Preface Page IX, line 7: Should be [GeR2] Section 1 page 16, line 2 from bottom: should be \sum^t_{i=0}\alpha^{r^{t-i}}_i page 16, line 2 from bottom: the last sum should be \sum_{i=0}^t \alpha_i \tau^{i-t} page 21, line 5: should be $$Df=f^\prime+fD\,,$$ page 29, line 6: symmetry'' should be replaced by reflexivity'' page 29, line 6: reciprocity'' should be replaced by symmetry'' Section 4 page 82, line -15: R should be {\cal F} page 83, line 14 should be: c\psi c^{-1} page 83, line 20 should read: scheme-theoretic kernel of $P$. page 98, line 14: should be $[D\colon L]=d^2$ page 110, line 23: should read: $$s=[L\colon \Bbb F_r(F)]\,.$$ page 126, line 12: should be \sum\limits^n_{i=0}\alpha_i \tau^i page 127, line 7: should read ..., we obtain'' Section 5 page 150, line 5 should be: \psi_{T} page 151, line 14: $\dim E=\rank_{L[\tau]}M(E)=w(E)r(E)$ page 158, line 14 should be: (\theta \tau^0+N_n) page 158, line 15 should be: (\theta \tau^0+N_n) page 161, line 6: b_j page 164, line 15 should read: where $e$ is page 166, line -2 should be: (b\mapsto m(f(b)))) page 171, line 11: should read: \Hom_{\overline{K}[T,\tau ]_0} page 173, line 13, both copies of (T+N_n) should be: (\theta +N_n) page 173, line -7, both copies of (T+N_n) should be: (\theta +N_n) Section 6 page 179, line 10: should be will follow'' page 179, line 16: should be R=\bigoplus\limits^\infty_{d=0} R_d page 185, line 3: section'' should be sections'' page 186, line 12, should read: \gamma\in H^0(\overline{X}-P,{\cal F}) page 187, line 14 from bottom: should read ...one can find an element...'' page 188, line -12 should be: R_d/R_{d-1} page 189, line 6: should be M=L\{\sigma\} page 190, line 10 from bottom: should be H^0({\cal F}_{n+1}/{\cal F}_n) page 192, line 5 should be: M_P Section 7 page 193, line -13 should be: \psi\colon {\bf A}\to L\{\tau\} page 199, line 4 should be: arbitrary homogeneous. We set page 201, line 17 should be: \lambda \psi \lambda^{-1} page 223, line 2 should be: \prodi_{i\in I,\, \deg i\leq N} page 226, line 6 should be: dj Section 8, page 242, line 16: should be order of $M$'' page 248: line 6 should be . We'' page 248, line 10 should be: f(s) page 249, line 2: should be Thus our'' page 250, line 15 should be: B_{i_j} page 252, line -14 should be: \sigma ({\bf V}) page 260, line -5 should be: c(a)a^{(-j-j_0)} page 262, line 8 should be: \cdots - j^h_\alpha page 263, line 8: should be f(u) page 264, line -3 should be: \deg (D) page 268, line -14 should be: and ${\bf H}^+$. page 269, line 15: should be the problem'' page 271, line 5 should be: 8.2.10 page 271, line 14: should be Witt ring of the finite'' page 271, line -4 should be: 7.5.5 page 334, line 15: should be Theorem 3.2 page 339, the fourth paragraph from bottom should be: Recall that the value field was defined as a subfield of ${\bf C}_\infty$. Let ${\bf K}_{\bf V}$ be the smallest extension of $\bf K$ containing $\{\langle I \rangle\}$ for $I\in \cal I$ (= ${\bf K}\cdot \bf V$ if $d_\infty=1$). Let ${\bf K}_{\bf V}(\rho)$ be the smallest extension of ${\bf K}_{\bf V}$ containing the coefficients of the characteristic polynomials of the Frobenius elements as discussed before 8.10.2. Thus, for each $y\in {\Bbb Z}_p$, the function $x\mapsto L(\rho,(x,y))$ is given by a power series in $x^{-1}$ with coefficients precisely in ${\bf K}_{\bf V}(\rho)$. Note that by construction, as $\rho$ is of Galois type, the field ${\bf K}_{\bf V}(\rho)$ is an unramified finite extension of ${\bf K}_{\bf V}$. Similarly we let ${\bf V}(\rho)$ be the smallest extension of ${\bf V}$ containing the above coefficients. Again, ${\bf V}(\rho)$ is a finite constant field extension of $\bf V$.'' page 341, line 11: should be \{\beta_1(y),\dots ,\beta_n(y)\} page 341, line 12: should be L_n((u^y)^{1/p^t} Section 9 page 356, line 6: should be $$2T^{36}+T^{28}+2T^{18}+T^2$$ page 364, line 5: should be \pi_1^{\deg D_j}D_j page 366, line 5: should be for all \$y...'' page 366, line 15: should be = {_t\xi}_{^{\scriptstyle u}}^{-1} Section 10 page 398, line 7: should be S_m(e_C(\xi x),1 page 398, line 20: should be \left\{S_m(e_C(\xi b/f),1)\right\}