arXiv subjects

Shinji Fukuhara

Publications and source records attributed to Shinji Fukuhara.

8 recordsLinked to original sources

arXiv2017

Kauffman-Jones polynomial of a curve on a surface

We introduce a Kauffman-Jones type polynomial $\mathcal{L}_γ(A)$ for a curve $γ$ on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial $\mathcal{L}_γ(A)$ is a Laurent polynomial in one variable $A$ and is an invariant of the homotopy class of $γ$. As an application, we obtain an estimate in terms of the span of $\mathcal{L}_γ(A)$ for the minimum self-intersection number of a curve within its homotopy class. We then give a chord diagrammatic description of $\mathcal{L}_γ(A)$ and show some computational results on the span of $\mathcal{L}_γ(A)$.

math.GT↗

arXiv2015

Generalized Kronecker formula for Bernoulli numbers and self-intersections of curves on a surface

We present a new explicit formula for the $m$-th Bernoulli number $B_m$, which involves two integer parameters $a$ and $n$ with $0\le a\le m\le n$. If we set $a=0$ and $n=m$, then the formula reduces to the celebrated Kronecker formula for $B_m$. We give two proofs of our formula. One is analytic and uses a certain function in two variables. The other is algebraic and is motivated by a topological consideration of self-intersections of curves on an oriented surface.

math.NT↗

arXiv2010

A new basis for the space of modular forms

Let $G_{2n}$ be the Eisenstein series of weight $2n$ for the full modular group $Γ=SL_2(\ZZ)$. It is well-known that the space $M_{2k}$ of modular forms of weight $2k$ on $Γ$ has a basis $\{G_{4}^αG_{6}^β |\ α,β\in\ZZ,\ α,β\geq 0,\ 4α+6β=2k\}$. In this paper we will exhibit another (simpler) basis for $M_{2k}$. It is given by $\{G_{2k}\}\cup\{G_{4i}G_{2k-4i}\ |\ i=1,2,\ldots,d_k\}$ if $2k\equiv 0\pmod 4$, and $\{G_{2k}\}\cup\{G_{4i+2}G_{2k-4i-2}\ |\ i=1,2,\ldots,d_k\}$ if $2k\equiv 2\pmod 4$ where $d_k+1=\dim_{\CC} M_{2k}$.

math.NT↗

arXiv2009

The elliptic Apostol-Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent polynomial reciprocity laws. Our construction is based on Machide's result on his elliptic Dedekind-Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain formulas by Ramanujan, van der Pol, Rankin and Skoruppa.

math.NT↗

arXiv2009

Twisted Hecke L-values and period polynomials

Let $f_1,...,f_d$ be an orthogonal basis for the space of cusp forms of even weight $2k$ on $Γ_0(N)$. Let $L(f_i,s)$ and $L(f_i,χ,s)$ denote the $L$-function of $f_i$ and its twist by a Dirichlet character $χ$, respectively. In this note, we obtain a ``trace formula'' for the values $L(f_i,χ,m)\overline{L(f_i,n)}$ 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 precisly the value of the ratio $L(f,χ,m)/L(f,n)$ for a Hecke eigenform $f$.

math.NT↗

arXiv2007

Period polynomials and explicit formulas for Hecke operators on Γ_0(2)

Let S_{w+2}(Γ_0(N)) be the vector space of cusp forms of weight w+2 on the congruence subgroup Γ_0(N). We first determine explicit formulas for period polynomials of elements in S_{w+2}(Γ_0(N)) by means of Bernoulli polynomials. When N=2, from these explicit formulas we obtain new bases for S_{w+2}(Γ_0(2)), and extend the Eichler-Shimura-Manin isomorphism theorem to Γ_0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ_0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on S_{w+2}(Γ_0(2)). As an application of our main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of S_{w+2}(Γ_0(2)).

math.NT↗

arXiv2006

Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials

Let S_{w+2} be the vector space of cusp forms of weight w+2 on the full modular group, and let S_{w+2}^* denote its dual space. Periods of cusp forms can be regarded as elements of S_{w+2}^*. The Eichler-Shimura isomorphism theorem asserts that odd (or even) periods span S_{w+2}^*. However, periods are not linearly independent; in fact, they satisfy the Eichler-Shimura relations. This leads to a natural question: which periods would form a basis of S_{w+2}^*. First we give an answer to this question. Passing to the dual space S_{w+2}, we will determine a new basis for S_{w+2}. The even period polynomials of this basis elements are expressed explicitly by means of Bernoulli polynomials. Next we consider three spaces--S_{w+2}, the space of even Dedekind symbols of weight w with polynomial reciprocity laws, and the space of even period polynomials of degree w. There are natural correspondences among these three spaces. All these spaces are equipped with compatible action of Hecke operators. We will find explicit form of period polynomials and the actions of Hecke operators on the period polynomials. Finally we will obtain explicit formulas for Hecke operators on S_{w+2} in terms of Bernoulli numbers B_k and divisor functions sigma_k(n), which are quite different from the Eichler-Selberg trace formula.

math.NT↗

arXiv2004

Hecke operators on weighted Dedekind symbols

Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind symbols. We prove that these newly introduced operators are compatible with Hecke operators on the space of modular forms. As an application, we present formulae to give Fourier coefficients of Hecke eigenforms. In particular we give explicit formulae for generalized Ramanujan's tau functions.

math.NT↗