Gunnar Carlsson has proved the Segal conjecture for finite groups: If \(G\) is a finite group, then the Segal map \(\pi^*_ G(S^ 0){\hat{\;}}\to \pi^*_ S(BG^+)\) is an isomorphism, where \(\pi^*_ G(S^ 0){\hat{\;}}\) denotes \(\pi^*_ G(S^ 0)\) completed at the augmentation ideal \(I(G)\) in the Burnside ring \(A(G)\). Carlsson's inductive argument starts from the correctness of the Segal conjecture for elementary abelian \(p\)-groups. This paper provides a proof of this step. The technique is to use the Adams spectral sequence, and so most of the work lies in computing the appropriate Ext or Tor groups over the Steenrod algebra \(A\). A map \(\Theta\) : \(L\to M\) is a Tor-equivalence if the induced map on \(Tor^ A({\mathbb{F}}_ p,-)\) is an isomorphism. A key result is that the map \(\epsilon\) : T(M)\(\to M\) of the Singer construction is a Tor- equivalence. If \(V\) is an elementary abelian \(p\)-group, \(S>\{0\}\) is a subset of \(H^ 2(BV; {\mathbb{F}}_ p)\), then the key theorem 1.5 reduces the calculation of the Ext groups of the localization \(H^*(BV; {\mathbb{F}}_ p)_ S\) to the unlocalized case for proper subgroups of \(V\). The extended introduction to the paper gives a valuable road map to the reader.