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Wei-Lun Tsai

Publications and source records attributed to Wei-Lun Tsai.

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.

math.NT↗
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.

math.NT↗
arXiv2026

AutoQ 2.0: From Verification of Quantum Circuits to Verification of Quantum Programs (Technical Report)

We present a verifier of quantum programs called AutoQ 2.0. Quantum programs extend quantum circuits (the domain of AutoQ 1.0) by classical control flow constructs, which enable users to describe advanced quantum algorithms in a formal and precise manner. The extension is highly non-trivial, as we needed to tackle both theoretical challenges (such as the treatment of measurement, the normalization problem, and lifting techniques for verification of classical programs with loops to the quantum world), and engineering issues (such as extending the input format with a~support for specifying loop invariants). We have successfully used AutoQ 2.0 to verify two types of advanced quantum programs that cannot be expressed using only quantum circuits: the \emph{repeat-until-success} (RUS) algorithm and the weak-measurement-based version of Grover's search algorithm. AutoQ 2.0 can efficiently verify all our benchmarks: all RUS algorithms were verified instantly and, for the weak-measurement-based version of Grover's search, we were able to handle the case of 100 qubits in $\sim$20 minutes.

cs.LO↗
arXiv2026

A Practical Specification Language for Automatic Quantum Program Verification (Technical Report)

Hoare-style verification provides a principled foundation for reasoning about the correctness of quantum programs, but existing approaches do not allow fully automatic verification. While automata-based verification scales well when specifications are given directly as automata, prior frameworks incur exponential blow-up when translating high-level set-based assertions into automata, which severely limits practicality. We introduce an extended set-based specification language and a specification-to-automata translation algorithm whose complexity is linear in the number of qubits, enabled by controlled automaton construction and qubit reordering. The resulting compact automata enable fully automatic Hoare-style verification of fixed-qubit quantum programs at previously infeasible scales, while substantially improving expressiveness without compromising efficiency.

cs.LO↗
arXiv2025

Do perfect powers repel partition numbers?

In 2013 Zhi-Wei Sun conjectured that $p(n)$ is never a power of an integer when $n>1.$ We confirm this claim in many cases. We also observe that integral powers appear to repel the partition numbers. If $k>1$ and $Δ_k(n)$ is the distance between $p(n)$ and the nearest $k$th power, then for every $d\geq 0$ we conjecture that there are at most finitely many $n$ for which $Δ_k(n)\leq d.$ More precisely, for every $\varepsilon>0,$ we conjecture that $$M_k(d):=\max\{n \ : \ Δ_k(n)\leq d\}=o( d^{\varepsilon}).$$ In $k$-power aspect with $d$ fixed, we also conjecture that if $k$ is sufficiently large, then $$ M_k(d)=\max \left\{ n \ : \ p(n)-1\leq d\right\}. $$ In other words, $1$ generally appears to be the closest $k$th power among the partition numbers.

math.CO↗
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 $(q;q)_{\infty}^3.$

math.NT↗
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

Verifying Quantum Circuits with Level-Synchronized Tree Automata (Technical Report)

We present a new method for the verification of quantum circuits based on a novel symbolic representation of sets of quantum states using level-synchronized tree automata (LSTAs). LSTAs extend classical tree automata by labeling each transition with a set of choices, which are then used to synchronize subtrees of an accepted tree. Compared to the traditional tree automata, LSTAs have an incomparable expressive power while maintaining important properties, such as closure under union and intersection, and decidable language emptiness and inclusion. We have developed an efficient and fully automated symbolic verification algorithm for quantum circuits based on LSTAs. The complexity of supported gate operations is at most quadratic, dramatically improving the exponential worst-case complexity of an earlier tree automata-based approach. Furthermore, we show that LSTAs are a promising model for parameterized verification, i.e., verifying the correctness of families of circuits with the same structure for any number of qubits involved, which principally lies beyond the capabilities of previous automated approaches. We implemented this method as a C++ tool and compared it with three symbolic quantum circuit verifiers and two simulators on several benchmark examples. The results show that our approach can solve problems with sizes orders of magnitude larger than the state of the art.

cs.LO↗
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.

math.NT↗
arXiv2023

An Automata-based Framework for Verification and Bug Hunting in Quantum Circuits (Technical Report)

We introduce a new paradigm for analysing and finding bugs in quantum circuits. In our approach, the problem is given by a triple $\{P\}\,C\,\{Q\}$ and the question is whether, given a set $P$ of quantum states on the input of a circuit $C$, the set of quantum states on the output is equal to (or included in) a set $Q$. While this is not suitable to specify, e.g., functional correctness of a quantum circuit, it is sufficient to detect many bugs in quantum circuits. We propose a technique based on tree automata to compactly represent sets of quantum states and develop transformers to implement the semantics of quantum gates over this representation. Our technique computes with an algebraic representation of quantum states, avoiding the inaccuracy of working with floating-point numbers. We implemented the proposed approach in a prototype tool and evaluated its performance against various benchmarks from the literature. The evaluation shows that our approach is quite scalable, e.g., we managed to verify a large circuit with 40 qubits and 141,527 gates, or catch bugs injected into a circuit with 320 qubits and 1,758 gates, where all tools we compared with failed. In addition, our work establishes a connection between quantum program verification and automata, opening new possibilities to exploit the richness of automata theory and automata-based verification in the world of quantum computing.

cs.LO↗
arXiv2023

Variants of Lehmer's speculation for newforms

In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of $τ(n)$ or is a Fourier coefficient $a_f(n)$ of any given newform $f(z)$. We offer a method, which applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $3\leq \ell\leq 37$ are not absolute values of coefficients of newforms with integer coefficients. For $τ(n)$ with $n>1$, we prove that $$τ(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 13, \pm 17, -19, \pm 23, \pm 37, \pm 691\},$$ and assuming GRH we show for primes $\ell$ that $$τ(n)\not \in \left \{ \pm \ell\ : \ 41\leq \ell\leq 97 \ {\textrm{with}}\ \left(\frac{\ell}{5}\right)=-1\right\} \cup \left \{ -11, -29, -31, -41, -59, -61, -71, -79, -89\right\}. $$ We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $\ell$, we prove that $\pm \ell^m$ is not a coefficient of any such newform $f$ with weight $2k>M^{\pm}(\ell,m)=O_{\ell}(m)$ and even level coprime to $\ell,$ where $M^{\pm}(\ell,m)$ is effectively computable.

math.NT↗
arXiv2022

AGM and jellyfish swarms of elliptic curves

The classical $\mathrm{AGM}$ produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. For finite fields $\mathbb{F}_q,$ with $q\equiv 3\pmod 4,$ we introduce a finite field analogue $\mathrm{AGM}_{\mathbb{F}_q}$ that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a $\mathit{jellyfish~swarm},$ as the 3D renderings of the connected components resemble $\mathit{jellyfish}$ (i.e. tentacles connected to a bell head). These swarms turn out to be more than the stuff of child's play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of $\mathbb{F}_q$-points, which can be used to prove that each swarm has at least $(1/2-\varepsilon)\sqrt{q}$ jellyfish. Additionally, this interpretation gives a description of the $\mathit{class~numbers}$ of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish.

math.NT↗
arXiv2022

Distributions of Hook lengths in integer partitions

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $μ_t(n)\sim \frac{\sqrt{6n}}π-\frac{t}{2}$ and variance $σ_t^2(n)\sim \frac{(π^2-6)\sqrt{6n}}{2π^3}.$ Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed $t\geq 4$ in partitions of $n$ converge to a shifted Gamma distribution with parameter $k=(t-1)/2$ and scale $θ=\sqrt{2/(t-1)}.$

math.NT↗
arXiv2022

Heights of points on elliptic curves over $\mathbb Q$

In this note we obtain effective lower bounds for the canonical heights of non-torsion points on $E(\mathbb{Q})$ by making use of suitable elliptic curve ideal class pairings $$Ψ_{E,-D}: E(\mathbb{Q})\times E_{-D}(\mathbb{Q})\mapsto \mathrm{CL}(-D).$$ In terms of the class number $H(-D)$ and $T_E(-D)$, a logarithmic function in $D$, we prove $$ \widehat{h}(P)> \frac{|E_{\mathrm{tor}}(\mathbb{Q})|^2}{\left( H(-D)+ |E_{\mathrm{tor}}(\mathbb{Q})|\right)^2}\cdot T_E(-D). $$

math.NT↗
arXiv2022

Limiting Betti distributions of Hilbert schemes on $n$ points

Hausel and Rodriguez-Villegas recently observed that work of Göttsche, combined with a classical result of Erdős and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $(\mathbb{C}^2)^{[n]}$ on $n$ points, as $n\rightarrow +\infty,$ is a \textit{Gumbel distribution}. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $((\mathbb{C}^2)^{[n]})^{T_{α,β}}$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdős and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $A\geq 2.$ Furthermore, if $p_k(A;n)$ denotes the number of partitions of $n$ with exactly $k$ parts that are multiples of $A$, then we obtain the asymptotic $$ p_k(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2π)^k}e^{2π\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, $$ a result which is of independent interest.

math.AG↗
arXiv2021

Even values of Ramanujan's tau-function

In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer $α$ is a value of $τ(n)$. For odd $α$, Murty, Murty, and Shorey proved that $τ(n)\neq α$ for sufficiently large $n$. Several recent papers have identified explicit examples of odd $α$ which are not tau-values. Here we apply these results (most notably the recent work of Bennett, Gherga, Patel, and Siksek) to offer the first examples of even integers that are not tau-values. Namely, for primes $\ell$ we find that $$ τ(n)\not \in \{ \pm 2\ell \ : \ 3\leq \ell< 100\} \cup \{\pm 2\ell^2 \ : \ 3\leq \ell <100\} \cup \{\pm 2\ell^3 \ : \ 3\leq \ell<100\ {\text {\rm with $\ell\neq 59$}}\}.$$ Moreover, we obtain such results for infinitely many powers of each prime $3\leq \ell<100$. As an example, for $\ell=97$ we prove that $$τ(n)\not \in \{ 2\cdot 97^j \ : \ 1\leq j\not \equiv 0\pmod{44}\}\cup \{-2\cdot 97^j \ : \ j\geq 1\}.$$ The method of proof applies mutatis mutandis to newforms with residually reducible mod 2 Galois representation and is easily adapted to generic newforms with integer coefficients.

math.NT↗
arXiv2021

Asymptotic Distribution of the Partition Crank

The partition crank is a statistic on partitions introduced by Freeman Dyson to explain Ramanujan's congruences. In this paper, we prove that the crank is asymptotically equidistributed modulo Q, for any odd number Q. To prove this, we obtain effective bounds on the error term from Zapata Rolon's asymptotic estimate for the crank function. We then use those bounds to prove the surjectivity and strict log-subadditivity of the crank function.

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