×

Hecke eigenvalues for real quadratic fields. (English) Zbl 1117.11304

Summary: We describe an algorithm to compute the trace of Hecke operators acting on the space of Hilbert cusp forms defined relative to a real quadratic field with class number greater than one. Using this algorithm, we obtain numerical data for eigenvalues and characteristic polynomials of the Hecke operators. Within the limit of our computation, the conductors of the orders spanned by the Hecke eigenvalue for any principal split prime ideal contain a nontrivial common factor, which is equal to a Hecke \(L\)-value.

MSC:

11F60 Hecke-Petersson operators, differential operators (several variables)
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11R42 Zeta functions and \(L\)-functions of number fields
PDFBibTeX XMLCite
Full Text: DOI Euclid EuDML

References:

[1] Dirichlet P. G. L., Vorlesungen über Zahlentheorie (1894)
[2] Doi K., Invent. Math. 9 pp 1– (1969) · Zbl 0182.54301 · doi:10.1007/BF01389886
[3] Doi K., Invent. Math. 134 pp 547– (1998) · Zbl 0924.11035 · doi:10.1007/s002220050273
[4] Iwasawa K., Lectures on p-Adic L-Functions (1972) · Zbl 0236.12001
[5] Miyake T., Modular Forms (1989)
[6] Naganuma H., J. Math. Soc. Japan 25 pp 547– (1973) · Zbl 0259.10023 · doi:10.2969/jmsj/02540547
[7] Neukirch J., Class Field Theory (1986) · Zbl 0587.12001 · doi:10.1007/978-3-642-82465-4
[8] Okazaki R., J. Math. Kyoto Univ. 31 pp 1125– (1991)
[9] Saito H., J. Math. Kyoto Univ. 24 pp 285– (1984) · Zbl 0547.10027 · doi:10.1215/kjm/1250521332
[10] Shimura G., Introduction to the Arithmetic Theory of Automorphic Functions (1971) · Zbl 0221.10029
[11] DOI: 10.1215/S0012-7094-78-04529-5 · Zbl 0394.10015 · doi:10.1215/S0012-7094-78-04529-5
[12] Shimura G., Duke Math. J. 63 pp 557– (1991) · Zbl 0752.11021 · doi:10.1215/S0012-7094-91-06324-6
[13] Shintani T., J. Fac. Sci. Univ. Tokyo Sect. I A Math. 23 pp 393– (1976)
[14] Siegel C. L., Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. pp 87– (1969)
[15] Weil A., Basic Number Theory (1967) · Zbl 0176.33601
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.