234.11066 |
| [26] | Stewart, C. L., Algebraic integers whose conjugates lie near the unit circle, Bull. Soc. Math. France106 (1978), 169-176. · Zbl 0396.12002 |
| [27] | Terras, A. A., Bessel series expansions of the Epstein zeta function and the functional equation, Trans. Amer. Math. Soc. 183 (1973), 477-486. · Zbl 0274.10039 |
| [28] | Terras, A., The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions, J. Number Theory12 (1980), 258-272. · Zbl 0432.10010 |
| [29] | Wielonsky, F., Séries d’Eisenstein, intégrales toroïdales et une formule de Hecke, Enseign. Math. 31 (1985), 93-135. · Zbl 0581.12006 |
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