# Automorphic Forms and Representations: Errata

This is the list of known errata to Automorphic Forms and Representations by Daniel Bump. If you find errors in the book, whether typos or historical or mathematical mistakes please! email me at bump@math.stanford.edu. Thanks to those of you who have already done so.

This list of errata is also available as a plain tex file or a tex dvi file.

Note: a negative line number is a line number counted upwards from the bottom of the page.

• p.4, line 8. `re' should not be italicized.
• p.6, bottom. The hypotheses on f do not imply that F is of locally bounded variation. Therefore add the hypothesis that f is of bounded total variation. This is true, for example, if f(x) is monotone for |x| large.
• p.9, fourth displayed formula (Fourier Inversion Formula). Omit subscript of t.
• p.11, line 7. ``Now substituting (1.19) ....'' It is not (1.19) which is substituted, but rather the formula at the bottom of p.10.
• p.18, line 12: omit space in `Z/ NZ'.
• p.19, line -10. Missing closing absolute value sign in |re(gamma(z))|<=1/2.
• p.20, formula (2.3). Comma should be outside the matrix.
• p.20, proof of Proposition 1.2.3. The bar is used in two different ways, which could be confusing. In the second usage (bar of gamma(F)) it means the topological closure.
• p.29, second displayed formula. The coefficient of q^5 should be 4830, not 2954.
• p.32, comments after Proposition 1.3.5. The estimate should be a_n <= Cn^{(k-1)/2+epsilon} for any positive epsilon.
• p.35, bottom. \$H^1\$ and \$H^2\$ should be \$H_1\$ and \$H_2\$.
• p.41, l. -10. In the Fourier expansion, the exponent of e should be multiplied by z.
• p.49, last line of proof of Theorem 1.4.4, Eq.(4.10) should be Eq.(4.11).
• p.52, Exercise 1.4.13, displayed formula. The last exponent of p should be k-1-2s, not -2s.
• p.52, bottom. Theorem 1.4.4 should be Theorem 1.4.5.
• p.53, line 11. ``theoretic'' should not be repeated.
• p.57, last line. (3.14) should be (3.16).
• p.58, line 2. Theorem 1.4.3 should be Theorem 1.4.4.
• p.60, beginning of paragraph before Theorem 1.5.1, `f in Gamma_0(N)' should be `f in S_k(Gamma_0(N),psi)'.
• p.66, third line from end of proof, ``follows from Eq. (6.5)'' should be ``follows from Eq. (6.6).
• p.67, after third displayed formula, omit the unnecessary ``... and the subsequent evaluation of the constant c.''
• p.69, bottom and p.71, first formula, infinity sign should be a subscript of Gamma.
• p.70, statment of Proposition 1.6.1, ``most simple poles'' should be ``at most simple poles.''
• p.72, first displayed formula: omit i in exponential.
• p.72, after statement of Theorem 1.6.2, (3.11) should be (3.13).
• p.74, line 3. Proposition 1.6.3 should be Theorem 1.6.2.
• p.76, line 2. Do not italicize ``re.''
• p.77, line 17. ``resulting the assumption from'' should be ``resulting from the assumption.''
• p.77, next to last line. The tensor product symbol should not be there: the index is supposed to be [o^x:o^x_+].
• p.90, Exercise 1.7.2. Insert space before (7.6).
• p.108. It is asserted that the Laplacian is positive definite. This should be ``semidefinite,'' and the reference should be to Exercise 2.1.8.
• p.129, first displayed formula, second partial derivative is with respect to z-bar, not z.
• p.129, (1.2) at bottom. In the definition of L_k, the partial derivative should be with respect to z-bar, not z.
• p.131, l. -12, the fraktur (German) h should be lower case.
• p.135, p.135, proof of Lemma 2.1.2. Not really an error, but replace ``closed manifold'' by ``compact manifold.''
• p.135, l. -4. In ``omega=u+iv=...'' omega should be w.
• p.170, Proposition 2.3.1 (iii), after the backslashes, insert G (twice).
• p.188, line 6. The function pi(g) Xf is automatically continuous, so this does not need to be assumed.
• p.245, near bottom. Replace X by D in this discussion, and note that pi(D)f is defined by (4.1) when D=X is in the Lie algebra g, and extended to U(g) by Proposition 2.2.3.
• p.291, line 16. L^2 should be L^2_0.
• p.310, line -8. Amend this to read: ``We will call H_G the Hecke algebra of G.''
• p.312, line 3. Amend this to read ``According to the notes in Knapp and Vogan, Flath had originally ...''
• p.317, line -11. ``This is a generalization of Theorem IV.6.6'' should read ``Theorem 4.6.3.'' Any theorem or proposition with a roman numeral should be suspected of being wrong. Let me know if you find any others!
• p.321, Theorem 3.5.1. The functional is of course only unique up to constant multiple.
• p.322, statement of Theorem 3.5.2. Add the assumption that (pi,V) is admissible.
• p.375, ``metaplectid'' should be ``metaplectic.''
• p.379, table. In the third (L-group) column n should be n+1 for the first three entries.
• p.383, line 21. ``hat pi is the Langlands L-function'' should be ``L(s,hat pi) is the Langland L-function.''
• p.383, l.-4 and -3. GL(2) should be GL(n) (twice) and GL(8) should be GL(n^2-1) (three times).
• p.385, Third line from bottom. (pi_1,V_0) should be (pi_1,V_1).
• p.426, formula (2.2). Omit parentheses from d_L(b); similarly, omit parenthesis from d_L(g) in following displayed formula.
• p.432. Not a correction, but it is useful to know that a stronger result than Proposition 4.2.7 is true. If there exists a single open subgroup K such that V_1^K and V_2^K are nonzero (hence simple H_K modules by Proposition 4.2.3), and if these are isomorphic as H_K modules, then V_1 and V_2 are isomorphic. To prove this, adapt the proof of Theorem 4.6.3 on p.493.
• p.436, second sentence of Section 4.3, ``this result'' should be ``these topics.''
• p.486, last displayed formula and p.487, top displayed formula. Domain of integration should be p^(-N).
• p.488, first displayed formula. The definition of L_2 is slightly wrong. The second term phi(1) should be multiplied by a function h(x) designed to make the statement that the integral is compactly supported actually true! For example, we can take:
```h(x)=|x|^-1 (chi_1^-1 chi_2)(x) if |x|>1
0                          if |x|<=1```
• p.493, Theorem 4.6.3. Not a correction, but note that this is a special case of the generalization of Proposition 4.2.7 described above on the note to p.432.
• p.540, line 14. Amend this to read ``After partial results towards Howe's conjecture were obtained by Howe and other authors, the conjecture was fully proved for local fields of odd residue characteristic by Waldspurger (1990).''
• p.541, Theorem 4.8.6. Since it is assumed here that E is a field, delete all references to the case E=F+F in the statement and proof of this theorem! The case where E=F+F is considered separately, later.
• p.550. The discussion in the second paragraph switches from GL(2) to GL(n) in a confusing way. Amend the third and fourth sentences to read ``The conjecture includes a hypothetical classification of the irreducible admissible representations of GL(n,F), where F is a local field, which has been proved in many cases. Over an archimedean field, the local Langlands conjecture (for an arbitrary reductive group) is a theorem of Langlands.''
• p.557, last paragraph. ``Theorem 4.9.3'' should be ``Proposition 4.9.3,'' and ``Theorem 4.9.4'' should be ``Theorem 4.9.1.''
• p.560. The paper of Doi and Naganuma was in vol. 9 of Inventiones, not vol. 19.

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