English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $N\geq 3$ be a composite, odd, and square-free integer and let $\Gamma$be the principal congruence subgroup of level $N$. Let $X(N)$ be the modularcurve of genus $g_{\Gamma}$ associated to $\Gamma$. In this article, we studythe Arakelov invariant $e(\Gamma)=\bar{\omega}^2/\varphi(N)$, with$\bar{\omega}^2$ denoting the self-intersection of the relative dualizing sheaffor the minimal regular model of $X(N)$, equipped with the Arakelov metric, and$\varphi(N)$ is the Euler's phi function. Our main result is the asymptotics$e(\Gamma) = 2g_{\Gamma}\log(N) + o(g_{\Gamma}\log(N))$, as the level $N$ tendsto infinity.

Self-intersection of the relative dualizing sheaf on modular curves X(N)