Heegner points, cycles and Maass forms.

*(English)*Zbl 0787.11016In the important paper [J. L. Waldspurger, J. Math. Pures Appl. 60, 375-484 (1981; Zbl 0431.10015)] it was proved that the value at the center of the critical strip of the \(L\)-series of a holomorphic newform \(f\) of even integral weight on \(\Gamma_ 0(N)\) twisted by a quadratic character of conductor \(t\) is proportional to the square of the \(t\)-th Fourier coefficient of a form \(g\) of half-integral weight corresponding to \(f\) under the Shimura lift. This phenomenon was studied later by several authors including D. Zagier and the present reviewer. In particular, in [W. Kohnen, Math. Ann. 271, 237-268 (1985; Zbl 0542.10018)] for \(N\) squarefree a generalization of Waldspurger’s theorem was given which relates the product of two Fourier coefficients of \(g\) to the integral of \(f\) over certain geodesic periods on the modular curve \(X_ 0(N)\).

In the present paper the authors study similar questions in the context of a non-holomorphic Maass cusp form \(\phi\) on \(SL_ 2(\mathbb{Z})\), relating both special values at complex multiplication points of \(\phi\) as well as integrals of \(\phi\) over closed geodesics to products of Fourier coefficients of non-holomorphic cusp forms of weight 1/2 corresponding to \(\phi\) by an analogous construction as the Shimura lift in the holomorphic case.

In the present paper the authors study similar questions in the context of a non-holomorphic Maass cusp form \(\phi\) on \(SL_ 2(\mathbb{Z})\), relating both special values at complex multiplication points of \(\phi\) as well as integrals of \(\phi\) over closed geodesics to products of Fourier coefficients of non-holomorphic cusp forms of weight 1/2 corresponding to \(\phi\) by an analogous construction as the Shimura lift in the holomorphic case.

Reviewer: W.Kohnen (Münster)

##### MSC:

11F37 | Forms of half-integer weight; nonholomorphic modular forms |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11F55 | Other groups and their modular and automorphic forms (several variables) |

##### Keywords:

integrals over closed geodesics; non-holomorphic cusp forms of half- integral weight; \(L\)-series; non-holomorphic Maass cusp form; special values at complex multiplication points; products of Fourier coefficients; Shimura lift
PDF
BibTeX
XML
Cite

\textit{S. Katok} and \textit{P. Sarnak}, Isr. J. Math. 84, No. 1--2, 193--227 (1993; Zbl 0787.11016)

Full Text:
DOI

##### References:

[1] | [AL] A. O. L. Atkin and J. Lehner, Hecke operators on \(\Gamma\)0(m), Math. Ann.185 (1970), 134–160. · Zbl 0185.15502 · doi:10.1007/BF01359701 |

[2] | [D] W. Duke,Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math.92 (1988), 73–90. · Zbl 0628.10029 · doi:10.1007/BF01393993 |

[3] | [GGP] I. M. Gelfand, M. I. Graev and I. I. Piatetskii-Shapiro,Representation Theory and Automorphic Functions, Saunders, Philadelphia, 1966. |

[4] | [GH] D. Goldfeld and J. Hoffstein,Eisenstein series of 1/2-integral weight and the mean value of real Dirichlet L-series, Invent. Math.80 (1985), 185–208. · Zbl 0564.10043 · doi:10.1007/BF01388603 |

[5] | [GR] I. Gradshtein and I. Ryzhik,Tables of Integrals, Series, and Products, Academic Press, New York, 1965. |

[6] | [H] D. Hejhal,Some Dirichlet series with coefficients related to periods of automorphic eigenforms I, II, Proc. Japan Acad., Ser. A58 (1982), 413–417;59 (1983), 335–338. · Zbl 0516.10018 · doi:10.3792/pjaa.58.413 |

[7] | [Ko1] W. Kohnen, Modular forms of half-integral weight on \(\Gamma\)0(4), Math. Ann.248 (1980), 149–266. · Zbl 0422.10015 · doi:10.1007/BF01420529 |

[8] | [Ko2] W. Kohnen,Fourier coefficients of modular forms of half-integral weight, Math. Ann.271 (1985), 237–268. · Zbl 0553.10020 · doi:10.1007/BF01455989 |

[9] | [KoZ] W. Kohnen and D. Zagier,Values of L-series of modular forms at the center of the critical strip, Invent. Math.64 (1981), 175–198. · Zbl 0468.10015 · doi:10.1007/BF01389166 |

[10] | [L] S. Lang, SL(2,R), Addison-Wesley, Reading, Mass. 1975. |

[11] | [M] H. Maass,Über die räumliche Verteilung der Punkte in Gittern mit indefiniter Metrik, Math. Ann.138 (1959), 287–315. · Zbl 0089.06102 · doi:10.1007/BF01344150 |

[12] | [Ma] W. Magnus and F. Oberhettinger,Formulas and Theorems for the Functions of Mathematical Physics, Springer, Berlin, Heidelberg, New York, 1966. · Zbl 0143.08502 |

[13] | [N] S. Niwa,Modular forms of half-integral weight and the integral of certain theta functions, Nagoya Math. J.56 (1974), 183–202. |

[14] | [N1] S. Niwa,On Shimura’s trace formula, Nagoya Math. J.66 (1977), 183–202. · Zbl 0351.10018 |

[15] | [S] G. Shimura,On modular forms of half-integral weight, Ann. Math., II Ser.97 (1973), 440–481. · Zbl 0266.10022 · doi:10.2307/1970831 |

[16] | [S2] G. Shimura,On Fourier coefficients of Hilbert modular forms of half-integral weight, preprint, 1991. |

[17] | [Si] C. L. Siegel,Lectures on Quadratic Forms, Tata Institute of Fundamental Research, Bombay, 1957. |

[18] | [Sh] T. Shintani,On construction of holomorphic cusp forms of half-integral weight, Nagoya Math. J.58 (1975), 83–126. · Zbl 0316.10016 |

[19] | [T] A. Terras,Harmonic Analysis on Symmetric Spaces and Applications, I, Springer-Verlag, New York, 1985. · Zbl 0574.10029 |

[20] | [W] J. L. Waldspurger,Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl.60 (1981), 375–484. · Zbl 0431.10015 |

[21] | [Z] D. Zagier,Eisenstein series and the Riemann Zeta function, inAutomorphic Forms, Representation Theory and Arithmetic, Tata Institute of Fundamental Research, Bombay, 1979, pp. 275–301. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.