Annhilators of local cohomology modules over modular invariant rings and Dickson polynomials

Let ${\mathbb{F}}_q$ be a finite field with $q=p^s$ elements. Let $V$ be a $d$ dimensional vector space over ${\mathbb{F}}_q$ and let $G$ be a subgroup of $GL(V)$. Let $R={\mathbb{F}}_q[V]={\text{Sym}}_{{\mathbb{F}}_q}(V^{\ast })$ and let $G$ act naturally on $R$. Set $S=R^G$. Let ${\mathbf{d}}_{d,0},{\mathbf{d}}_{d,1},\dots ,{\mathbf{d}}_{d,d-1}\in S$ be the Dickson polynomials with $\mathrm{°}{\mathbf{d}}_{d,i}=q^d-q^i$. Let $I$ be a homogeneous ideal of $S$ and let $H_I^i(S)$ be the $i^{th}$-local cohomology module of $S$ with respect to $I$. Let $J_i=\sqrt{\text{ann}{H}_I^i(S)}$. Assume $J_i\mathrm{\ne }0$ and $\dim S\mathrm{/}{J}_i=d-g$. Then we show that ${\mathbf{d}}_{d,0},\dots ,{\mathbf{d}}_{d,d-g+1}\in J_i$. We give several applications of our results. An application is a considerably simpler proof of Landweber-Stong conjecture.