arXiv subjects

Nina Sakharova

Publications and source records attributed to Nina Sakharova.

3 recordsLinked to original sources

arXiv2018

The modular Cauchy kernel for the Hilbert modular surface

In this paper we construct the modular Cauchy kernel on the Hilbert modular surface $Ξ_{\mathrm{Hil},m}(z)(z_2-\bar{z_2})$, i.e. the function of two variables, $(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$, which is invariant under the action of the Hilbert modular group, with the first order pole on the Hirzebruch-Zagier divisors. The derivative of this function with respect to $\bar{z_2}$ is the function $ω_m (z_1, z_2)$ introduced by Don Zagier in \cite{Za1}. We consider the question of the convergence and the Fourier expansion of the kernel function. The paper generalizes the first part of the results obtained in the preprint \cite{Sa}

math.AG↗

arXiv2018

Modular Cauchy kernel corresponding to the Hecke curve

In this paper we construct the modular Cauchy kernel $Ξ_N(z_1, z_2)$, i.e. the modular invariant function of two variables, $(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$, with the first order pole on the curve $$D_N=\left\{(z_1, z_2) \in \mathbb{H} \times \mathbb{H}|~ z_2=γz_1, ~γ\in Γ_0(N) \right\}.$$ The function $Ξ_N(z_1, z_2)$ is used in two cases and for two different purposes. Firstly, we prove generalization of the Zagier theorem ([La], [Za3]) for the Hecke subgroups $Γ_0(N)$ of genus $g>0$. Namely, we obtain a kind of "kernel function" for the Hecke operator $T_N(m)$ on the space of the weight 2 cusp forms for $Γ_0(N)$, which is the analogue of the Zagier series $ω_{m, N}(z_1,\bar{z_2}, 2)$. Secondly, we consider an elementary proof of the formula for the infinite Borcherds product of the difference of two normalized Hauptmoduls, $J_{Γ_0(N)}(z_1)-J_{Γ_0(N)}(z_2)$, for genus zero congruence subgroup $Γ_0(N)$.

math.AG↗

arXiv2016

Convergence of the Zagier type series for the Cauchy kernel

In 1975 prof. Don Zagier derived a preliminary formula for the trace of the Hecke operators acting on the space of cusp forms (\cite{5}, \cite{6}). Actually, it is an expression in terms of an integral over a fundamental domain of $SL_2(\mathbb{Z}).$ His theorem tells us that if $f$ is a cusp form of weight $k$, then we can identify the Peterson scalar product of $f$ and a certain series $ω_m(z_1,\bar{z_2}, k)$ with the action of the Hecke operator $T(m)$ on the function $f$, up to a constant that depends only on $k$ and $m$. It follows that $ω_m(z_1,\bar{z_2}, k)$ is kind of "kernel function" for the operator $T(m)$. Don Zagier proved this theorem using the Rankin-Selberg method. Other evidence was proposed by prof. A. Levin. He suggested to construct a Cauchy kernel. Formally, the Cauchy kernel expressed by the series, which doesn't converge absolutely. The main purpose of this paper is to extend this series to the edge of convergence by analytic continuation. The second part of the paper is devoted to getting an expression for differential form of logarithm of difference of two $j$-invariant values $|j(z_1)-j(z_2)|$.

math.AG↗