Browsing by Author "Fukuhara, Shinji"

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A basis for S (k) (I" (0)(4)) and representations of integers as sums of squares
(2012-05-01)
Fukuhara, Shinji
;
Yang, Yifan
;
應用數學系
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Department of Applied Mathematics
In this paper, we find a basis for the space S (k) (I" (0)(4)) of cusp forms of even weight k for the congruence subgroup I" (0)(4) in terms of Eisenstein series. As an application, we obtain formulas for r (4s) (n), the number of ways to represent a nonnegative integer n as a sum of 4s integral squares.
BASES FOR S-k(Gamma(1)(4)) AND FORMULAS FOR EVEN POWERS OF THE JACOBI THETA FUNCTION
(2013-12-01)
Fukuhara, Shinji
;
Yang, Yifan
;
應用數學系
;
Department of Applied Mathematics
We find a basis for the space S-k(Gamma(1)(4)) of cusp forms of weight k for the congruence subgroup G1(4) in terms of Eisenstein series. As an application, we obtain formulas for r(2k)(n), the number of ways to represent a non-negative integer n as sums of 2k integer squares.
Period polynomials and explicit formulas for Hecke operators on Gamma(0)(2)
(2009-03-01)
Fukuhara, Shinji
;
Yang, Yifan
;
應用數學系
;
Department of Applied Mathematics
Let S(w)+(2) (Gamma(0)(N)) be the vector space of cusp forms of weight w + 2 on the congruence subgroup Gamma(0)(N). We first determine explicit formulas for period polynomials of elements in S(w+2)(Gamma(0)(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for S(w+2)(Gamma(0)(2)), and extend the Eichler-Shimura-Manin isomorphism theorem to Gamma(0)(2). This implies that there are natural correspondences between the spaces of cusp forms on Gamma(0)(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on S(w+2)(Gamma(0)(2)). As an application of main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of S(w+2)(Gamma(0)(2)).
Twisted Hecke L-values and period polynomials
(2010-04-01)
Fukuhara, Shinji
;
Yang, Yifan
;
應用數學系
;
Department of Applied Mathematics
Let f(1),..., f(d) be an orthogonal basis for the space of cusp forms of even weight 2k on Gamma(0)(N). Let (f(i), s) and L(f(i), chi, s) denote the L-function of f(i) and its twist by a Dirichlet character chi, respectively. In this note, we obtain a "trace formula" for the values L(f(i), chi, m)<(L(f(i), n))over bar> at integers m and n with 0 < m, n < 2k and proper parity. In the case N = 1 or N = 2, the formula gives us a convenient way to evaluate precisely the value of the ratio L(f, chi, m)/L(f, n) for a Hecke eigenform f. (C) 2009 Elsevier Inc. All rights reserved.