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.
 [B]
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 ed.,
Cambridge University Press, Cambridge, 1997. MR 1628193
(99e:11068)
 [G1]
Benedict
H. Gross, Arithmetic on elliptic curves with complex
multiplication, Lecture Notes in Mathematics, vol. 776, Springer,
Berlin, 1980. With an appendix by B. Mazur. MR 563921
(81f:10041)
 [G2]
Benedict
H. Gross, Minimal models for elliptic curves with complex
multiplication, Compositio Math. 45 (1982),
no. 2, 155–164. MR 651979
(84j:14044)
 [Go]
Fernando
Q. Gouvêa, The squarefree sieve over number fields, J.
Number Theory 43 (1993), no. 1, 109–122. MR 1200814
(93m:11045), http://dx.doi.org/10.1006/jnth.1993.1012
 [GM]
F.
Gouvêa and B.
Mazur, The squarefree sieve and the rank of
elliptic curves, J. Amer. Math. Soc.
4 (1991), no. 1,
1–23. MR
1080648 (92b:11039), http://dx.doi.org/10.1090/S08940347199110806487
 [HR]
H.
Halberstam and H.E.
Richert, Sieve methods, Academic Press [A subsidiary of
Harcourt Brace Jovanovich, Publishers], LondonNew York, 1974. London
Mathematical Society Monographs, No. 4. MR 0424730
(54 #12689)
 [HW]
G.
H. Hardy and E.
M. Wright, An introduction to the theory of numbers, 5th ed.,
The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
(81i:10002)
 [J]
Jorge
JimenezUrroz, Nontrivial zeroes for quadratic twists of
HasseWeil 𝐿functions, J. Number Theory 77
(1999), no. 2, 331–335. MR 1702208
(2000d:11088), http://dx.doi.org/10.1006/jnth.1999.2391
 [KL]
V.
A. Kolyvagin and D.
Yu. Logachëv, Finiteness of the ShafarevichTate group and the
group of rational points for some modular abelian varieties, Algebra i
Analiz 1 (1989), no. 5, 171–196 (Russian);
English transl., Leningrad Math. J. 1 (1990), no. 5,
1229–1253. MR 1036843
(91c:11032)
 [MR]
Hugh
L. Montgomery and David
E. Rohrlich, On the 𝐿functions of canonical Hecke
characters of imaginary quadratic fields. II, Duke Math. J.
49 (1982), no. 4, 937–942. MR 683009
(84e:12014)
 [MY]
Stephen
D. Miller and Tonghai
Yang, Nonvanishing of the central derivative of canonical Hecke
𝐿functions, Math. Res. Lett. 7 (2000),
no. 23, 263–277. MR 1764321
(2001i:11058), http://dx.doi.org/10.4310/MRL.2000.v7.n3.a2
 [N]
T. Nagel, Généralisation d'un théorème de Tchebycheff, J. Math. Pures. Appl. (8) 4 (1921), 343356.
 [R]
David
E. Rohrlich, A modular version of Jensen’s formula,
Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 1,
15–20. MR
727075 (85d:11043), http://dx.doi.org/10.1017/S0305004100061259
 [RVY]
Fernando
Rodriguez Villegas and Tonghai
Yang, Central values of Hecke 𝐿functions of CM number
fields, Duke Math. J. 98 (1999), no. 3,
541–564. MR 1695801
(2000j:11074), http://dx.doi.org/10.1215/S0012709499098174
 [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), no. 4, 943–973. MR 1290234
(95m:11055), http://dx.doi.org/10.1090/S08940347199512902345
 [Y]
Tonghai
Yang, Nonvanishing of central Hecke 𝐿values and rank of
certain elliptic curves, Compositio Math. 117 (1999),
no. 3, 337–359. MR 1702416
(2001a:11093), http://dx.doi.org/10.1023/A:1000934108242
 [B]
 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
 [HW]
 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
 [J]
 J. Jiménez Urroz, Nontrivial zeroes for quadratic twists of HasseWeil Functions, J. Number Theory 77 (1999), 331335. MR 2000d:11088
 [KL]
 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
 [MR]
 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
 [N]
 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
