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 $M(\mathit{odd})\subset Z/2[[x]]$ be the space of odd mod~2 modular formsof level $\Gamma_{0}(3)$. It is known that the formal Hecke operators$T_{p}:Z/2[[x]]\rightarrow Z/2[[x]]$, $p$ an odd prime other than $3$,stabilize $M(\mathit{odd})$ and act locally nilpotently on it. So$M(\mathit{odd})$ is an $\mathcal{O} = Z/2[[t_{5},t_{7}, t_{11},t_{13}]]$-module with $t_{p}$ acting by $T_{p}$, $p\in \{5,7,11,13\}$. We show:  (1) Each $T_{p}:M(\mathit{odd})\rightarrow M(\mathit{odd})$, $p\ne 3$, ismultiplication by some $u$ in the maximal ideal, $m$, of $\mathcal{O}$.  (2) The kernel, $I$, of the action of $\mathcal{O}$ on $M(\mathit{odd})$ is$(A^{2},AC,BC)$ where $A,B,C$ have leading forms $t_{5}+t_{7}+t_{13},\,t_{7},\, t_{11}$.  We prove analogous results in level $\Gamma_{0}(5)$. Now $\mathcal{O}$ is$Z/2[[t_{3},t_{7},t_{11},t_{13}]]$, and the leading forms of $A,B,C$ are$t_{3}+t_{7}+t_{11},\, t_{7},\, t_{13}$.  Let $\mathit{HE}$, "the shallow mod~2 Hecke algebra (of level $\Gamma_{0}(3)$or $\Gamma_{0}(5)$)" be $\mathcal{O}/I$. (1) and (2) above show that$\mathit{HE}$ is a 1 variable power series ring over the 1-dimensional localring $Z/2[[A,B,C]]/(A^{2},AC,BC)$. For another approach to all these results,based on deformation theory, see Deo and Medvedovsky, "Explicit old componentsof mod-2 Hecke algebras with trivial $\bar{\rho}$."

Generators and relations for the shallow mod 2 Hecke algebra in levels $\Gamma_{0}(3)$ and $\Gamma_{0}(5)$