Heegner zeros of theta functions
Authors:
Jorge JimenezUrroz and Tonghai Yang
Journal:
Trans. Amer. Math. Soc. 355 (2003), 41374149
MSC (2000):
Primary 11G05, 11M20, 14H52
Published electronically:
June 18, 2003
MathSciNet review:
1990579
Fulltext PDF Free Access
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References 
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Additional Information
Abstract: Heegner divisors play an important role in number theory. However, little is known on whether a modular form has Heegner zeros. In this paper, we start to study this question for a family of classical theta functions, and prove a quantitative result, which roughly says that many of these theta functions have a Heegner zero of discriminant . This leads to some interesting questions on the arithmetic of certain elliptic curves, which we also address here.
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 W. E. H. Berwick, Modular invariants expressible in terms of quadratic and cubic irrationals, Math. Ann. 96 (1927), 5369.
 [BKO]
 J. Bruinier, W. Kohnen, and K. Ono, The arithmetic of the values of modular functions and the divisors of modular forms, Compositio Math., accepted for publication.
 [C]
 J. E. Cremona, Algorithms for modular elliptic curves, 2nd edition, Cambridge University Press, 1997. MR 99e:11068
 [G1]
 B. Gross, Arithmetic of elliptic curves vith complex multiplication, Lecture Notes in Mathematics 776, SpringerVerlag, Berlin, 1980. MR 81f:10041
 [G2]
 , Minimal models for elliptic curves with complex multiplication, Compositio Math. 45 (1982), 155164. MR 84j:14044
 [Go]
 F. Gouvêa, The squarefree sieve over number fields, J. Number Theory 43 (1993), 109122. MR 93m:11045
 [GM]
 F. Gouvêa and B. Mazur, The squarefree sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), 123. MR 92b:11039
 [HR]
 H. Halberstam and H.E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press, London, 1974. MR 54:12689
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 G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th edition, The Clarendon Press, Oxford Univ. Press, 1979. MR 81i:10002
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 J. Jiménez Urroz, Nontrivial zeroes for quadratic twists of HasseWeil Functions, J. Number Theory 77 (1999), 331335. MR 2000d:11088
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 V. A. Kolyvagin and D. Yu. Logachev, Finiteness of the ShafarevichTate group and the group of rational points for some modular abelian varieties (Russian), Algebra i Analiz 1 (1989), no. 5, 171196. Translation in Leningrad Math. J. 1 (1990), 12291253. MR 91c:11032
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 H. Montgomery and D. Rohrlich, On the Lfunctions of canonical Hecke characters of imaginary quadratic fields, Duke Math. J. 49 (1982), 937942. MR 84e:12014
 [MY]
 S. Miller and T. H. Yang, Nonvanishing of the central derivative of canonical Hecke Lfunctions, Math. Res. Letters 7 (2000), 263277. MR 2001i:11058
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 T. Nagel, Généralisation d'un théorème de Tchebycheff, J. Math. Pures. Appl. (8) 4 (1921), 343356.
 [R]
 D. E. Rohrlich, A modular version of Jensen's formula, Math. Proc. Cambridge Philos. Soc. 95 (1984), 1520. MR 85d:11043
 [RVY]
 F. Rodriguez Villegas and T. H. Yang, Central values of Hecke functions of CM number fields, Duke Math. J. 98 (1999), 541564. MR 2000j:11074
 [ST]
 C. L. Stewart and J. Top, On ranks of twists of elliptic curves and powerfree values of binary forms, J. Amer. Math. Soc. 8 (1995), 943973. MR 95m:11055
 [Y]
 T. H. Yang, Nonvanishing of central Hecke values and rank of certain elliptic curves, Compositio Math. 117 (1999), 337359. MR 2001a:11093
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Additional Information
Jorge JimenezUrroz
Affiliation:
Departamento de Matemática Aplicada IV, ETSETB, Universidad Politecnica de Catalunya, 08034 Barcelona, España
Email:
jjimenez@mat.upc.es
Tonghai Yang
Affiliation:
Department of Mathematics, University of Wisconsin Madison, Madison, Wisconsin 53717
Email:
thyang@math.wisc.edu
DOI:
http://dx.doi.org/10.1090/S000299470303277X
PII:
S 00029947(03)03277X
Keywords:
Theta functions,
elliptic curves,
Heegner points
Received by editor(s):
February 25, 2002
Received by editor(s) in revised form:
December 20, 2002
Published electronically:
June 18, 2003
Additional Notes:
The first author was partially supported by PB900179 and Ramon y Cajal program of MCYT. The second author was partially supported by NSF grant DMS0070476
Article copyright:
© Copyright 2003 American Mathematical Society
