User kestutis cesnavicius - MathOverflow

Answer by Kestutis Cesnavicius for Etale Cohomology of Punctured Spectra of Local Rings

2013-05-09T00:50:40Z

With your particular choice of $R$, the $H^2$ is $0$. More generally, if $R$ is a strictly Henselian regular local ring of dimension $2$, then by the purity for the Brauer group (in this particular case it is known and due to, I believe, Grothendieck; for a proof see Grothendieck "Le groupe de Brauer II", Prop. 2.3) $H^2_{et}(R \setminus \{ \mathfrak{m} \}, \mathbf{G}_m) = H^2_{et}(R, \mathbf{G}_m) = 0$ . The first equality is due to the purity because you're removing a closed subscheme of codimension $2$; the second equality is because $R$ is strictly Henselian and $\mathbf{G}_m$ is smooth: by Grothendieck "Le groupe de Brauer III" appendix, Thm. 11.7 2), the cohomology can thus be computed over the separably closed residue field, where it vanishes.

More generally assuming that $R$ is a strictly Henselian regular local ring of dimension $\ge 2$, the $H^2$ is always supposed to be $0$ by the aforementioned purity and the argument as above. This is open (as far as I know), although many cases are known, including $\dim R \le 3$. For a brief survey on precisely this question see Gabber "On purity for the Brauer group" in Oberwolfach report No. 37/2004.

Answer by Kestutis Cesnavicius for a question of Galois cohomology

2013-05-03T02:51:21Z

The claimed triviality holds (the nonabelian cohomology set is not a group though), and I don't think you need $Char(K) \neq 2$. To argue this, I will use the long exact nonabelian cohomology sequence of the central extension $1 \rightarrow \mathbf{G}_m \rightarrow GL_2 \rightarrow PGL_2 \rightarrow 1$, a segment of which reads $H^1(K, GL_2) \rightarrow H^1(K, PGL_2) \rightarrow H^2(K, \mathbf{G}_m)$. Firstly, $H^1(K, GL_2)$ is the one-point set because it classifies rank 2 vector bundles over $Spec(K)$, of which there is only the trivial one. Secondly, $K$ is a $C_1$ field by a theorem of Lang (see Serre "Galois cohomology", p. 80, II.3.3 c)), hence is of $dim \le 1$, so its Brauer group $H^2(K, \mathbf{G}_m)$ vanishes (loc. cit. for more details).

The same argument shows that $H^1(K, PGL_n)$ is the one-point set for any $n$ and any $C_1$ field $K$.

Answer by Kestutis Cesnavicius for ramification of discrete valuation field

2013-04-24T12:48:36Z

Yes, there is. Inertia is a subgroup of the decomposition group which by definition preserves the extended valuation. Consequently, the action of $I$ on $\overline{K}$ preserves both $\mathfrak{m}^r$ and $\mathfrak{m}^{r+}$, hence induces the action on the quotient. The induced action on the quotient preserves the $k = \mathcal{O}_{\overline{K}}/\mathfrak{m}$-vector space structure because by definition the action induced by $I$ on the residue field $k$ is trivial.

Is there excision for fppf cohomology?

2013-02-21T16:55:15Z

I am wondering whether the analogue of III.1.27 in Milne's "Etale cohomology" holds true if one works with fppf cohomology with supports instead of etale cohomology with supports. More precisely, let $f\colon X^{\prime} \rightarrow X$ be fppf, let $Z \hookrightarrow X$ be a closed subscheme, and let $Z^{\prime} \hookrightarrow X^{\prime}$ be its preimage, i.e., $Z^{\prime} = Z \times_X X^{\prime}$. Suppose that $f|_{Z^{\prime}}\colon Z^{\prime} \rightarrow Z$ is an isomorphism and also that $f(X^{\prime} - Z^{\prime}) \subset X - Z$. Let $F$ be a sheaf on $X_{fppf}$. Is it true that the natural morphism of fppf cohomology groups with supports $H^i_{Z}(X, F) \rightarrow H^i_{Z^{\prime}}(X^{\prime}, f^*F)$ is an isomorphism?

As in loc. cit., one is reduced to considering $i = 0$, in which case the proof of injectivity continues to work. Imitating the proof of surjectivity in loc. cit., one would take $\gamma^{\prime} \in \Gamma_{Z^{\prime}}(X^{\prime}, F) \subset \Gamma(X^{\prime}, F)$ and would attempt to glue it with the zero section on $X - Z$. To check the glueing condition, one would need to check in particular that the two pull-backs of $\gamma^{\prime}$ to $X^{\prime} \times_X X^{\prime}$ agree. This step is not explained explicitly in loc. cit., presumably because it is clear in the view of II.3.11. The latter says that the push-forward establishes an equivalence of the category of etale sheaves on $Z$ onto the full subcategory of etale sheaves on $X$ consisting of those sheaves that are supported on $Z$. That brings me to my second question: is there an analogue of this result for the fppf topology?

Answer by Kestutis Cesnavicius for Image of isogeny of elliptic curves determines kernel?

2012-08-14T11:37:39Z

It is possible. Suppose $E$ has complex multiplication defined over a number field $F$ by the ring of integers $\mathcal{O}_K$ in a quadratic imaginary field $K$ (and we take $K$ that only has $\pm 1$ as roots of unity, so that your $Aut(E) = \pm 1$ condition is satisfied). Then one can take $E^{\prime} = E$ and the following for kernels of (infinitely many) distinct cyclic isogenies $E \rightarrow E$: for each rational prime $p$ split in $K$ take a prime $\mathfrak{p}|p$ of $K$ and let the kernel of the isogeny be $E[\mathfrak{p}] := \{ P \in E(\bar{F}): \alpha P = 0, \text{ for all } \alpha \in \mathfrak{p}\}$ . By the theory of complex multiplication, $E[\mathfrak{p}] \cong \mathcal{O}_K/\mathfrak{p}$ as $End_F E = \mathcal{O}_K$-modules, so the kernels of the isogenies are indeed cyclic.

Answer by Kestutis Cesnavicius for what is the maximum number of rational points of a curve of genus 2 over the rationals

2012-07-27T17:08:39Z

I believe it is 642. See http://www.mathe2.uni-bayreuth.de/stoll/recordcurve.html

Answer by Kestutis Cesnavicius for Blackbox Theorems

2012-06-14T05:02:14Z

The existence of Neron models. This gets used all the time when one talks of abelian varieties, but familiarity with the proof is almost never needed.

Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies

2012-02-25T03:13:51Z

Let $K$ be a $p$-adic field, that is a complete discrete valuation ring of characteristic $0$ with a perfect residue field $k$ of characteristic $p > 0$ (to simplify one could also take $K$ to be a finite extension of $\mathbb{Q}_p$). Let $\mathbb{C}_K$ be the completion of a (fixed) algebraic closure $\overline{K}$ of $K$. Then one of Faltings' theorems in $p$-adic Hodge theory says for any smooth proper $K$-scheme $X$ there is a natural isomorphism of $\mathbb{C}_K$-semilinear $\mathrm{Gal}(\overline{K}/K) = \mathrm{Gal}(\mathbb{C}_K/K)$-representations

$\mathbb{C}_K \otimes_{\mathbb{Q}_p} H^n_{et}(X_{\overline{K}}, \mathbb{Q}_p) \cong \bigoplus_{q\in\mathbb{Z}} (\mathbb{C}_K(-q) \otimes_K H^{n - q}(X, \Omega_{X/K}^q)).$

Here $X_{\overline{K}}$ is the base change of $X$ to the algebraic closure, whereas $\mathbb{C}_K(s)$ stands for the usual $s$-th order Tate twist by the cyclotomic character describing the action of the absolute Galois group on the $p$-power roots of unity.

My question is: is there an integral version of the above isomorphism? Let me be more precise and explain what I mean by this: let $\mathcal{X}$ be a smooth proper scheme over the valuation ring $\mathcal{O}_K$ of $K$ and let $\mathcal{O}_{\overline{K}}$ be the valuation ring of the algebraic closure. Is there an isomorphism similar to the one above relating, say, $H^n_{et}(\mathcal{X}\times_{\mathcal{O}_K} \mathcal{O}_{\overline{K}}, \mathbb{Z}_p)$ and the $H^{n - q}(\mathcal{X}, \Omega^q_{\mathcal{X}/\mathcal{O}_K})$ 's?

Separable sigma-algebra: equivalence of two definitions

2010-06-08T18:18:19Z

The two definitions alluded to in the title can be found here: http://en.wikipedia.org/wiki/Separable_sigma_algebra (one is that the $\sigma$-algebra is countably generated, the other is pretty much the standard usage of the word separable wrt the semi-metric given by the measure). Why are they equivalent?

Answer by Kestutis Cesnavicius for Degree of finite group schemes

2012-05-18T17:44:28Z

This can be seen from the existence of the quotient $G/H$ as a finite flat $S$-scheme (and an $H$-torsor). One shows that the natural map $G \rightarrow G/H$ is finite flat of order $[H : S]$; the conclusion then follows from the product formula $[G : S] = [G : G/H] [G/H : S]$. Let me give you a reference where all this is spelled out in detail (and which I am basically copying):

Tate, John. Finite flat group schemes. Modular forms and Fermat's last theorem (Boston, MA, 1995), 121–154, Springer, New York, 1997.

What you need is section (3.5) (see also (3.4) and (3.1)).

What's the minimum number of generators for the wild inertia?

2012-05-05T16:00:58Z

Suppose $K$ is a finite extension of $\mathbb{Q}_p$ and $K^{nr}$ the maximal unramified extension of $K$ in some fixed algebraic closure. Let $G_K$ be the absolute Galois group of $K$ and let $I_w$ be the wild inertia subgroup (recall that it is pro-$p$). What is the minimum number of topological generators for $I_w$? In other words, what is the $\mathbb{F}_p$-dimension of $H^1(I_w, \mathbb{Z}/p \mathbb{Z})$?

What I would like to compute is $H^1(K^{nr}, \mathbb{Z}/p\mathbb{Z})^{Gal(K^{nr}/K)}$. Does this follow once one knows the answer to the first question? Note that $H^1(K^{nr}, \mathbb{Z}/p\mathbb{Z}) \cong H^1(I_w, \mathbb{Z}/p\mathbb{Z})$.

Status of conjectures in Serre's 1969 expose on Galois representations on l-adic cohomology

2012-02-17T00:03:15Z

[S]: Serre, Jean-Pierre. Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Seminaire Delange-Pisot-Poitou, 1969-70

Serre presents nine conjectures C$_1$-C$_9$ concerning Galois representations on $l$-adic cohomology of nonsingular projective algebraic varieties defined over a local or global field. What is the status of those conjectures today? The first two of them are part of the Weil conjectures but how about the rest? What (partial) progress has been made towards their resolution? Let me recall the conjectures below (skipping the first two).

Let $K_v$ be a local field of residue characteristic $p$, let $Y$ be a nonsingular projective variety over $K_v$, and let $m\ge 0$. If $G_v$ is the absolute Galois group of $K_v$ the functoriality of $l$-adic cohomology ($l \neq p$) gives us a representation $\rho_l\colon G_v \rightarrow H^m(\overline{Y}, \mathbb{Q}_l) =: V$ (here $\overline{Y}$ is the base change of $Y$ to the separable closure of $K_v$). One can measure how ramified $\rho_l$ is by introducing $\epsilon = \dim V - \dim V^{I_v}$ ($I_v$ is the inertia in $G_v$) and $\delta$ which is a little bit more involved to define (it is the inner product of $\mathrm{Tr }\ \rho_l|_{I_v}$ with the "Swan character" of $\rho_l$, see 2.1 in [S]). The conductor exponent of $\rho_l$ is then $f = \epsilon + \delta$.

C$_3$: The integers $\epsilon$, $\delta$ and $f$ are independent of $l$.

Status: ???

C$_4$: $\mathrm{Tr }\ \rho_l|_{I_v}$ takes values in $\mathbb{Z}$ and is independent of $l$.

Status: ???

Consider the geometric Frobenius $\pi \in G_v/I_v$. Then $\rho_l(\pi)$ acts on $V^{I_v}$ and we get a polynomial $P_{\rho_l}(T) = \det(1 - \rho_l(\pi) T)$.

C$_5$: $P_{\rho_l}$ has coefficients in $\mathbb{Z}$ and is independent of $l$.

Status: ???

Assuming the latter conjecture split $P_{\rho_l}(T) = \prod (1 - \lambda_\alpha T)$ and let $Nv$ denote the cardinality of the residue field of $K_v$.

C$_6$: For each $\alpha$ there is an integer $m(\alpha)$ between $0$ and $m$ such that $|\alpha| = (Nv)^{m(\alpha)/2}$.

Status: ???

C$_7$: If $\epsilon = 0$ (i.e., if $\rho_l$ is unramified) then all $m(\alpha)$ are equal to $m$.

Status: ???

C$_8$: Let $g$ be an element of $G_v$ whose image in $G_v/I_v$ is an integral power of Frobenius. Then the characteristic polynomial of $\rho_l(g)$ has coefficients in $\mathbb{Q}$ and is independent of $l$.

Status: ???

The last conjecture C$_9$ concerns the zeta function $\zeta(s)$ of a nonsingular projective variety $X$ defined over a global field $K$ (and fixed $m \ge 0$ as above), as well as the completed version $\xi(s)$ of $\zeta(s)$. It's a bit of a trek to define $\zeta(s)$ and $\xi(s)$ so I'll refer to [S] $\S$3, $\S$4 for that. Both $\zeta(s)$ and $\xi(s)$ are holomorphic functions on some right half-plane.

C$_9$: $\zeta(s)$ and $\xi(s)$ admit meromorphic continuations to the complex plane. In addition, $\xi(s)$ satisfies the functional equation $\xi(s) = w\xi(m + 1 - s)$ with $w = \pm 1$.

Status: Open. Afterall, a special case of this is meromorphic continuation of $L$ functions of elliptic curves over number fields.

As the answers come in feel free to edit the status fields above adding more information (and I will try to do that myself).

Answer by Kestutis Cesnavicius for Irreducibility of compositions of polynomials

2012-02-03T18:45:53Z

Yes, there does. In fact, we will show that a polynomial $q(x) = x^2 + d$ works for some $d \in \mathbb{Q}$. The conclusion will follow from Hilbert's Irreducibility Theorem once we show that $p(x^2 + d)$ is irreducible (as a polynomial in two variables).

Suppose, therefore, that $p(x^2 + d) = (d^a f_a(x) + \dotsb)(d^bg_b(x) + \dotsb)$ splits. Both $a$ and $b$ are positive (else you could plug in some value of $x$ in $\mathbb{C}$ to make the RHS vanish regardless of $d$, which is impossible by starring at the LHS). But then $f_a$ and $g_b$ are nonzero and so there is a rational $x$ which is a zero of neither of them. Plug it in and get a nontrivial splitting of (a shift of) $p(x)$. Contradiction, as $p(x)$ was assumed to be irreducible.

Distance functions on elliptic curves over number fields

2011-07-30T01:16:19Z

My question originates from the book of Silverman "The Aritmetic of Elliptic Curves", 2nd edition (call it [S]). On p. 273 of [S] the author is considering an elliptic curve $E/K$ defined over a number field $K$ and he introduces the notion of a $v$-adic distance from $P$ to $Q$. This is done as follows:

Firstly, let's fix an absolute value (archimedean or not) $v$ of $K$ and a point $Q\in E(K_v)$ (here $K_v$ is the completion of $K$ at $v$). Next let's pick a function $t_Q \in K_v(E)$ defined over $K_v$ which vanishes at $Q$ to the order $e$ but has no other zeroes. Now the $v$-adic distance from $P \in E(K_v)$ to $Q$ is defined to be $d_v(P, Q) := \min (|t_Q(P)|_v^{1/e}, 1)$. We will say that $P$ goes to $Q$, written $P~\xrightarrow{v}~ Q$, if $d_v(P, Q) \rightarrow 0$. Later in the text (among other places in the proof of IX.2.2) the author considers a function $\phi\in K_v(E)$ which is regular at $Q$ and claims that this means that $|\phi(P)|_v$ is bounded away from $0$ and $\infty$ if $P~\xrightarrow{v}~ Q$.

I have a couple of questions about this:

How does one choose a $t_Q$ that works? In the footnote in [S] it is demonstrated how one could use Riemann-Roch to pick a $t_Q$ that has a zero only at $Q$. It seems to me however that such a procedure will not make sure that $t_Q$ is defined over $K_v$ since $K_v$ is not algebraically closed.
For $\phi$ as above which does not vanish nor has a pole at $Q$, how does one see that $|\phi(P)|_v$ is bounded away from $0$ and $\infty$ as $P~\xrightarrow{v}~ Q$?
Do these $d_v$ have anything to do with defining a topology on $E(K_v)$? I assume not, since I don't see how to make sense of it; but then on the other hand they are called "distance functions"...

Module of Kahler differentials of rings of integers of number fields

2011-07-12T00:27:08Z

How does one prove that if $L/K$ is an extension of number fields with rings of integers $B/A$, then the module of Kahler differentials $\Omega^1_{B/A}$ can be generated by one element as a $B$-module? When one proves that the annihilator of $\Omega^1_{B/A}$ is the different of the extension, one localizes and completes since both the different and the module of Kahler differentials are compatible with localization and completion. In the local complete case one has $B = A[b]$ for some $b\in B$ and both the claim about the different and my question are easy. What I don't see is how to pick a $b\in B$ before localization and completion which would "work at all finite places" (assuming this approach could be used to solve my question, to begin with)?

Answer by Kestutis Cesnavicius for Metrizability of $\mathfrak{a}$-adic topology

2011-05-16T14:12:07Z

No, it is not true in general. Take $\mathfrak{a} = A$ and $A$ nonzero to get that $A$ equipped with the $\mathfrak{a}$-adic topology is not Hausdorff and hence not a metric space (since $|A| > 1$). More generally, your topology will be non-Hausdorff whenever $\bigcap_{n\ge 1} \mathfrak{a}^n \neq 0$ yielding many more counterexamples.

Answer by Kestutis Cesnavicius for Does completion commute with localization?

2011-05-09T16:26:54Z

It is true. $(\widehat{A}, \widehat{m})$ is a Noetherian local ring so your left hand side could be simplified replacing it by $\widehat{A}$. Now let's just use the definitions: $\widehat{A} = \varprojlim A/\mathfrak{m}^n$, whereas $\widehat{A_{\mathfrak{m}}} = \varprojlim A_{m}/(\mathfrak{m}A_{\mathfrak{m}})^n$. The desired equality is the result of localization being exact so that $A_{\mathfrak{m}}/(\mathfrak{m}A_{\mathfrak{m}})^n = (A/ \mathfrak{m}^n)_{\mathfrak{m}}$ and the fact that in $A/\mathfrak{m^n}$ everything outside the maximal ideal is already invertible, so that $(A/\mathfrak{m^n})_{\mathfrak{m}} = A/\mathfrak{m}^n$.

Categories of descent data

2011-03-02T20:34:44Z

Let us work over the etale site $\mbox{Aff}/S$ (for the sake of definiteness) for some fixed base scheme $S$, where the covers are jointly surjective etale maps $\{ U_i \rightarrow U\}_{i\in I}$ (and $I$ is finite if you like). Let us also consider a prestack $F$ fibred in groupoids. Recall the definition of a category of descent data $F(\{ U_i \rightarrow U\}_{i \in I})$ associated to a cover $\{ U_i \rightarrow U\}_{i \in I}$. The objects of this category are collections of elements $\xi_i\in F(U_i)$ together with morphisms $\phi_{ij}$ between their appropriate pullbacks satisfying the cocycle condition. This definition is the one appearing on p. 15 of "Champs algebriques" by Laumon & Moret-Bailly (among other sources, e.g., Vistoli's notes in "FGA explained").

However, one can also consider coverings $U^\prime \rightarrow U$ consisting of one element only. In $\mbox{Aff}/S$ starting with any covering one can obtain one of such form by taking $U^\prime := \bigsqcup U_i$. However, it is not clear to me how this passage to a cover with a single morphism interacts with the associated categories of descent data. I suspect, they should be equivalent (in "Champs algebriques" for instance, the authors switch to the latter when exhibiting the stackification of a prestack in (3.2)) but on the other hand I don't see how is one supposed to get a single $\xi \in F(U^\prime)$ starting off with the $\xi_i$ as above, let alone an equivalence of categories between $F(\{ U_i \rightarrow U\}_{i \in I})$ and $F(\{ U^\prime \rightarrow U\})$. Are these categories equivalent and what is a functor exhibiting this equivalence? And if not, why is one allowed to consider only coverings of the form $U^\prime \rightarrow U$ when constructing the stackification?

Cesaro convergence implies weak convergence of a subsequence

2010-07-19T14:24:47Z

Suppose a bounded sequence $(x_n)$ converges to $x$ in the Cesaro sense (i.e., $\frac{1}{n}(x_1 + x_2 + \dots + x_n)\rightarrow x$) in a separable Hilbert space $H$. How to prove that some subsequence $(x_{n_k})$ converges weakly to $x$?

Subspaces of finite codimension in Banach spaces

2010-07-07T10:15:31Z

Is every finite codimensional subspace of a Banach space closed? Is it also complemented? I know how to answer the same questions for finite dimensional subspaces, but couldn't figure out the finite codimension case.

Hanner's inequalities: the intuition behind them

2010-06-10T13:21:56Z

Hanner's inequalities in the theory of $L^p$ spaces (see http://en.wikipedia.org/wiki/Hanner's_inequalities) look hard to come-up with at the first glance. Their proof (say, the one in Lieb & Loss "Analysis", Theorem 2.5.) gives no intuition (at least for me) how they come about. How does one see that these inequalities turn up naturally? Do you know a proof which at the same time hints to how one starts considering Hanner inequalities. I hope this is not too vague of a question. Both Wikipedia and Lieb & Loss mention that Hanner's inequalities have to do with uniform convexity of $L^p$ spaces, but from that alone I cannot see how they arise "naturally".

Countable discrete abelian group amenable

2010-06-12T19:15:14Z

For me the definition of amenability of an at most countable discrete group (with counting measure) is existence of a Folner sequence. Assuming this, why is every countable discrete abelian group amenable? What is the Folner sequence that does the job?

A question on a proof that fine sheaves are soft

2010-05-12T13:49:16Z

Let's open R.O.Wells "Differential Analysis on Complex Manifolds" p. 53 and have a look at the Proposition 3.5 stating that all fine sheaves are soft (over a paracompact Hausdorff $X$). In the proof we consider the covering of a closed $S \subset X$ by open $U_i$. Why can't we just take a single $U_1$ covering $S$? A section $s$ over $S$ by definition is an element of a direct limit, so it should have a representative in some neighborhood of $S$ and we could just set $U_1$ to be that neighborhood. Or couldn't we? But the proof is more complicated than that and I'm confused...

Characterization of combinatorial manifolds in terms of links

2010-05-10T09:01:20Z

I need to reference the following result. Do you know a good source?

The following conditions on an $n$-dimensional simplicial complex $S$ are equivalent: a) $S$ is an $n$ manifold; b) The link of every vertex of $S$ is homeomorphic to the $(n - 1)$-sphere; c) The link of every $k$-simplex is homeomorphic to the $(n - k - 1)$-sphere.

A question on a Davis complex of a Coxeter group

2010-05-07T16:57:11Z

Let us have a look at p. 64 of M. Davis book "The Geometry and Topology of Coxeter Groups". The discussion preceeding Definition 5.1.3. shows that $\mathcal{U}(G, X)/G$ is homeomorphic to $X$. Theorem 7.2.4. says that $\mathcal{U}(W, K)$ is $W$-equivariantly homeomorphic to the Davis complex $\Sigma$. So, $\Sigma/W$ is homeomorphic to $K$. $K$ is the cone on the barycentric subdivision of the nerve $L$. $L$ can have topological type of any polyhedron. So $K$ can be a cone on any polyhedron (up to homeomorphism). But the action of $W$ on $\Sigma$ is cocompact (p. 4, bottom). So $\Sigma/W$ is compact, i.e., $K$ is compact. So a cone on any polyhedron is compact. What's wrong?

Noncompact homology spheres?

2010-04-20T15:59:15Z

Are all homology spheres compact? Are all generalized homology spheres compact? By a homology sphere I mean an $n$-manifold $X$ with same homology as the $n$-sphere. By a generalized homology sphere I mean the same with the assumption "$n$-manifold" replaced by "homology $n$-manifold".

If that helps, assume further that the spaces under consideration are simply connected.

Comment by Kestutis Cesnavicius

2013-05-03T02:53:21Z

A more informative title wouldn't hurt, I think.

Comment by Kestutis Cesnavicius

2013-04-24T13:10:36Z

Yes, it can. See Serre "Proprietes galoissienes..." Prop. 7 in section 1.8.

Comment by Kestutis Cesnavicius

2013-04-23T16:04:37Z

When the base in integral and geom. unibranch, abelian schemes are projective; see Raynaud "Faisceaux amples..." XI.1.4.

Comment by Kestutis Cesnavicius

2013-02-21T19:56:28Z

Thanks for your comment! As you explain, in the Noetherian case one can replace $X^{\prime}$ by an open $V \subset X^{\prime}$ containing $Z^{\prime}$ and replace $X$ by its image $U = f(V)$ to reduce to the case where $f$ is etale. In this case, for the purpose of showing that the two pull-backs of $\gamma^{\prime}$ to $X^{\prime} \times_X X^{\prime}$ agree, one can restrict the attention to the small etale site, which is the case treated in Milne.

Comment by Kestutis Cesnavicius

2013-02-04T19:35:08Z

You may find the Appendix of Gille, Pianzola "Isotriviality and etale cohomology of Laurent polynomial rings" relevant (though I don't think it addresses your question precisely).

Comment by Kestutis Cesnavicius

2013-01-27T12:30:03Z

What precisely do you want to know about it?

Comment by Kestutis Cesnavicius

2012-12-05T11:49:29Z

Are you working with small or big etale sites? Your 1. (1) only makes sense to me in the big etale site, in which case the agreement that you want results from the definition of $i^*G$.

Comment by Kestutis Cesnavicius

2012-09-03T12:33:09Z

No, you don't. Even if $\mathfrak{p}$ is not principal, $E[\mathfrak{p}]$ is Galois invariant of order $\#(\mathcal{O}_K/\mathfrak{p}) = p$. Hence it is the kernel of some $p$-isogeny $E \rightarrow E$.

Comment by Kestutis Cesnavicius

2012-08-30T13:41:23Z

For the second question: there are CM elliptic curves over Q. Their base changes to the field are hand are CM as well. E.g., you could take y^2 = x^3 - x over any number field. On the other hand there are non-CM curves as well: just take the j-invariant to lie in the number field you want but to not be an algebraic integer.

Comment by Kestutis Cesnavicius

2012-08-04T07:09:33Z

I'm not sure whether this answers your question completely, but you may find Volkov, Maja, Les représentations l-adiques associées aux courbes elliptiques sur Qp. J. Reine Angew. Math. 535 (2001), 65–101. and Volkov, Maja, A class of p-adic Galois representations arising from abelian varieties over Qp. Math. Ann. 331 (2005), no. 4, 889–923. relevant.

Comment by Kestutis Cesnavicius

2012-05-31T23:27:08Z

Thanks everyone!

Comment by Kestutis Cesnavicius

2011-10-18T16:05:46Z

I don't understand what you mean by 'Does this imply easily that the covering ... 'splits'?' ? It seems to me that Deligne's theory of cohomological descent fits to what you're asking for in your first paragraph: see Brian Conrad's notes math.stanford.edu/~conrad/papers/hypercover.pdf (esp. Thm 7.2 can be interpreted to say that 'there is descent for something that usually is far from being a hypercover') and the references given there (Expose Vbis in SGA4, Deligne's Theorie des Hodge III) for details.

Comment by Kestutis Cesnavicius

2011-10-12T22:42:59Z

But I also have $\tau_{\ge n} \mathcal{G}^{\bullet}$ for which I would like to "apply the induction hypothesis" and it's only bounded below.

Comment by Kestutis Cesnavicius

2011-07-12T13:13:02Z

Thanks! For those who might read this in the future: I found en.wikipedia.org/wiki/… useful to understand the "by the Chinese remainder theorem" part.

Comment by Kestutis Cesnavicius

2011-03-19T16:49:57Z

As for compositions of immersions being immersions, that is easy to see. All you need is to show that any f, which is an open immersion followed by a closed immersion can be factored as a closed immersion followed by an open immersion. The image of your open immersion is an open subset of im f and therefore comes as an intersection of an open set U with im f. Well, in your new factorization U will be give an open immersion and im f \cap U will give a closed immersion (into U).