On mod \(p\) cohomology of the space \(X_{\Gamma{}}\) and mod \(p\) trace formula for Hecke operators
Let \(M_ 2\mathbb{Z}\) denote the set of \(2\times 2\) matrices over \(\mathbb{Z}\). It acts on the 2-torus \(T^ 2\) in the standard way. For any subgroup \(\Gamma\subset M_ 2 \mathbb{Z}\) one can define the semi-direct product \(\Gamma\ltimes T^ 2\) with classifying space \(X_ \Gamma=B(\Gamma\ltimes T^ 2)\). The purpose of the paper is to study, for \(\Gamma=SL_ 2\mathbb{Z}\) or a congruence subgroup, the cohomology \(H^*(X_ \Gamma;\mathbb{F}_ p)\), and apply the results obtained to get a formula on the mod \(p\) value of the trace of Hecke operators acting on cusp forms. More precisely, let \(T(\ell)\) be a Hecke operator acting on the space \(S_ k\) of weight \(k\) cusp forms. With respect to a suitable base of \(S_ k\), the \(T(\ell)\) can be represented by integral matrices. The formula alluded to gives an expression \(\mod p\) for \[ H_ t(\ell)=\sum_{k\geq 0} \text{Trace}[T(\ell)| S_ k]t^ k\in\mathbb{Z}t. \] The main part of the formula involves a sum of traces of Hecke operators acting on certain spaces \(H^ 0(SL_ 2\mathbb{Z};Q_ *)\) and \(H^ 1(SL_ 2\mathbb{Z};Q_ *)\). As a matter of fact, one can easily express the \(H^*(X_ \Gamma;M)\), where \(M\) is any \(\mathbb{Z}[1/d]\)-module and \(d\) is an integer depending on \(\Gamma\), e.g. one may take \(d=6\) for \(\Gamma=SL_ 2\mathbb{Z}\), in terms of \(H^ 0(\Gamma;\mathbb{Z}[x,y]_ n \otimes M)\) and \(H^ 1(\Gamma;\mathbb{Z}[x,y]_ n\otimes M)\). Here \(\mathbb{Z}[x,y]_ n\) denotes the module of homogeneous polynomials in the standard generators \(x\) and \(y\) of \(H^ 2(BT^ 2,\mathbb{Z})\) of degree \(n\). For a prime \(p\) with \((p,d)=1\) one has \(\oplus H^{2n}(X_ \Gamma,\mathbb{F}_ p)\cong H^ 0(\Gamma; \mathbb{F}_ p[x,y])\cong \mathbb{F}_ p[x,y]^ \Gamma\), the ring of Dickson invariants, and for \(H^{2n+1}(X_ \Gamma;\mathbb{F}_ p)\) one finds that its \(\mathbb{F}_ p\)-dimension is equal to \(\dim_{\mathbb{F}_ p} \mathbb{F}_ p[x,y]^ \Gamma_ n+\dim_ \mathbb{R} S_{n+2}+\nu\), where \(\nu\) is the number of equivalence classes of cusps of \(\Gamma\). Assume \(\Gamma=SL_ 2 \mathbb{Z}\) and \(p\geq 5\) in the sequel. Then \(\mathbb{F}_ p[x,y]^ \Gamma=\mathbb{F}_ p[x,y]^{SL_ 2 \mathbb{Z}} = \mathbb{F}_ p[q,q_ 1]\) with \(q=xy^ p-x^ p y\in\mathbb{F}_ p[x,y]_{p+1}\) and \[ q_ 1=x^{p(p-1)} + x^{(p-1)^ 2} y^{p-1} +\dots+x^{p-1} y^{(p- 1)^ 2}+ y^{p(p-1)}\in\mathbb{F}_ p [x,y]_{p(p-1)}. \] Define the graded \(\mathbb{F}_ p[M_ 2\mathbb{Z}]\)-module \(Q_ *\) by \(Q_ *=\mathbb{F}_ p[x,y]/(q)\). Then one has a periodicity results: For \(n,m\geq p+1\) and \(n\equiv m\bmod(p-1)\), \(Q_ n\cong Q_ m\) as \(\mathbb{F}_ p[M_ 2\mathbb{Z}]\)- modules. This implies a periodicity for the \(H^ i(\Gamma;Q_ k)\), \(i=0,1\), and thus for the traces of the \(T(\ell)\) acting on these spaces occurring in the formula for \(H_ t(\ell)\). Explicit descriptions for \(H^ 0(SL_ 2\mathbb{Z},Q_ *)\) and \(H^ 1(SL_ 2\mathbb{Z},\mathbb{F}_ p[x,y])\) are derived. As an application one finds e.g. \(H_ t(5)\equiv 0\bmod 5\).
scientific article; zbMATH DE number 169029