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Dongxi Ye

Publications and source records attributed to Dongxi Ye.

At least 19 recordsLinked to original sources
arXiv2026

Difference of the modular function $ω_{2}(τ)$, revisited

Adapting the analytic method of Gross and Zagier, Roskam proved a prime-factorization formula for the norm of the difference of two level-two Weber singular moduli. Independently, Yang and Yin obtained an equivalent formula using Borcherds lifts. More precisely, the formula concerns the norm of \[ ω_{2}\left(\frac{-1+\sqrt{d_{1}}}{2}\right) - ω_{2}\left(\frac{-1+\sqrt{d_{2}}}{2}\right) \] for coprime negative fundamental quadratic discriminants $d_{1},d_{2}\equiv1\pmod 8$, where \[ ω_{2}(τ) = 2^{12}\frac{η(2τ)^{24}}{η(τ)^{24}}, \] and $η(τ)$ denotes the Dedekind eta function. In this work, we revisit this formula from the perspective of arithmetic intersection theory and give a new proof.

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arXiv2026

On the Petersson norm $\langleθ_ψ,θ_ψ\rangle$

Let $K$ be an imaginary quadratic field of discriminant~$-d<-11$, and for $\ell\geq1$, let $ψ$ be a Hecke character of $K$ with trivial finite conductor and infinite type~$(2\ell,0)$. In this work we prove that for any prime $p>3$, the quotient $$ \frac{\langleθ_ψ,θ_ψ\rangle}{Ω_{K}^{4\ell}}, $$ is $λ$-integral for a prime ideal $λ$ over $p$, where $\langleθ_ψ,θ_ψ\rangle$ is the Petersson norm of the theta function $θ_ψ=θ_ψ(τ)$ attached to $ψ$, and $Ω_{K}$ denotes the Chowla--Selberg period attached to~$K$, and the product $$ \prod_{i=1}^{h_{K}}\frac{\langleθ_{ψ_{i}},θ_{ψ_{i}}\rangle}{Ω_{K}^{4\ell}} $$ is rational, where the product is over all $h_{K}$ of the underlying Hecke characters.

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arXiv2026

Vanishing Coefficients in Products of Quintuple Products

Explicit arithmetic progressions modulo primes $p \equiv 1 \pmod{4}$ are derived in which the coefficients in the expansions of products of quintuple products vanish. In particular, if $p = m^{2} + n^{2}$, and $b$ is a positive integer, and $$\sum_{n=0}^{\infty} a_{n}q^{n} = \frac{(q^{2bm},q^{p-2bm};q^{2bn},q^{p-2bn};q^p)_{\infty}}{(q^p,-q^{b m},-q^{p-bm},-q^{bn},-q^{p-bn};q^p)_{\infty}^2},$$ we determine $α= α(m,n,p)$ such that $a_{pt+ α}=0$. Our results are proven using involutive transformations on integer lattices.

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arXiv2026

Enumeration of modular forms for $Γ_1(N)$

This paper considers holomorphic modular forms for $Γ_1(N)$ of integral weight of the form $$f^{(N)}_{\mathbf a}(τ) =q^{s} (q^{N};q^{N})_{\infty}^{a_0}\prod_{j=1}^{\lfloor N/2 \rfloor}(q^j,q^{N-j};q^N)_\infty^{a_j}, \quad \mathbf a = (a_1, \ldots, a_{\lfloor N/2 \rfloor}),$$ for fixed $a_0=2k \in 2 \Bbb Z_{\ge 0}$. We show that the number of relevant exponent vectors $\mathbf a$ is finite and characterize them in terms of the $\mathbb{Q}$-rational cuspidal divisor class group of $X_{1}(N)$. Effective procedures are given for counting the admissible exponents by enumerating the corresponding polytopes. This leads to formulas for the number of exponent vectors in terms of quasipolynomials in $k$.

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arXiv2026

The $p$-Dissection of a Product of Quintuple Products

Let $p \equiv 1 \pmod{4}$ be prime, let $m$ and $n$ be integers such that $p=m^2+n^2$, and let $b$ be a positive integer. Let $Q(z,q) = (z,q/z,q;q)_{\infty}(qz^2,q/z^2;q^2)_{\infty}$ denote the product appearing in the quintuple product identity. We derive explicit formulae for the $p$-dissection of $Q(q^{bm},q^p)Q(q^{bn},q^p)$, and determine sign patterns in length-$p$ arithmetic progressions of the Taylor series coefficients of the associated quotient $Q(q^{bm},q^{p})Q(q^{bn},q^p)/(q^p;q^p)_{\infty}^2$. Some combinatorial applications of the $p$-dissection formulae are also given.

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arXiv2026

Quadratic irrational analogues of Ramanujan's series for $1/π$

About 40 years ago Jonathan and Peter Borwein discovered the series identity $$ \sum_{n=0}^\infty \frac{(-1)^n(6n)!}{(3n)!(n!)^3} \frac{(A+nB)}{C^{n+1/2}} = \frac{1}{12π} $$ where \begin{align*} A&=1657145277365+212175710912\sqrt{61},\cr B&=107578229802750+13773980892672\sqrt{61},\cr C&=\left(5280(236674+30303\sqrt{61})\right)^3 \end{align*} which adds roughly 25 digits of accuracy per term. They noted that if each of the quadratic irrationals $A$, $B$ and $C$ is replaced by their conjugates, that is, each number $a+b\sqrt{61}$ is changed to $a-b\sqrt{61}$, then the resulting series also converges to a rational multiple of $1/π$. They gave several other examples of quadratic irrational series for $1/π$, and noted that the conjugate series converges to another rational multiple of $1/π$ or in some cases the conjugate series diverges. The purpose of this work is to provide an explanation and classification of such series. Our classification includes Ramanujan's 17 original series, as well as series of the Borweins, Chudnovskys, Sato and others. We extend the classification to genus-zero subgroups $Γ_0(\ell)+$, that is, for each $\ell \in \big\{1,2,3,\ldots,36,38,39,41,42,44,45,46,47,49,50,51,54,55,56,59,60,62,66,69,70, 71,78,87,92,94,95,105,110,119\big\}$ we calculate the Hauptmoduls, associated weight two modular forms, and the corresponding rational and real quadratic irrational series for $1/π$. The classification reveals many interrelations among the different series. For example, we show that the Borweins' series above, and its conjugate, are equivalent by hypergeometric transformation formulas to the level~7 rational series $$ \sum_{n=0}^\infty \left\{\sum_{j=0}^n {n \choose j}^2{2j \choose n} {n+j \choose j}\right\} (11895n+1286) \frac{(-1)^n}{22^{3n+3}} = \frac{1}{π\sqrt{7}}. $$

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arXiv2025

Identical Vanishing of Coefficients in the Series Expansion of Eta Quotients, modulo 4, 9 and 25

Let $A(q)=\sum_{n=0}^{\infty}a_n q^n$ and $B(q)=\sum_{n=0}^{\infty}b_n q^n$ be two eta quotients. Previously, we considered the problem of when \[ a_n=0 <=> b_n=0. \] Here we consider the ``mod $m$'' version of this problem, i.e. eta quotients $A(q)$ and $B(q)$ and integers $m>1$ such that \[ a_n \equiv 0 \pmod m <=> b_n \equiv 0 \pmod m? \] We found results for $m=p^2$, $p=2, 3$ and $5$. For $m=4,9$, we found results which apply to infinite families of eta quotients. For example: Let $A(q)$ have the form \begin{equation} A(q) = f_1^{3j_1+1}\prod_{3\nmid i}f_i^{3j_i}\prod_{3|i}f_i^{j_i} =: \sum_{n=0}^{\infty}a_nq^n,\,\,B(q) = \frac{f_3}{f_1^3}A(q) =: \sum_{n=0}^{\infty}b_nq^n \end{equation} with $f_{k}=\prod_{n=1}^{\infty}(1-q^{kn})$. Then \begin{align*} a_{3n}-b_{3n}&\equiv 0\pmod 9,\\ 2a_{3n+1}+b_{3n+1}&\equiv0\pmod 9,\\ a_{3n+2}+2b_{3n+2}&\equiv0\pmod 9. \end{align*} Some of these theorems also had some combinatorial implications, such as the following: Let $p_2^{(3)}(n)$ denote the number of bipartitions $(π_1, π_2)$ of $n$ where $π_1$ is 3-regular. Then \begin{equation*} p_2^{(3)}(n)\equiv0\pmod 9 <=> n\text{ is not a generalized pentagonal number}. \end{equation*} In the case of $m=25$, we do not have any general theorems that apply to an infinite family of eta quotients. Instead we give two tables of results that appear to hold experimentally. We do prove some individual results (using theory of modular forms), such as the following: Let the sequences $\{c_n\}$ and $\{d_n\}$ be defined by \begin{equation*} f_1^{10}=:\sum_{n=0}^{\infty}c_nq^n, \hspace{25pt} f_1^{5}f_5=:\sum_{n=0}^{\infty}d_nq^n. \end{equation*} Then \begin{equation*} c_n \equiv 0 \pmod{25} <=> d_n \equiv 0 \pmod{25}. \end{equation*}

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arXiv2025

Sign-patterns of Certain Infinite Products

The signs of Fourier coefficients of certain eta quotients are determined by dissecting expansions for theta functions and by applying a general dissection formula for certain classes of quintuple products. A characterization is given for the coefficient sign patterns for \[ \frac{(q^i;q^i)_{\infty}}{(q^p;q^p)_{\infty}} \] for integers \( i > 1 \) and primes \( p > 3 \). The sign analysis for this quotient addresses and extends a conjecture of Bringmann et al. for the coefficients of \( (q^2;q^2)_{\infty}(q^5;q^5)_{\infty}^{-1} \). The sign distribution for additional classes of eta quotients is considered. This addresses multiple conjectures posed by Bringmann et al.

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arXiv2025

Central $L$ values of congruent number elliptic curves

Let $E_n$ be the congruent number elliptic curve $y^2=x^3-n^2x$, where $n$ is square-free and not divisible by primes $p\equiv 3\pmod 4$. In this paper, we prove that $L(E_n,1)$ can be expressed as the square of CM values of some simple theta functions, generalizing two classical formulas of Gauss. Our result is meaningful in both theory and practical computation.

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arXiv2025

On a Conjecture of Yui and Zagier II

Yui and Zagier made some fascinating conjectures on the factorization on the norm of the difference of Weber class invariants $ f(\mathfrak a_1) - f(\mathfrak a_2)$ based on their calculation in \cite{YZ}. Here $\mathfrak a_i$ belong two diferent ideal classes of discrimants $D_i$ in imagainary quadratic fields $\mathbb{Q}(\sqrt{D_i})$. In \cite{LY}, we proved these conjectures and their generalizations when $(D_1, D_2) =1$ using the so-called big CM value formula of Borcherds lifting. In this sequel, we prove the conjectures when $\mathbb{Q}(\sqrt{D_1}) =\mathbb{Q}(\sqrt{D_2})$ using the so-called small CM value formula. In addition, we give a precise factorization formula for the resultant of two different Weber class invariant polynomials for distinct orders.

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arXiv2024

Euler-type recurrences for $t$-color and $t$-regular partition functions

We give Euler-like recursive formulas for the $t$-colored partition function when $t=2$ or $t=3,$ as well as for all $t$-regular partition functions. In particular, we derive an infinite family of ``triangular number" recurrences for the $3$-colored partition function. Our proofs are inspired by the recent work of Gomez, Ono, Saad, and Singh on the ordinary partition function and make extensive use of $q$-series identities for $(q;q)_{\infty}$ and $(q;q)_{\infty}^3.$

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arXiv2024

The Modularity of Z.-W. Sun's Conjectural Formulas for $\frac{1}π$

In this work, we establish modular parameterizations for two general formulas for $\frac{1}π$ that subsume conjectural Ramanujan type formulas due to Z.-W. Sun, which have remained open since 2011. As an application of this, in a conceptual way we interpret how Sun's conjectural formulas arise and can be verified, as well as recover other cases that were proved by Cooper, Wan and Zudilin.

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arXiv2024

Lucas congruences using modular forms

In this work, we prove that many Apéry-like sequences arising from modular forms satisfy the Lucas congruences modulo any prime. As an implication, we completely affirm four conjectural Lucas congruences that were recently posed by S. Cooper and reinterpret a number of known results.

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arXiv2024

Supercongruences via Beukers' method

Recently, using modular forms F. Beukers posed a unified method that can deal with a large number of supercongruences involving binomial coefficients and Apéry-like numbers. In this paper, we use Beukers' method to prove some conjectures of the first author concerning the congruences for $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k^3}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k\binom{6k}{3k}}{m^k}, \ \sum_{n=0}^{p-1}\frac{V_n}{m^n},\ \sum_{n=0}^{p-1}\frac{T_n}{m^n},\ \sum_{n=0}^{p-1}\frac{D_n}{m^n} $$ and $\sum_{n=0}^{p-1}(-1)^nA_n$ modulo $p^3$, where $p$ is an odd prime representable by some suitable binary quadratic form, $m$ is an integer not divisible by $p$, $V_n=\sum_{k=0}^n\binom{2k}k^2\binom{2n-2k}{n-k}^2$, $T_n=\sum_{k=0}^n\binom nk^2\binom{2k}n^2$, $D_n=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2n-2k}{n-k}$ and $A_n$ is the Apéry number given by $A_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2$.

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arXiv2024

Ramanujan type congruences for quotients of Klein forms

In this work, Ramanujan type congruences modulo powers of primes $p \ge 5$ are derived for a general class of products that are modular forms of level $p$. These products are constructed in terms of Klein forms and subsume generating functions for $t$-core partitions known to satisfy Ramanujan type congruences for $p=5,7,11$. The vectors of exponents corresponding to products that are modular forms for $Γ_{1}(p)$ are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for $Γ_{1}(p)$ of weights $1\le k \le 5$ for primes $5\le p \le 19$ and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including $p> 19$. Canonical bases for modular forms of level $5\le p \le 19$ are constructed by summing weight one Hecke Eisensten series of levels $5\le p \le 19$ and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for $Γ_{1}(p)$ and $Γ(p)$ are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight $1$ for $Γ_{1}(p)$ as a direct sum of subspaces of modular forms for $Γ(p)$ of the form $q^{r/p}\Bbb Z[[q]]$. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences.

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arXiv2017

Series for $1/π$ of signature 20

Properties of theta functions and Eisenstein series dating to Jacobi and Ramanujan are used to deduce differential equations associated with McKay Thompson series of level 20. These equations induce expansions for modular forms of level 20 in terms of modular functions.The theory of singular values is applied to derive expansions for $1/π$ of signature $20$ analogous to those formulated by Ramanujan.

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arXiv2017

Level $17$ Ramanujan-Sato series

Two level 17 modular functions $$ r=q^{2}\prod_{n=1}^{\infty}(1-q^{n})^{\left(\frac{n}{17}\right)},\quad s=q^{2}\prod_{n=1}^{\infty}\frac{(1-q^{17n})^{3}}{(1-q^{n})^{3}}, $$ are used to construct a new class of Ramanujan-Sato series for $1/π$. The expansions are induced by modular identities similar to those level of 5 and 13 appearing in Ramanujan's Notebooks. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived.

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