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The proof of Corollary 1.7 is fine. I had misunderstood his proof. His proof uses in a crucial way his assumption (ii) which appears on the top of p. 41. As is explained on p. 41, this assumption impl …

I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication). I think there is a mistake in his Corollary 1.7 and I'm wo …

Let $n\geq 2$ and $\mathbb{P}^n(\mathbf{C})$ be the complexe projective space of dimension $n$. Let $H\subseteq \mathbb{P}^n(\mathbf{C})$ be a hypersurface of degree $d$ where the coordinates in $\ma …

Let $X\subseteq\mathbb{R}^k$ be a non-closed, unbounded semi-algebraic subset. Then it seems to me that Proposition 5.49 on p. 189 of the book Algorithms in Real Algebraic Geometry still holds true fo …

This is just an after thought about my question. At the end it is really a linear algebra problem. Starting with the polynomials $g_1,g_2,\ldots,g_n,g_{n+1}=f$ one may ask if it is possible to find an …

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a p …

In Weil's short note entitled "On the Riemann hypothesis in function-fields" he mentions the notion of the complementary correspondence associated to a given correspondence $T:C\rightarrow C$ where $C …

Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us as …

Let $k$ be an arbitrary closed field (of arbitrary characteristic). Assume that we have a short exact sequence of k-algebraic abelian connected groups $$ 1\rightarrow K\rightarrow G \rightarrow H\rig …

On p. 205 of Katz's paper entitled "p-adic L-functions for CM fields" Katz says that "Shimura's algebraicity theorem, in our context, is an easy consequence of the fact that Hodge decomposition of th …

So this question is directly related to a comment made by David Mumford in his Lecture 1 given at U. Michigan in 1974 entitled: What is a curve and how explicitly can we describe them ? Mumford cla …

Let $X\subseteq \mathbb{P}^n(\mathbf{C})$ be a quasi-projective variety. Q: Is $X$ necessarily quasi-compact in the Zariski topology (if yes then how to prove it)?

Let $U$ be a smooth quasi-projective variety over $\mathbf{C}$. Let $U^{\infty}$ be $U$ but thought of as a smooth manifold. Q1: Is there a simple proof (so it should avoid Hironaka's desingulariza …

So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and $Y …

Let $p$ be a prime number and $X_0(p)/\mathbf{Q}$ be the classical modular curve for $\Gamma_0(p)$. Let $\tilde{X}_0(p)/\mathbf{Z}$ be the projective arithmetic surface corresponding to the normalizat …