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Yifan Yang

Publications and source records attributed to Yifan Yang.

359 records · Page 20Linked to original sources
arXiv2013

Ramanujan-type identities for Shimura curves

In 1914, Ramanujan gave a list of 17 identities expressing $1/π$ as linear combinations of values of hypergeometric functions at certain rational numbers. Since then, identities of similar nature have been discovered by many authors. Nowadays, one of the standard approaches to this kind of identities uses the theory of modular curves. In this paper, we will consider the case of Shimura curves and obtain Ramanujan-type formulas involving special values of hypergeometric functions and products of Gamma values. The product of Gamma values are related to periods of elliptic curves with complex multiplication by Q(\sqrt{-3}) and Q(\sqrt{-4}).

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arXiv2012

Computing modular equations for Shimura curves

In the classical setting, the modular equation of level $N$ for the modular curve $X_0(1)$ is the polynomial relation satisfied by $j(τ)$ and $j(Nτ)$, where $j(τ)$ is the standard elliptic $j$-function. In this paper, we will describe a method to compute modular equations in the setting of Shimura curves. The main ingredient is the explicit method for computing Hecke operators on the spaces of modular forms on Shimura curves developed in [13].

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arXiv2011

Schwarzian differential equations and Hecke eigenforms on Shimura curves

Let $X$ be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on $X$ in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation $$_2F_1(1/24,7/24,5/6, -\frac{2^{10}\cdot3^3\cdot5}{11^4})=\sqrt6 \sqrt[6]{\frac{11}{5^5}} $$ and other similar identities.

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arXiv2011

Jacobsthal identity for Q(sqrt(-2))

Let $p$ be a prime congruent to 1 or 3 modulo 8 so that the equation $p=a^2+2b^2$ is solvable in integers. In this paper, we obtain closed-form expressions for $a$ and $b$ in terms of Jacobsthal sums. This is analogous to a classical identity of Jacobsthal.

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arXiv2011

Modular forms of half-integral weights on SL(2,Z)

In this paper, we prove that, for an integer $r$ with $(r,6)=1$ and $0<r<24$ and a nonnegative even integer $s$, the set {η(24τ)^rf(24τ):f(τ)\in M_s(1)} is isomorphic to S_{r+2s-1}^{\text{new}}(6,-(\frac8r),-(\frac{12}r))\otimes(\frac{12}\cdot) as Hecke modules under the Shimura correspondence. Here $M_s(1)$ denotes the space of modular forms of weight $s$ on $Γ_0(1)=\mathrm{SL}(2,\Z)$, $S_{2k}^{\text{new}}(6,ε_2,ε_3)$ is the space of newforms of weight $2k$ on $Γ_0(6)$ that are eigenfunctions with eigenvalues $ε_2$ and $ε_3$ for Atkin-Lehner involutions $W_2$ and $W_3$, respectively, and the notation $\otimes(\frac{12}\cdot)$ means the twist by the quadratic character $\frac{12}\cdot)$. There is also an analogous result for the cases $(r,6)=3$.

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arXiv2009

Congruences of the partition function

Let $p(n)$ denote the partition function. In this article, we will show that congruences of the form $$ p(m^j\ell^kn+B)\equiv 0\mod m \text{for all} n\ge 0 $$ exist for all primes $m$ and $\ell$ satisfying $m\ge 13$ and $\ell\neq 2,3,m$. Here the integer $k$ depends on the Hecke eigenvalues of a certain invariant subspace of $S_{m/2-1}(Γ_0(576),χ_{12})$ and can be explicitly computed. More generally, we will show that for each integer $i>0$ there exists an integer $k$ such that for every non-negative integers $j\ge i$ with a properly chosen $B$ the congruence $$ p(m^j\ell^kn+B)\equiv 0\mod m^i $$ holds for all integers $n$ not divisible by $\ell$.

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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 $f$.

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arXiv2007

Modular unit and cuspidal divisor class groups of X_1(N)

In this article, we consider the group $F_1^\infty(N)$ of modular units on $X_1(N)$ that have divisors supported on the cusps lying over $\infty$ of $X_0(N)$, called the $\infty$-cusps. For each positive integer $N$, we will give an explicit basis for the group $F_1^\infty(N)$. This enables us to compute the group structure of the rational torsion subgroup $C_1^\infty(N)$ of the Jacobian $J_1(N)$ of $X_1(N)$ generated by the differences of the $\infty$-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the $p$-primary part of $C_1^\infty(p^n)$ for a regular prime $p$.

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arXiv2007

Apery limits and special values of L-functions

We describe a general method to determine the Apery limits of a differential equation that have a modular-function origin. As a by-product of our analysis, we discover a family of identities involving the special values of L-functions associated with modular forms. The proof of these identities is independent of differential equations and Apery limits.

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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)).

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arXiv2007

Defining equations of $X_0(2^{2n})$

In this note we obtain defining equations of modular curves $X_0(2^{2n})$. The key ingredient is a recursive formula for certain generators of the function fields on $X_0(2^{2n})$.

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arXiv2007

Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds

In this paper we are concerned with the monodromy of Picard-Fuch differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy relative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds in the paper. Furthermore, we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgroups of Sp(4,Z) of finite index whose levels are related to the geometric invariants of the Calabi-Yau threefolds

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arXiv2006

Differential equations satisfied by modular forms and K3 surfaces

We study differential equations satisfied by modular forms associated to $Γ_1\timesΓ_2$, where $Γ_i (i=1,2)$ are genus zero subgroups of $SL_2(\mathbf R)$ commensurable with $SL_2(\mathbf Z)$, e.g., $Γ_0(N)$ or $Γ_0(N)^*$. In some examples, these differential equations are realized as the Picard--Fuch differential equations of families of K3 surfaces with large Picard numbers, e.g., $19, 18, 17, 16$. Our method rediscovers some of the Lian--Yau examples of ``modular relations'' involving power series solutions to the second and the third order differential equations of Fuchsian type in [14, 15].

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