$p$-adic interpolation of square roots of central values of Hecke $L$-functions

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$p$-adic interpolation of square roots of central values of Hecke $L$-functions

Adriana Sofer

Source: Duke Math. J. Volume 83, Number 1 (1996), 51-78.

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Primary Subjects: 11F67

Secondary Subjects: 11F85

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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077244247
Mathematical Reviews number (MathSciNet): MR1388843
Zentralblatt MATH identifier: 00912120
Digital Object Identifier: doi:10.1215/S0012-7094-96-08303-9

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References

[1] R. Damerell, $L$-functions of elliptic curves with CM, I, Acta Arith. 17 (1970), 287–301.

Mathematical Reviews (MathSciNet): MR44:2758

Zentralblatt MATH: 0209.24603

[2] R. Damerell, $L$-functions of elliptic curves with CM, II, Acta Arith. 19 (1971), 311–317.

Mathematical Reviews (MathSciNet): MR53:2954

Zentralblatt MATH: 0229.12015

[3] E. de Shalit, Iwasawa Theory of Elliptic Curves with Complex Multiplication, Perspectives in Mathematics, vol. 3, Academic Press, Boston, 1987.

Mathematical Reviews (MathSciNet): MR89g:11046

Zentralblatt MATH: 0674.12004

[4] B. Dwork, On the Tate constant, Compositio Math. 61 (1987), no. 1, 43–59.

Mathematical Reviews (MathSciNet): MR88e:14030

Zentralblatt MATH: 0622.14016

[5] F. Gouvêa, Arithmetic of $p$-adic Modular Forms, Lecture Notes in Math., vol. 1304, Springer-Verlag, Berlin, 1988.

Mathematical Reviews (MathSciNet): MR91e:11056

Zentralblatt MATH: 0641.10024

[6] B. Gross, Arithmetic on Elliptic Curves with Complex Multiplication, Lecture Notes in Math., vol. 776, Springer-Verlag, Berlin, 1980.

Mathematical Reviews (MathSciNet): MR81f:10041

Zentralblatt MATH: 0433.14032

[7] B. Gross, Minimal models for elliptic curves with complex multiplication, Compositio Math. 45 (1982), no. 2, 155–164.

Mathematical Reviews (MathSciNet): MR84j:14044

Zentralblatt MATH: 0541.14010

[8] N. Jochnowitz, Congruences between modular forms of integral weights, and implications for class numbers and elliptic curves, to appear in Invent. Math.

[9] N. Jochnowitz, A $p$-adic theory for half-integral weight modular forms, to appear.

[10] N. Katz, $p$-adic properties of modular schemes and modular forms, Modular Functions of One Variable III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., vol. 350, Springer-Verlag, Berlin, 1973, pp. 69–190.

Mathematical Reviews (MathSciNet): MR56:5434

Zentralblatt MATH: 0271.10033

Digital Object Identifier: doi:10.1007/978-3-540-37802-0_3

[11] N. Katz, $p$-adic $L$-functions via moduli of elliptic curves, Algebraic Geometry (Humboldt State Univ., Arcata, Calif., 1974), Proc. Sympos. Pure Math., vol. 29, Amer. Math. Soc., Providence, 1975, pp. 479–506.

Mathematical Reviews (MathSciNet): MR55:5635

Zentralblatt MATH: 0317.14009

[12] N. Katz, Higher congruences between modular forms, Ann. of Math. (2) 101 (1975), 332–367.

Mathematical Reviews (MathSciNet): MR54:5120

Zentralblatt MATH: 0356.10020

Digital Object Identifier: doi:10.2307/1970994

JSTOR: links.jstor.org

[13] N. Katz, $p$-adic interpolation of real analytic Eisenstein series, Ann. of Math. (2) 104 (1976), no. 3, 459–571.

Mathematical Reviews (MathSciNet): MR58:22071

Zentralblatt MATH: 0354.14007

Digital Object Identifier: doi:10.2307/1970966

JSTOR: links.jstor.org

[14] N. Katz, The Eisenstein measure and $p$-adic interpolation, Amer. J. Math. 99 (1977), no. 2, 238–311.

Mathematical Reviews (MathSciNet): MR58:5602

Zentralblatt MATH: 0375.12022

Digital Object Identifier: doi:10.2307/2373821

JSTOR: links.jstor.org

[15] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984.

Mathematical Reviews (MathSciNet): MR86c:11040

Zentralblatt MATH: 0553.10019

[16] N. Koblitz, $p$-adic congruences and modular forms of half integer weight, Math. Ann. 274 (1986), no. 2, 199–220.

Mathematical Reviews (MathSciNet): MR88a:11043

Zentralblatt MATH: 0571.10030

Digital Object Identifier: doi:10.1007/BF01457070

[17] S. Lang, Cyclotomic Fields, I and II, Graduate Texts in Math., vol. 121, Springer-Verlag, New York, 1990.

Mathematical Reviews (MathSciNet): MR91c:11001

Zentralblatt MATH: 0704.11038

[18] S. Lang, Elliptic Functions, Graduate Texts in Math., vol. 112, Springer-Verlag, New York, 1987.

Mathematical Reviews (MathSciNet): MR88c:11028

Zentralblatt MATH: 0615.14018

[19] N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972.

Mathematical Reviews (MathSciNet): MR50:2568

Zentralblatt MATH: 0271.33001

[20]¹ J. Lubin, One-parameter formal Lie groups over $p$-adic integer rings, Ann. of Math. (2) 80 (1964), 464–484.

Mathematical Reviews (MathSciNet): MR29:5827

Zentralblatt MATH: 0135.07003

Digital Object Identifier: doi:10.2307/1970659

JSTOR: links.jstor.org

[20]² J. Lubin, Correction to: “One-parameter formal Lie groups over $p$-adic integer rings”, Ann. of Math. (2) 84 (1966), 372.

Mathematical Reviews (MathSciNet): MR34:179

Zentralblatt MATH: 0146.25702

Digital Object Identifier: doi:10.2307/1970451

[21] Y. Manin and M. Vishik, $p$-adic Hecke series of imaginary quadratic fields, Mat. Sb. (N.S.) 95(137) (1974), 357–383, 471.

Mathematical Reviews (MathSciNet): MR51:8078

Zentralblatt MATH: 0352.12013

[22] F. Rodriguez-Villegas, On the square root of special values of certain $L$-series, Invent. Math. 106 (1991), no. 3, 549–573.

Mathematical Reviews (MathSciNet): MR93a:11098

Zentralblatt MATH: 0773.11034

Digital Object Identifier: doi:10.1007/BF01243924

[23] F. Rodriguez-Villegas and D. Zagier, Square roots of central values of Hecke $L$-series, Advances in Number Theory (Kingston, ON, 1991), Oxford Sci. Publ., Clarendon Press, Oxford, 1993, pp. 81–99.

Mathematical Reviews (MathSciNet): MR96j:11069

Zentralblatt MATH: 0791.11060

[24] K. Rubin, The one-variable main conjecture for elliptic curves with complex multiplication, $L$-Functions and Arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 353–371.

Mathematical Reviews (MathSciNet): MR92j:11055

Zentralblatt MATH: 0741.11028

Digital Object Identifier: doi:10.1017/CBO9780511526053.015

[25] J. P. Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer, Berlin, 1973, 191–268. Lecture Notes in Math., Vol. 350.

Mathematical Reviews (MathSciNet): MR53:7949a

Zentralblatt MATH: 0277.12014

Digital Object Identifier: doi:10.1007/978-3-540-37802-0_4

[26] J. P. Serre, Cours d'arithmétique, Presses Universitaires de France, Paris, 1977.

Mathematical Reviews (MathSciNet): MR58:16473

Zentralblatt MATH: 0376.12001

[27] G. Shimura, On some arithmetic properties of modular forms of one and several variables, Ann. of Math. (2) 102 (1975), no. 3, 491–515.

Mathematical Reviews (MathSciNet): MR58:10758

Zentralblatt MATH: 0327.10028

Digital Object Identifier: doi:10.2307/1971041

[28] G. Shimura, Modular forms of half-integral weight, Modular Functions of One Variable I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., vol. 320, Springer-Verlag, Berlin, 1973, pp. 57–74.

Mathematical Reviews (MathSciNet): MR58:519a

Zentralblatt MATH: 0268.10014

Digital Object Identifier: doi:10.1007/978-3-540-38509-7_3

[29] G. Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481.

Mathematical Reviews (MathSciNet): MR48:10989

Zentralblatt MATH: 0266.10022

Digital Object Identifier: doi:10.2307/1970831

JSTOR: links.jstor.org

[30] W. Sinnott, On the $\mu$-invariant of the $\Gamma$-transform of a rational function, Invent. Math. 75 (1984), no. 2, 273–282.

Mathematical Reviews (MathSciNet): MR85g:11112

Zentralblatt MATH: 0531.12004

Digital Object Identifier: doi:10.1007/BF01388565

[31] A. Sofer, $p$-adic interpolation of square roots of central values of Hecke $L$ -functions, Ph.D. thesis, the Ohio State University, 1993.

[32] A. Sofer, $p$-adic interpolation of half-integral weight modular forms, Arithmetic Geometry (Tempe, AZ, 1993), Contemp. Math., vol. 174, Amer. Math. Soc., Providence, 1994, pp. 119–128.

Mathematical Reviews (MathSciNet): MR95f:11032

Zentralblatt MATH: 0869.11041

[33] H. Stark, $L$-Functions at $s=1$. IV. First Derivatives at $s=0$, Adv. in Math. 35 (1980), no. 3, 197–235.

Mathematical Reviews (MathSciNet): MR81f:10054

Zentralblatt MATH: 0475.12018

Digital Object Identifier: doi:10.1016/0001-8708(80)90049-3

[34] G. Stevens, $\Lambda$-adic modular forms of half-integral weight and a $\Lambda$-adic Shintani lifting, Arithmetic Geometry (Tempe, AZ, 1993), Contemp. Math., vol. 174, Amer. Math. Soc., Providence, 1994, pp. 129–151.

Mathematical Reviews (MathSciNet): MR95h:11051

Zentralblatt MATH: 0869.11042

[35] H. Weber, Lehrbuch der Algebra, Vol. 3, Braunschweig, 1908, chapters II and III.

[36] R. Yager, On two variable $p$-adic $L$-functions, Ann. of Math. (2) 115 (1982), no. 2, 411–449.

Mathematical Reviews (MathSciNet): MR84b:14020

Zentralblatt MATH: 0496.12010

Digital Object Identifier: doi:10.2307/1971398

JSTOR: links.jstor.org

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