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Algebraic theta functions and the $$p$$-adic interpolation of Eisenstein-Kronecker numbers. (English) Zbl 1205.11076
Let $$\Gamma$$ be a lattice in $$\mathbb C$$, and let $$A$$ denote the area of its fundamental domain divided by $$\pi$$. Then for integers $$a$$ and $$b > a+2$$, and for complex numbers $$z$$ and $$w$$, the Eisenstein-Kronecker numbers (see in particular, A. Weil’s book [Elliptic functions according to Eisenstein and Kronecker. Berlin: Springer (1999; Zbl 0955.11001)]) are defined by $e_{a,b}^*(z,w) = \sum_{\gamma \in \Gamma \setminus \{-z\}} \frac{(\overline{z} + \overline{\gamma})^a}{(z+\gamma)^b}\, \langle \gamma, w \rangle_\Gamma,$ where $\langle z, w \rangle_\Gamma = \exp\Big(\frac{z\overline{w} - w \overline{z}}A\Big).$ The sum converges for all $$b > a-2$$, but can be given a meaning for any $$a \geq 0$$ and $$b > 0$$ by analytic continuation. The Eisenstein-Kronecker numbers can be used to express special values of Hecke $$L$$-functions associated to the field of complex multiplication of a lattice $$\Gamma$$, just as generalized Bernoulli numbers can be used to express special values of Dirichlet $$L$$-functions.
In Section 1, the authors introduce theta functions attached to the Poincaré bundle of elliptic curves, and show that the Kronecker theta function is a generating function for the Eisenstein-Kronecker numbers. Section 2 covers algebraic and $$p$$-adic properties of reduced theta functions on abelian varieties with complex multiplication, and in Section 3 these are applied to the construction of a $$p$$-adic measure and R. I. Yager’s [Ann. Math. (2) 115, 411–449 (1982; Zbl 0496.12010)] $$p$$-adic $$L$$-function. Section 4 is devoted to the calculation of the Laurent expansion of the Kronecker theta function. In a subsequent article, the authors intend to study the $$p$$-divisibility of critical values of Hecke $$L$$-functions associated to Hecke characters of complex quadratic number fields.

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14K25 Theta functions and abelian varieties 14K22 Complex multiplication and abelian varieties 14K05 Algebraic theory of abelian varieties
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##### References:
  K. Bannai, On the $$p$$-adic realization of elliptic polylogarithms for CM-elliptic curves , Duke Math. J. 113 (2002), 193–236. · Zbl 1019.11018 · doi:10.1215/S0012-7094-02-11321-0  K. Bannai, H. Furusho, and S. Kobayashi, $$p$$-adic Eisenstein-Kronecker functions and the elliptic polylogarithm for CM elliptic curves , · Zbl 1336.11050 · arxiv.org  K. Bannai and S. Kobayashi, Integral structures on $$p$$-adic Fourier theory , · arxiv.org  -, Divisibility by inert primes of the critical values of Hecke $$L$$-functions associated to imaginary quadratic fields , in preparation.  K. Bannai, S. Kobayashi, and T. Tsuji, On the de Rham and $$p$$-adic realizations of the elliptic polylogarithm for CM elliptic curves , to appear in Ann. Sci. École Norm. Sup. (4), · Zbl 1197.11073 · arxiv.org  I. Barsotti, “Considerazioni sulle funzioni theta” in Symposia Mathematica , Vol. III (INDAM, Rome, 1968/69) , Academic Press, London, 1970, 247–277. · Zbl 0194.52201  A. Beilinson and A. Levin, “The elliptic polylogarithm” in Motives (Seattle, WA, 1991) , Proc. Sympos. Pure Math. 55 , pt. 2 (1994), 123–190. · Zbl 0817.14014  D. Bernardi, C. Goldstein, and N. Stephens, Notes $$p$$-adiques sur les courbes elliptiques , J. Reine Angew. Math. 351 (1984), 129–170. · Zbl 0529.14018 · crelle:GDZPPN002201658 · eudml:152649  C. Birkenhake and H. Lange, Complex abelian varieties , 2nd ed., Grundlehren Math. Wiss. 302 , Springer, Berlin, 2004. · Zbl 1056.14063  J. L. Boxall, A new construction of $$p$$-adic $$L$$-functions attached to certain elliptic curves with complex multiplication , Ann. Inst. Fourier (Grenoble) 36 (1986), 31–68. · Zbl 0608.14015 · doi:10.5802/aif.1068 · numdam:AIF_1986__36_4_31_0 · eudml:74738  -, $$p$$-adic interpolation of logarithmic derivatives associated to certain Lubin-Tate formal groups , Ann. Inst. Fourier (Grenoble) 36 (1986), 1–27. · Zbl 0587.12007 · doi:10.5802/aif.1056 · numdam:AIF_1986__36_3_1_0 · eudml:74724  M. Candilera and V. Cristante, Bi-extensions associated to divisors on abelian varieties and theta functions , Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 10 (1983), 437–491. · Zbl 0576.14043 · numdam:ASNSP_1983_4_10_3_437_0 · eudml:83913  M. Chellali, Congruences entre nombres de Bernoulli-Hurwitz dans le cas supersingulier , J. Number Theory 35 (1990), 157–179. · Zbl 0705.11008 · doi:10.1016/0022-314X(90)90110-D  J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer , Invent. Math. 39 (1977), 223–251. · Zbl 0359.14009 · doi:10.1007/BF01402975 · eudml:142468  -, On $$p$$-adic $$L$$-functions and elliptic units , J. Austral. Math. Soc. Ser. A 26 (1978), 1–25. · doi:10.1017/S1446788700011459  P. Colmez and L. Schneps, $$p$$-adic interpolation of special values of Hecke $$L$$-functions , Compositio Math. 82 (1992), 143–187. · Zbl 0777.11049 · numdam:CM_1992__82_2_143_0 · eudml:90151  V. Cristante, Theta functions and Barsotti-Tate groups , Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 181–215. · Zbl 0438.14027 · numdam:ASNSP_1980_4_7_2_181_0 · eudml:83835  -, $$p$$-adic theta series with integral coefficients , Astérisque 119 –120. (1984), 169–182. · Zbl 0559.14030  R. M. Damerell, $$L$$-functions of elliptic curves with complex multiplication, I , Acta Arith. 17 (1970), 287–301. · Zbl 0209.24603 · matwbn.icm.edu.pl · eudml:204958  -, $$L$$-functions of elliptic curves with complex multiplication, II , Acta Arith. 19 (1971), 311–317. · Zbl 0229.12015 · eudml:205036  E. De Shalit, Iwasawa theory of elliptic curves with complex multiplication , Boston, Academic Press, 1987. · Zbl 0674.12004  E. De Shalit and E. Z. Goren, On special values of theta functions of genus two , Ann. Inst. Fourier (Grenoble), 47 (1997), 775–799. · Zbl 0974.11027 · doi:10.5802/aif.1580 · numdam:AIF_1997__47_3_775_0 · eudml:75244  L. Fourquaux, Fonctions $$L$$ $$p$$-adiques des courbes elliptiques de type CM , preprint, 2001.  Y. Fujiwara, On divisibilities of special values of real analytic Eisenstein series , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1988), 393–410. · Zbl 0669.10049  C. Goldstein and N. Schappacher, Séries d’Eisenstein et fonctions $$L$$ des courbes elliptiques à multiplication complexe , J. Reine Angew. Math. 327 (1981), 184–218. · Zbl 0456.12007 · crelle:GDZPPN002198800 · eudml:152388  N. M. Katz, $$p$$-adic interpolation of real analytic Eisenstein series , Ann. of Math. 104 (1976), 459–571. JSTOR: · Zbl 0354.14007 · doi:10.2307/1970966 · links.jstor.org  -, Divisibilities, congruences, and Cartier duality, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1982), 667–678.  S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), 1–36. · Zbl 1047.11105 · doi:10.1007/s00222-002-0265-4  M. Kurihara, On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction, I , Invent. Math. 149 (2002), 195–224. · Zbl 1033.11028 · doi:10.1007/s002220100206  S. Lang, Introduction to Algebraic and abelian functions , 2nd ed., Grad. Texts Math. 89 , Springer, New York, 1982. · Zbl 0513.14024  -, Complex multiplication , Grundlehren Math. Wiss. 255 , Springer, New York, 1983.  -, Cyclotomic fields I and II , combined 2nd ed., with an appendix by Karl Rubin, Grad. Texts Math. 121 , Springer, New York, 1990. · Zbl 0704.11038  B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves , Invent. Math. 25 (1974), 1–61. · Zbl 0281.14016 · doi:10.1007/BF01389997 · eudml:142281  B. Mazur and J. Tate, The $$p$$-adic sigma function , Duke. Math. J. 62 (1991), 663–688. · Zbl 0735.14020 · doi:10.1215/S0012-7094-91-06229-0  D. Mumford, Tata lectures on theta, III , with the collaboration of Madhav Nori and Peter Norman, Progr. Math. 97 , Birkhäuser, Boston, 1991. · Zbl 1124.14043  P. Norman, $$p$$-adic theta functions , Amer. J. Math. 107 (1985), 617–661. JSTOR: · Zbl 0587.14028 · doi:10.2307/2374372 · links.jstor.org  B. Perrin-Riou, Arithmétique des courbes elliptiques et théorie d’Iwasawa , Mém. Soc. Math. France (N.S.) 17 (1984), 1–130. · Zbl 0599.14020 · numdam:MSMF_1984_2_17__1_0 · eudml:94856  A. Polishchuk, Abelian varieties, theta functions and the Fourier transform , Cambridge Tracts Math. 153 , Cambridge Univ. Press, Cambridge, 2003. · Zbl 1018.14016  G. Robert, Unités elliptiques et formules pour le nombre de classes , C. R. Acad. Sci. Paris Sér. A–B 277 (1973), A1143–A1146. · Zbl 0311.12006  P. Schneider and J. Teitelbaum, $$p$$-adic Fourier theory , Doc. Math. 6 (2001), 447–481. · Zbl 1028.11069 · eudml:123109  G. Shimura, Theta functions with complex multiplication , Duke Math. J. 43 (1976), 673–696. · Zbl 0371.14022 · doi:10.1215/S0012-7094-76-04353-2  -, On the derivatives of theta functions and modular forms , Duke Math. J. 44 (1977), 365–387. · Zbl 0371.14023 · doi:10.1215/S0012-7094-77-04416-7  -, Abelian Varieties with Complex Multiplication and Modular Functions , Princet. Math. Ser. 46 , Princeton Univ. Press, Princeton, 1998. · Zbl 0908.11023  J. Tilouine, Fonctions $$L$$ $$p$$-adiques à deux variables et $$\bbZ_p^2$$-extensions , Bull. Soc. Math. France 114 (1986), 3–66.  M. M. Višik and J. I. Manin, $$p$$-adic Hecke series for imaginary quadratic fields , Mat. Sb. (N.S.) 95 (137) (1974), 357–383., 471.  A. Weil, Elliptic Functions according to Eisenstein and Kronecker , Springer, Berlin, 1976. · Zbl 0318.33004  E. T. Whittaker and G. N. Watson, A Course of Modern Analysis , 4th ed., Cambridge Univ. Press, Cambridge, 1996. · Zbl 0951.30002  R. I. Yager, On two variable $$p$$-adic $$L$$-functions , Ann. of Math. 115 (1982), 411–449. JSTOR: · Zbl 0496.12010 · doi:10.2307/1971398 · links.jstor.org  S. Yamamoto, On $$p$$-adic $$L$$-functions for CM elliptic curves at supersingular primes , M.S. thesis, University of Tokyo, Tokyo, 2002.  D. Zagier, Periods of modular forms and Jacobi theta functions , Inv. Math. 104 (1991), 449–465. · Zbl 0742.11029 · doi:10.1007/BF01245085 · eudml:143891
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