Some remarks on Landau-Siegel zeros. (English) Zbl 1008.11033

Sarnak, P.; Zaharescu, A.

Duke Math. J. 111, No. 3, 495-507 (2002).

The authors study the possible “exceptional” zero of a Dirichlet \(L\)-function \(L(s, \chi _D)\) for a real primitive character \(\chi _D \pmod{|D|}\) under certain hypotheses. Let \(E\) be an elliptic curve over \(\mathbb Q\). “Hypothesis H” assumes that all zeros of \(L(s, \chi _d)\) and \(L(s, E\otimes \chi _D)\) are either real or lie on the critical line. It is shown that this hypothesis (for Dirichlet \(L\)-functions only) implies the ineffective lower bound \(L(1, \chi _D) \geq c(\varepsilon)(\log |D|)^{-\varepsilon }\). Further, the full hypothesis H gives the effective lower bound \(L(1, \chi _D) \geq c(\eta)|D|^{-\eta }\) for any \(\eta >2/5\) if \(\chi _D(37)=-1\). Also, if \(L(s,E)\) has a zero of order \(m\) at \(s=1/2\), then the effective estimate \(L(1, \chi _D) \geq c(E, \varepsilon)|D|^{-(2+\varepsilon)/(m+1)}\) holds.
The general scheme of the argument is to start from a suitable product of \(L\)-functions and to apply corresponding “explicit formulae” connecting arithmetic sums with zeros.

Reviewer: Matti Jutila (Turku)

MSC:

11M20	Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
11M26	Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses

Keywords:

\(L\)-functions; Siegel zero; elliptic curves

References:

[1]	E. Bombieri, Remarks on Weil’s quadratic functional in the theory of prime numbers, I , Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000), 183–233. · Zbl 1008.11034
[2]	C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q : Wild \(3\)-adic exercises, J. Amer. Math. Soc. 14 (2001), 843–939. JSTOR: · Zbl 0982.11033
[3]	H. Davenport, Multiplicative Number Theory , 2d ed., revised by Hugh L. Montgomery, Grad. Texts in Math. 74 , Springer, New York, 1980. · Zbl 0453.10002
[4]	M. Deuring, Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins , Invent. Math. 5 (1968), 169–179. · Zbl 0155.38001
[5]	L. Dirichlet, Recherches sur diverse applications de l’analyse infinitésimale à la théorie des nombres, I , J. Reine Angew. Math. 19 (1839), 324–369.; II , 21 (1840), 1–12. · ERAM 021.0655cj
[6]	W. Duke, Some old problems and new results about quadratic forms, Notices Amer. Math. Soc. 44 (1997), 190–196. · Zbl 0969.11002
[7]	D. M. Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Ann. Scuola Norm. Sup. Pisa (4) 3 (1976), 624–663. · Zbl 0345.12007
[8]	B. Gross and D. Zagier, Points de Heegner et dérivées de fonctions \(L\) , C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), 85–87. · Zbl 0538.14023
[9]	D. R. Heath-Brown, Prime twins and Siegel zeros, Proc. London Math. Soc. (3) 47 (1983), 193–224. · Zbl 0517.10044
[10]	H. Heilbronn, On the class-number in imaginary quadratic fields, Quart. J. Math. Oxf. Ser. (2) 5 (1934), 150–160. · Zbl 0009.29602
[11]	Y. Ihara, “Discrete subgroups of PL\((2, k_\wp)\)” in Algebraic Groups and Discontinuous Subgroups (Boulder, 1965) , Proc. Sympos. Pure. Math. 9 , Amer. Math. Soc., Providence, 1966, 272–278.
[12]	H. Iwaniec, course lecture notes, Rutgers, 1999.
[13]	H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of L-functions , Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55–131. · Zbl 1012.11041
[14]	E. Landau, Bemerkungen zum Heilbronnschen Satz, Acta Arith. 1 (1935), 1–18. · Zbl 0011.00902
[15]	J. E. Littlewood, On the class-number of the corpus \(P(\sqrt -k)\) , Proc. London Math. Soc. (2) 27 (1928), 358–372. · JFM 54.0206.02
[16]	R. Martin and W. McMillen, An elliptic curve over \(\mathbf Q\) with rank at least \(24\) , preprint, 2000, http://listserv.nodak.edu/scripts/wa.exe?A2=ind0005&L=nmbrthry&F=&S=&P=521
[17]	H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis , CBMS Reg. Conf. Ser. Math. 84 , Amer. Math. Soc., Providence, 1994. · Zbl 0814.11001
[18]	J. Oesterlé, Nombres de classes des corps quadratiques imaginaires , Astérisque 121 –. 122 (1985), 309–323., Séminaire Bourbaki, 1983/84, exp. no. 631. · Zbl 0551.12003
[19]	Z. Rudnick and P. Sarnak, Zeros of principal \(L\)-functions and random matrix theory , Duke Math. J. 81 (1996), 269–322. · Zbl 0866.11050
[20]	A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. · Zbl 0072.08201
[21]	C. L. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1935), 83–86. · Zbl 0011.00903
[22]	T. Tatuzawa, On a theorem of Siegel, Jap. J. Math. 21 (1951), 163–178. · Zbl 0054.02302