From Bhupendra Nath Tiwari (corrections in red).
p.16 Stem Fields ----- F[x]= F[X]/ f(X)
p.25 the field construction in this lemma is ...
p.51 Example 4.9---- It is irreducible by .... resolvent cubic is g(X)= X^3-8X + 16, which ...
p.81 5-1 ... is not isomorphic go G_{a_j}.
p.70, Remark 5.26(a). I should clarify that in this remark it is not necessary to assume that F contains an nth root of 1. As Chad Schoen points out: "The confusing feature here is that in the section on cyclic extensions the assumption is that the field F contains a primitive n-th root of 1, yet this assumption appears to be forgotten in Remark 5.26(a). The condition given when 4|n is unnecessary if indeed there is a 4-th root of unity in the base field. If the field element, a, satisfies the given condition, then a is a square and this case is covered by the first condition. Note that the reference to Lang is to a theorem which holds even if the base field does not contain a primitive n-th root of 1. If one is allowed to assume that F contains a primitive n-th root of 1, then Remark 5.26(a) with the redundant condition when 4|n removed is not a difficult exercise. By contrast the theorem in Lang is considerably more involved."
p. 90, Aside 6.1. The smallest group with an element of the commutator group that is not itself a commutator has order 96, not 94 (Leandro Vendramin ... I checked the computation myself using GAP). For discussions of this question, see stackexchange and mathoverflow.
From Bhupendra Nath Tiwari (corrections in red).
p.12 proposition 1.11, S1 ab,and
p.53 ... is PSL3(F2), which as order 168 ...
p.58 ORBITS. Let G act on X.
p.68 Example 4.29 ... and the partition attached of n attached to
p.72 Example 4.41 ... : if it stabilized
p.73 Exercise 4-2 (c) Give an example of a proper subset S ...
p.111 (a) .. to exactly of of the S_i
p.112 called the character of \rho. Note that ...
From Bhupendra Nath Tiwari (corrections in red).
p.3 (Introduction)
One see easily that this is again an ideal, ... ----> One sees easily that this is again an ideal, ...
p.5 (Introduction) When m=3, this can proved by the method ... --->
When m=3, this can be proved by the method ...
p.23 (2.11 PROOF.) ... for f (see FT p.19), and so ... ---> ... for f (see FT
p.16), and so ...
p.45 (3.13) ... for all ideals nonzero prime ideals p of A. --->
... for all nonzero prime ideals p of A.
p.51 (3.29 PROOF.) ... a Dedekind domain (p.3.3). ---> ... a Dedekind domain
(see 3.3).
p.70 (4.16 b) When \Lambda /superset Lambda^{\prime} are two full lattices R^n, ... ---->
When \Lambda /superset Lambda^{\prime} are two full lattices of R^n, ...
p.76 (4.27 PROOF.) In order to have \mu(X(t)) \ge 2^n . \mu(D), we need ... --->
In order to have \mu(X(t)) \ge 2^n . \mu(D), we need (see 4.22, 4.26)...
p.82 (5.7) Recall that we previously considered the map
\sigma: K -> R^r \times C^s, \alpha |-> (\sigma_1 \alpha, ... , \sigma_r \alpha, \sigma_{r+1} \alpha, ... , \sigma_{r+s} \alpha) ------->
\sigma: K -> R^r \times C^s, \alpha |-> (\sigma_1 \alpha, ... ,
\sigma_r \alpha, \sigma_{r+1} \alpha, ... , \overline{\sigma}_{r+s} \alpha)
p.89 (5.14) The number 1, 4, and ... are square modulo 29, but 2 is not, hence m must be 0. Since..
----> The number 1, 4, and ... are square modulo 29, but 22 is not, hence m must be 0. Since..
p.90 (5-1) Fix an m and and M. ---> Fix an m and M.
p.100 (6.11 PROOF) (a) 1= \zeta^{2r}; but then (*) says ---->
(a) 1= \zeta^{2r}; but then (16) says
p.100 (6.11 PROOF ) (b) 1= \zeta^{2r-1}; then \zeta=\zeta^{2r}, and (*) says --->
(b) 1= \zeta^{2r-1}; then \zeta=\zeta^{2r}, and (16) says
p.100 (6.11 PROOF) (c) 1= \zeta^{2r}; but then (*) says ---->
(c) 1= \zeta^{2r}; but then (16) says
p.105 (7.9) ... and we checked (3.26 et seq.) that this implies that equality holds if ... --->
... and we checked (see 3.26) that this implies that equality holds if ...
p.109 (7.19 PROOF) For i \ge 2, |a_r|_i, ... .... as r -> 0. ------->
For i \ge 2, |a_r|_i, ... as r -> \infty. (because for i \ge 2, |a|_i <1)
p.118 (7.38) ... . Then | | extends uniquely to a discrete valuation | |_L on L, ... ---->
... . Then | |_K extends uniquely to a discrete valuation | |_L on L, ...
p.119 (7.38 PROOF.) Hence, | |_p extends uniquely to a discrete valuation | | on L. --->
Hence, | |_p extends uniquely to a discrete valuation | |_L.
p.119 (7.38 PROOF.) For each \sigma \in Gal(L/K),
consider the map L -> C, \beta |-> |\sigma \beta|.
This is again a valuation on L, and so the uniqueness implies that
| \beta|= | \sigma \beta |. Now
| Nm(\beta) | = | \prod \sigma \beta | = | \beta |^n
----->
For each \sigma \in Gal(L/K),
consider the map L -> C, \beta |-> |\sigma \beta |_L{\prime}.
This is again a valuation on L, and so the uniqueness implies that
| \beta |_L= | \sigma \beta |_L{\prime}. Now
| Nm_{L/K}(\beta)|_K = | \prod \sigma \beta |_L{\prime} = | \beta |^n_L
p.119 (7.40) ... . Then | | extends in a unique way to a valuation on | | on \Omega.
PROOF. The theorem shows that | | extends in a unique way to any finite subextension of \Omega, ...
----> ... . Then | |_K extends in a unique way to a valuation on | |_{\Omega} on \Omega.
PROOF. The theorem shows that | |_K extends in a unique way to any finite subextension of \Omega, ...
p.134 (8.7 PROOF.) ... =^{7.38} \prod_{i=1}^g | \alpha |_i ^{n(i)} =^{8.6} .... ---->
... =^{7.38} \prod_{i=1}^g | \alpha |_i ^{n_i} =^{8.6} ....
p.138 ( The Frobenius element)
(b) .... , where q is the number of elements the residue field O_K/p, .... ---->
(b) .... , where q is the number of elements in the residue field O_K/p, ....
p.143 (Cubic Polynomials)
For example, the calculations on p. 61 show that ... ---->
For example, the calculations on p. 58, 59 show that ...
p.147 (8.40 (c)) Proposition 8.39 is not true without the Galois assumption: --->
(c) Corollary 8.39 is not true without the Galois assumption:
p.150 (4-2) Write p B = \prod P_i^{e(P_i/p)} and P_i C = \prod Q_{ij}^{e(Q_{ij}/P_i)}. --->
In the p B, P_i and p should be in fraktur notation in rhs.
In the P_i C, Q_{ij} and P_i should be in fraktur notation in rhs.
p.151 (4-7) ... , only the divisors of (5) (in Z[i]) can ramify in _K, and hence only.... --->
... , only the divisors of (5) (in Z[i]) can ramify in O_K, and hence only....
p.152 (6-1) ... , and so _K = Z + (\alpha + 1) _K. The proof that O_K= Z[\alpha] ... ---->
... , and so O_K = Z + (\alpha + 1) O_K. The proof that O_K= Z[\alpha] ...
General: Many places you need to replace df by def, or add in the list of notation: def=df.
On page 98, in the definition of the transpose of a correspondence, X and X^\prime should be switched. (Alex Ghitza)
Saikat Biswas points out that, in the first display on p113, the power of \zeta should be \chi{\sigma} not \chi{m}.
From Hendrik Verhoek See pdf file (3 pages)
From Bhupendra Nath Tiwari (corrections in red).
p.2 Then I(\alpha z)= .... -----> I(\alpha(z)) = ....
p.13 (1.8) ... are equivalent if and only they define ... ------>
... are equivalent if and only if they define ...
p.17 (1.21) ... , then L(K-D)=0 (because, for ... -------> then l(K-D)=0 (because, for ...
(as the space L(K-D) can be \phi but not zero, only is dimension can be zero.
p.19 (1.28) ... Thus
2-2g(Y)=(2-2g(X)) - \sum (e_P-1) ---> 2- 2g(Y) = (2- 2g(X))m - \sum (e_P-1)
p.21 (2.1) ... Let $\gamma$ be automorphism $H$. -----> Let $\gamma$ be automorphism of $H$.
p.24 (bottom) ... is implies that SL_2(Z) is dense in ... -------> implies that SL_2(Z) is dense in ...
p.33 (2.22) ... index of P^{\prime} over is 3. ----> ... index of P^{\prime} over
P is 3.
p.38 (3.6) ... On eliminating $\alpha$ from between the two equations, we find that
c \tau^2+ (d-a)\tau + b=0 -----> c \tau^2+ (d-a)\tau -b =0
p.38 (3.6) ... On eliminating $\tau$ from between the two equations, we find that
\alpha^2 -(a+d)\alpha +bc=0 ----> \alpha^2 -(a+d)\alpha +ad- bc=0
p.41 (3.12) ... that z |--> \Delta(z, Z+ Z) is a modular function for... ------>
that z |--> \Delta(z Z+ Z) is a modular function for ... (no , between z and Z).
p.46 (3.12) ... For a subgroup of finite index in $\Gamma(1)$... ----->
For a subgroup $Gamma $ of finite index in $\Gamma(1)$...
p.46 (3.12) ... and at a cusp, and therefore define holomorphic .... ----->
... and at a cusp, therefore define holomorphic .... (no need for the and)
p.47 (4.11 PROOF.) ... (iii) In this case, p is... ----> (c) In this case, p is...
p.48 (4.12 PROOF.) ... ; ord_Q(f)= ord_P(\omega) -k for Q a cusp;... ---->
... ; ord_Q(f)= ord_P(\omega) +k for Q a cusp;...
p.52 (4.20 PROOF. Last line) ... \sum_{n=1}^\infty \sigma_{2k-1} (a). q^n. ---->
... \sum_{n=1}^\infty \sigma_{2k-1} (n). q^n.
(this is with the given definition of \sigma_k(n), as on page 51 just above the Proposition 4.20,
but later you represented it by this notation as well, so it may be clear from the context).
p.57 (The Geometry of H) ... The group PSL_2(R) = ^{def} SL_2(R)/ \pm I plays the same role ... ----> The group PSL_2(R) = ^{def} SL_2(R)/ { \pm I } plays the same role ...
p.64 (5.6) ... if p is prime n \ge 1. Just as in the case of $\Delta$,... ----->
... if p is prime and n \ge 1. Just as in the case of $\Delta$, ....
p.66 (5.10) ... then apply (a) of the theorem. ----> ... then apply (a) of the
proposition (or (5.9)).
p.72 (5.24) ... right orbits M(p)/ (1). ----> ... right orbits M(p)/ Gamma(1).
p.73 (5.25) ... ,with eigenvalue is \sigma_{2k-1} (n). ----> ... ,with
eigenvalues \sigma_{2k-1} (n).
p.74 (5.27 PROOF.) ... The matrix of T(n) with respect to a basis for M_k (Z) integer coefficients, ...
----> ... The matrix of T(n) with respect to a basis for M_k (Z) has integer coefficients, ...
p.75 (3rd para) ... in it, and so we can we can write... ---> ... in it, and so
we can write...
p.76 (5.31) ... A correspondence is a is a "many valued mapping", .... ---->
... A correspondence is a "many valued mapping", ....
p.78 (5.33 PROOF.) ... T^N(n) . T^N(m) . f = \sum_{ d | m, n} d^{2k-1}. ... --->
... T^N(n) . T^N(m) . f = \sum_{ d | gcd(m, n)} d^{2k-1}. ...
p.88 (7.5) ... the ideal (\partial F/ partial X, \partial F/ partial X) mod F(X,Y). ---->
... the ideal (\partial F/ partial X, \partial F/ partial Y) mod F(X,Y).
p.92 (The curve Y_0(N)_Q as a moduli variety) ... S(k^{al}) is cyclic subgroup of
S(k^{al}) of order N. -----> ... S(k^{al}) is cyclic subgroup of E(k^{al}) of order N.
p.93 (8.7) The map H -> E_N(C), z |-> (C / / \Lamda(z,1), (z,1) mod \Lamda(z,1)) induces... --->
The map H -> E_N(C), z |-> (C /
\Lamda(z,1), \Lamda(z,1) mod \Lamda(z,1)) induces...
p.95 (9.2) ... and R(x) >0. (**) ---> ... and R(s) >0. (**)
p.99 (The Hecke correspondence) ... and j^{\prime}_i = j(E^{\prime}/ \alpha S_i). ---->
... and j^{\prime}_i = j(E^{\prime}/ \alpha (S_i)).
p.105 (Review of elliptic curves) First, let W= Tgt_0(E). ----> First, let W= Tgt_O(E),
the tangent space at O to E. (It should be mentioned here, as in the 12.22 on p.119)
p.106 (The zeta function of X_)(N): case of genus 1)
When one of the integer 11, 14, 15, 17, 19, 20, 21, 24, 17, 32, 36, or, 49, the curve... --->
When one of the integer 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, or, 49, the curve...
p.116 (12.9) ... Then j(z) is a algebraic. ---> ... Then j(z) is algebraic.
p.116 (The integrality of j)
When we are given a isomorphism i: R -> End(E), .... ---->
When we are given an isomorphism i: R -> End(E), ....
p.122. Give short notion adelic version of the main theorem (or atleast state),
instead of Omitted.
General: Many places you need to replace df by def, or add in the list of notation: def=df.
From Tobias Barthel See pdf file (2 pages)
From Shaul Zemel In the proof of Theorem 10.15, p49, concerning the map from Hom(A,B) \tensor Z_l to the module Hom(T_l A,T_l B) (over Z_l), you start by proving that if e_1,...,e_n are linearly independent over Z in Hom(A,B) then their images are linearly independent over Z_l in Hom(T_l A,T_l B). But this immediately proves that n cannot exceed the rank of the latter over Z_l, i.e., 4dimAdimB (as can be even more clearly seen in Hom(V_l A,V_l B) over Q_l, after tensoring the latter with Q_l). Hence you immediately obtain the finiteness of the rank of Hom(A,B), and the desired bound, without the need to involve decomposition into simple Abelian varieties, different topologies, and polynomials. This is in fact similar (as you have indicated there for something else) to the fact that showing that if a (clearly torsion-free) subgroup of a real vector space of dimension n is discrete then it's free of rank not exceeding n.
Timo Keller points out that, in the proof of Theorem 10.15, p49, M should be defined to be a Z-submodule of End^0(A), not End(T_l(A)) (the degree map P is defined on End^0(A), not on End(T_l(A)). Also, when I write "Now choose the e_i to be a Q-basis for End^0(A)." I seem to be assuming that End^0(A) is finite dimensional over Q, which is what I'm trying to prove. The proof should be replaced by this.
From Everett Howe In Prop. 13.2(b), I found a small typo, probably carried over from copying the result from [1986b] and not changing all of the notation: the "f" in the exponent should be an alpha.
Tim Dokchitser points out that I prove Zarhin's trick (13.12) only
over an algebraically closed field , and then immediately apply it in (13.13) over a finite field.
This is doubly confusing because (13.10) is certainly false over nonalgebraically closed field (over such a field
an abelian variety need not be isogenous to a principally polarized abelian variety).
However, I believe everything is O.K. Specifically, the proof of Zarhin's trick requires only (13.8), and,
because this holds over an algebraically closed field, it holds over every perfect field (see my 1986
Storr's article Abelian Varieties 16.11 and 16.14).
From Sunil Chetty. Near the start of I 14 (Rosati involution): in (\alpha\beta)^\dagger = \beta\alpha there should be a dagger on each of \beta and \alpha.
From Roy Smith (on proofs of Torelli's theorem III 13)
You ask on your website for advice on conceptual proofs of Torelli. ... here goes.
There are many, and the one you give there is the least conceptual one, due I believe to Martens.
Of course you also wanted short, ....well maybe these are not all so short.
The one due to Weil is based on the fact that certain self
intersections of a jacobian theta divisor are reducible, and is
sketched in mumford's lectures on curves given at michigan. Indeed
about 4 proofs are sketched there.
The most geometric one, due to Andreotti - Mayer and Green is to
intersect at the origin of the jacobian, those quadric hypersurfaces
occurring as tangent cones to the theta divisor at double points, thus
recovering the canonical model of the curve as their base locus, with
some few exceptions.
To show this works, one can appeal to the deformation theoretic
results of Kempf. i.e. since the italians proved that a canonical
curve is cut out by quadrics most of the time, one needs to know that
the ideal of all quadrics containing the canonical curve is generated
by the ones coming as tangent cones to theta. the ones which do arise
that way cut out the directions in moduli of abelian varieties where
theta remains singular in codimension three.
But these equisingular deformations of theta embed into the
deformations of the resolution of theta by the symmetric product of
the curve, which kempf showed are equal to the deformations of the
curve itself. hence every equisingular deformation of theta(C) comes
from a deformation of C, and these are cut out by the equations in
moduli of abelian varieties defined by quadratic hypersurfaces
containing C. hence the tangent cones to theta determine C.
This version of Green's result is in a paper of smith and varley,
in compositio 1990.
Perhaps the shortest geometric proof is due to andreotti, who
computed the branch locus of the canonical map on the theta divisor,
and showed quite directly it equals the dual variety of the canonical
curve. this is explained in andreotti's paper from about 1958, and
quite nicely too, with some small errata, in arbarello, cornalba,
griffiths, and harris' book on geometry of curves.
There are other short proofs that torelli holds for general
curves, simply from the fact that the quadrics containing the
canonical curve occur as the kernel of the dual of the derivative of
the torelli map from moduli of curves to moduli of abelian varieties.
this is described in the article on prym torelli by smith and varley
in contemporary mat. vol. 312, in honor of c.h. clemens, 2002, AMS.
there is also a special argument there for genus 4, essentially using
zariski's main theorem on the map from moduli of curves to moduli of
jacobians.
There are also inductive arguments, based on the fact that the
boundary of moduli of curves of genus g contains singular curves of
genus g-1, and allowing one to use lower genus torelli results to
deduce degree torelli for later genera.
Then of course there is matsusaka's proof, derived from torelli's
original proof that given an isomorphism of polarized jacobians, the
theta divisor defines the graph of an isomorphism between their
curves.
For shortest most conceptual, I recommend the proof in Arbarello,
Cornalba, Griffiths Harris, i.e. Andreotti's, for conceptualness and
completeness in a reasonably short argument..
p94, Theorem 14.9. Although this particular spectral sequence is commonly called the Hochschild-Serre spectral sequence, it would be more appropriate to call it the Cartan-Leray spectral sequence (see Serre, Notices AMS, Sept 2011, p1087).
p16. Proposition 2.1 should say that the projection map W-->V is etale, as is clear from the line 4th from the bottom of the proof. (Rex Cheung)
p20. Ming-chang Kang suggests another proof of Proposition 2.9: a flat unramified homomorphism A->B of local rings with the same residue fields induces an isomorphism on the completions. By a result on flatness (Bourbaki, Commutative Algebra, p. 227; Matsumura, Commutative Ring Theory, p. 74), the induced maps M_A^n/M_A^{n+1}->M_B^n/M_B^{n+1} are isomorphisms. Thus the maps A/M_A^n->B/M_B^n are isomorphisms. On passing to the limit over n, we find that the maps on the completions are isomorphisms.
From Michiel Kosters
Page 12: neighborhood (V,x) of V -> neighborhood (V,x) of x
Page 44: If Z is has a group structure -> if Z has a group structure
Page 59: At the bottom, Hom_{X_{et}}(F,G) -> Hom_{X_{et}}(F,\pi_*G)
Page 66: is the analogous statementfor -> is the analogous statement for
p24. Immanuel Stampfli points out that Corollary 2.16 is nonsense. For example, x->x^2 and x->x^3 are etale maps from the complement of 0 in A^1 to A^1 that agree at 1 but aren't equal. The corollary should be a statement about maps over a base variety S (as in my Etale Cohomology book I 3.13) such that X is etale and separated over S. (A simpler counterexample is provided by the maps from A^1 to A^1 sending x to +x and to -x (Ming-chang Kang)).
p27, line 10. I twice omit the bar over x in \pi_1(X,x). (Ming-chang Kang).
p40, Definition 5.2. I should require that F commutes with fibre products.(Ming-chang Kang)
p42, Example 6.3. Since this looks strange, I add some explanation. When f(T) is irreducible and separable and K=k[T]/(f) is its splitting field, then K/k is Galois (by standard Galois theory) with Galois group G a transitive subgroup of order n of the group of permutations of the n roots of f. (Ming-chang Kang)
p119. It's not necessarily true that H^1(X,Z)=0, as I claim. For example, it need not be true for a curve X with a node (see SGA4 XII 2). However, the argument I give is correct for a normal scheme. (Akhil Mathew)
p128, 21.7. In the diagram, the arrow for i points the wrong way.
p156 The expression appearing in the middle of the page (bounded by two powers of $q$) should be $|\alpha|$. (Eric Moorhouse).
From Michiel Kosters
Page 76: footnote, that is it -> that it is
Page 77: regarded as a G-modules -> regarded as G-modules
From Tom Bachmann
Page 148 (size narrow-class groups) you write "h_m = h or 2h,
according to as epsilon and epsilonbar have the same or different
signs". However, it is just the other way round: if the signs are the
same then h_m = 2h, whereas if the signs differ then h_m = h. Indeed
this is also what you state about the specific examples in the next
paragraph.
I, Theorem 16.21. The centre of a reductive group is of multiplicative type, but need not be reduced (e.g., SL_p). The correct statement is that the radical of G (which is a torus) is the reduced connected group attached to the centre. (There may be other errors of this type, where I incorrectly translated from the characteristic zero case to the general case.) There will be a new version of the notes probably in March 2012.