Searcharxiv⌕ Search

arXiv subjects

Nobuki Takeda

Publications and source records attributed to Nobuki Takeda.

6 recordsLinked to original sources
arXiv2026

Harder's conjecture and Hermitian automorphic forms

Let $k\geq 4$ and $j\geq 2$ be integers with $j$ even. Harder's conjecture predicts a congruence between a primitive elliptic Hecke cusp form of weight $2k+j-2$ and a degree $2$ vector-valued Siegel Hecke cusp form of weight ${\det}^{k}\mathrm{Sym}^{j}$. In this paper, we prove this conjecture under several explicit arithmetic hypotheses on a congruence prime and an auxiliary imaginary quadratic field. We construct Hermitian spin lifts from degree $2$ Siegel cusp forms to Hermitian cusp forms on $\mathrm{U}_{2,2}$, and use congruences between Hermitian Klingen-Eisenstein lifts and Hermitian cusp forms to obtain the congruence predicted by Harder's conjecture.

math.NT↗
arXiv2026

Fourier Coefficients of Siegel-Eisenstein Series of Degree $2m$ and Weight $m+1$

We study Fourier coefficients of the Siegel-Eisenstein series $E_{m+1}^{(2m)}(Z)$ for $m\equiv1\pmod4$ and $m\geq5$. Using the Fourier expansion formula due to Mizumoto, we determine the constant term and the exceptional non-zero Fourier coefficients. The result gives a higher-degree analogue of the degree-two formulas of Kohnen and Nagaoka, which were later rederived by Haruki.

math.NT↗
arXiv2026

Congruences between Klingen-Eisenstein series and cusp forms on $\mathrm{U}_{n,n}$

In this paper, we study congruences of Hecke eigenvalues between Hermitian Klingen-Eisenstein series and cusp forms on the unitary group $\mathrm{U}_{n,n}$ defined over the rational number field $\mathbb{Q}$. We also prove the rationality of the space of Hermitian automorphic forms and the integrality of their Hecke eigenvalues.

math.NT↗
arXiv2025

Differential operators on Hermitian modular forms on $\mathrm{U}(n, n)$

We construct explicit differential operators on hermitian modular forms, extending methods developed for Siegel modular forms. These differential operators are closely related to the two-variable spherical pluriharmonic polynomials. We construct explicit bases for the space of such polynomials and use them to build concrete operators. As an application, we derive an exact pullback formula for hermitian Eisenstein series.

math.NT↗