We consider a family of weight one Eisenstein series associated with the imaginary quadratic fields. The subspace of the Eisenstein series associated with the Kronecker symbol is characterized by the quadratic forms generated from the genus theory of Gauss; we will derive a family of identities connected with these quadratic fields.
SHEN Li-chien
. Weight one Eisenstein series associated with imaginary quadratic fields[J]. Journal of East China Normal University(Natural Science), 2019
, 2019(2)
: 7
-20
.
DOI: 10.3969/j.issn.1000-5641.2019.02.002
[1] COHN H. Advanced Number Theory[M]. New York:Dover, 1980.
[2] BOREVICH Z I, SHAFAREVICH I R. Number Theory[M]. New York:Academic Press, 1966.
[3] SHEN L C. On Eisenstein series generated from twisting of the geometric series[J]. Journal of East China Normal University (Natural Science), 2017, 6:1-24.
[4] SHEN L C. On a class of q-series related to quadratic forms[J], Bulletin of the Institute of Mathematics Academia Sinica, 1998, 26(2):111-126.
[5] SIEGEL C L. Lectures on Advanced Analytic Number Theory[M]. Bombay:Tata Institute, 1961.
[6] SHEN L C. On separation of quadratic forms on the imaginary quadratic field in its Hilbert class fields[J]. Proceedings of the American Mathematical Society, 2008, 136(9):3061-3067.
[7] DIAMOND F, SHURMAN J. A First Course in Modular Forms[M]. New York:Springer, 2005.
[8] COX D. Primes of the forms x2 +ny2[M]//Fermat, Class Field Theory, and Complex Multiplication. New York:Wiley, 1989.
[9] KOBLITZ N. Introduction to Elliptic Curves and Modular Forms[M]. 2nd ed. New York:Springer-Verlag, 1993.