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Shuji Horinaga

Publications and source records attributed to Shuji Horinaga.

9 recordsLinked to original sources
arXiv2026

Worst-Case Quantum Algorithm for Optimal Polynomial Intersection Beyond Decoded Quantum Interferometry

The Optimal Polynomial Intersection (OPI) problem asks us to find a low-degree polynomial over a finite field whose values lie in prescribed subsets on as many given inputs as possible. Decoded quantum interferometry (DQI) gives a quantum algorithm for OPI in parameter regimes beyond those achieved by the best known classical heuristics. Follow-up works improve the parameter regimes, but their analyses are limited to average-case settings. Recently, Sun and Wootters showed that, even in the worst case, OPI has a solution in a larger parameter regime than the one covered by DQI. However, they left open whether one can design a quantum algorithm that solves OPI in the worst-case beyond the DQI regime. We give such a quantum algorithm. As a byproduct, we also improve the existential bound of Sun and Wootters in certain parameter regimes. In particular, when each subset contains roughly half of the field elements, our algorithm finds a solution with satisfaction rate $s=1$ whenever the rate satisfies $R>0.75$. This matches the previous average-case bound, whereas DQI cannot achieve $s=1$ unless $R=1$. Our existential bound guarantees the existence of a solution when $R> 0.7158$, improving over the previous threshold $R>0.7495$. More generally, our existential results extend to the Max-LINSAT problem with respect to arbitrary maximum distance separable (MDS) codes. The corresponding algorithmic results apply only to MDS codes whose dual admits an efficient list decoder. Our results are obtained through a novel application of a Brascamp--Lieb-type inequality in the MDS setting, which may have further applications.

quant-ph↗
arXiv2026

Holomorphic differential forms on some orthogonal modular varieties

We construct holomorphic differential forms of many degrees, including the minimum possible one, on the modular varieties associated to the even lattices of signature $(2, n)$ with $n\equiv 1, 3$ mod $8$ and discriminant $-2$ in the range $n\geq 25$. This is the first example of holomorphic differential forms of non-top degree on orthogonal modular varieties. The proof uses the Arthur multiplicity formula in the theory of automorphic representations.

math.AG↗
arXiv2025

The Kodaira dimension of even-dimensional ball quotients

We prove that, up to scaling, there exist only finitely many isometry classes of Hermitian lattices over $O_E$ of signature $(1,n)$ that admit ball quotients of non-general type, where $n>12$ is even and $E=\mathbb{Q}(\sqrt{-D})$ for an odd discriminant $-D<-3$. Furthermore, we show that even-dimensional ball quotients, associated with arithmetic subgroups of $\mathrm{U}(1,n)$ defined over $E$, are always of general type if $n > 207$, or $n>12$ and $D>2557$. To establish these results, we construct a nontrivial full-level cusp form of weight $n$ on the $n$-dimensional complex ball. A key ingredient in our proof is the use of Arthur's multiplicity formula from the theory of automorphic representations.

math.AG↗
arXiv2025

On $A$-parameters containing unitary lowest weight representations of $\mathrm{U}(p, q)$

In this paper, we determine all the Arthur packets containing an irreducible unitary lowest weight representation $π$ of real unitary group $G = \mathrm{U}(p, q)$, including non-scalar cases. Our methods are the Barbasch-Vogan parametrization of representations of $G$ and Trapa's algorithm to calculate the cohomologically induced representations. In particular, we show that an Arthur packet has at most one irreducible unitary lowest weight representation of $G$. As a consequence, if an irreducible unitary lowest weight representation $π$ exists in the Arthur packet of $ψ$, we give an explicit formula of the lowest $K$-type of $π$.

math.RT↗
arXiv2021

The special values of the standard $L$-functions for $\mathrm{GSp}_{2n} \times \mathrm{GL}_1$

We prove the expected algebraicity property for the critical values of character twists of the standard $L$-function associated to vector-valued holomorphic Siegel cusp forms of archimedean type $(k_1, k_2, \ldots, k_n)$, where $k_n \geq n+1$ and all $k_i$ are of the same parity. For the proof, we use an explicit integral representation to reduce to arithmetic properties of differential operators on vector-valued nearly holomorphic Siegel cusp forms. We establish these properties via a representation-theoretic approach.

math.NT↗
arXiv2019

Nearly holomorphic automorphic forms on $\mathrm{SL}_2$

We define the space of nearly holomorphic automorphic forms on a connected reductive group $G$ over $\mathbb{Q}$ such that the homogeneous space $G(\mathbb{R})^1/ K_\infty^\circ$ is a Hermitian symmetric space. By Pitale, Saha and Schmidt's study, there are the classification of indecomposable $(\mathfrak{g},K_\infty)$-modules which occur in the space of nearly holomorphic elliptic modular forms and Siegel modular forms of degree $2$. This paper studies global representations of the adele group $G(\mathbb{A}_\mathbb{Q})$ which occur in the space of nearly holomorphic Hilbert modular forms. In the case of elliptic modular forms, the result of this paper is an adelization of Pitale, Saha and Schmidt's result.

math.NT↗