This paper concerns problems in the study of higher dimensional ramification. Let A be a regular local ring with maximal ideal \({\mathfrak m}_ A\) and let G be a finite group of automorphisms of A. For \(\sigma\in G\), let \(I_{\sigma}\) be the ideal of A generated by \(\{\) a- \(\sigma\) (a)\(| a\in A\}\). Assume that: \((1)\quad A^ G=\{a\in A| \quad a=\sigma (a)\) for all \(\sigma\in G\}\) is a noetherian ring and A is a finitely generated module over \(A^ G\); \((2)\quad For\) any \(\sigma\in G\setminus \{1\}\), \(A/I_{\sigma}\) is of finite length over A; \((3)\quad The\) map \(A^ G/A^ G\cap {\mathfrak m}_ A\to A/{\mathfrak m}_ A\) is an isomorphism. - Define the Serre function \(a_ G:\quad G\to {\mathbb{Z}}\) by \(a_ G(\sigma)=-length_ A(A/I_{\sigma})\) for \(\sigma\in G\setminus \{1\}\) and \(a_ G(1)=-\sum_{\sigma \in G,\sigma \neq 1}a_ G(\sigma).\) The authors prove that under the additional assumptions that \(\dim (A)=2\) and A is of equal characteristic the following Serre conjecture is true: \(a_ G\) is a character of a \({\mathbb{Q}}_{\ell}\)-rational representation of G for any prime number \(\ell\) which is invertible in A. - Using the same methods, they also prove that the conductors defined by G. Laumon and later by S. Saito for not necessarily isolated ramification on algebraic surfaces coincide.