arXiv subjects

Mingkuan Zhang

Publications and source records attributed to Mingkuan Zhang.

4 recordsLinked to original sources

arXiv2026

Twisted Siegel-Weil formulas for $\mathrm{GL}_2$ over non-Galois quartic CM fields

We establish twisted Siegel-Weil formulas for non-Galois quartic CM fields, identifying the twisted theta integral against a quadratic character with the Doi-Naganuma lift of Hecke's integral. This implies the base change of Jacquet-Langlands correspondence for certain Hecke characters from $\mathbb{Q}$ to real quadratic fields via Doi-Naganuma lift is an isomorphism. As an application, we prove that the twisted CM values of Borcherds forms are algebraic multiple of logarithm of units, explicitly described by the Fourier coefficients of twisted theta integrals.

math.NT↗

arXiv2025

Hilbert Eisenstein series as Doi-Naganuma lift

In this paper, we show that incoherent Hilbert Eisenstein series for a real quadratic fields can be expressed as the Doi-Naganums lift of an incoherent Eisenstein series over $\mathbb{Q}$. As an application, we show when $N$ is odd and square-free, the values at Heegner points of Borcherds product on $X_0(N)^2$ with effective divisors are not integral units when the discriminants are sufficiently large. This generalizes a result of the first author to higher levels. In the process, we explicitly describe the Rankin-Selberg type L-function that appeared in the work of Bruinier-Kudla-Yang when the quadratic space has signature (2, 2), and give a new construction of fundamental invariant vectors appearing in Weil representations of finite quadratic modules.

math.NT↗

arXiv2024

Hilbert Poincaré series and kernels for products of $L$-functions

We study Hilbert Poincaré series associated to general seed functions and construct Cohen's kernels and double Eisenstein series as series of Hilbert Poincaré series. Then we calculate the Rankin-Cohen brackets of Hilbert Poincaré series and Hilbert modular forms and extend Zagier's kernel formula to totally real number fields. Finally, we show that the Rankin-Cohen brackets of two different types of Eisenstein series are special values of double Eisenstein series up to a constant.

math.NT↗

arXiv2023

On the non-vanishing of Hilbert Poincaré series

We prove that if $ν$ has small norm with respect to the level and the weight, the $ν$-th Hilbert Poincaré series does not vanish identically. We also prove Selberg's identity on Kloosterman sums in the case of number fields, which implies certain vanishing and non-vanishing relations of Hilbert Poincaré series when the narrow class number is $1$. Finally, we pass to the adelic setting and interpret the problem via Hecke operators.

math.NT↗