On linear relations for \(L\)-values over real quadratic fields. (English) Zbl 1431.11064
Summary: In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The paper focuses on the Hilbert modular forms over real quadratic fields. We will state a construction of relations between the special values of \(L\)-functions, especially at 0, and arithmetic functions. We will also give a relation between the sum of squares functions with underlying fields \(\mathbb{Q}(\sqrt{D})\) and \(\mathbb{Q}\).
MSC:
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
References:
| [1] | Cohen, H.: Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann. 217, 217-285 (1975) · Zbl 0311.10030 · doi:10.1007/BF01436180 |
| [2] | Eichler, M.: On the class number of imaginary quadratic fields and the sums of divisors of natural numbers. J. Indian Math. Soc. 19, 153-180 (1955) · Zbl 0068.03302 |
| [3] | Hiraga, K., Ikeda, T.: On the Kohnen plus space for Hilbert modular forms of half-Integral weight I. Compos. Math. 149, 1963-2010 (2013) · Zbl 1368.11044 · doi:10.1112/S0010437X13007276 |
| [4] | Kohnen, W.: Modular forms of half-integral weight on \[\Gamma_0(4)\] Γ0(4). Math. Ann. 248, 249-266 (1980) · Zbl 0416.10023 · doi:10.1007/BF01420529 |
| [5] | Kojima, H.: An arithmetic of Hermitian modular forms of degree two. Invent. Math. 69, 217-227 (1982) · Zbl 0502.10011 · doi:10.1007/BF01399502 |
| [6] | Okazaki, R.: On evaluation of \[L\] L-functions over real quadratic fields. J. Math. Kyoto Univ. 31, 1125-1153 (1991) · Zbl 0776.11071 · doi:10.1215/kjm/1250519681 |
| [7] | Shintani, T.: On evaluation of zeta functions of totally real algebraic number fields at nonpositive integer. J. Fac. Sci. Univ. Tokyo Sec. IA 23, 393-417 (1976) · Zbl 0349.12007 |
| [8] | Su, R.-H.: Eisenstein series in the Kohnen plus space for Hilbert modular forms. Int J Number Theory 12, 691-723 (2016) · Zbl 1368.11047 · doi:10.1142/S1793042116500469 |
| [9] | Weil, A.: Basic Number Theory. Springer, Berlin (1974) · Zbl 0326.12001 · doi:10.1007/978-3-642-61945-8 |
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.
