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 Yearling
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May
18
comment Some questions about reduction of elliptic curves
Seems like you're asking about the reduction of $X_0(n)$. Have a look at Katz-Mazur's book Arithmetic moduli of elliptic curves and Brian Conrad's article in J. inst. Jussieu with a similar title.
May
15
comment Some questions about reduction of elliptic curves
Any elliptic curve $E$ over a dense open $U \subset \mathbb A^1_{\mathbb Q}$ extends to an elliptic curve over a dense open $\mathcal U \subset \mathbb A^1_{\mathbb Z}$. That should help you do what you want for large $p$. You won't find a curve over $S$ whose fiber at $s$ is isomorphic to $\tilde{E}_s$ simply because $\mathbb Q$ and $\mathbb F_p$ are not isomorphic...
May
15
comment Some questions about reduction of elliptic curves
Consider an elliptic curve $E \to \mathbb A^1_{\mathbb Q} - \{0,1\}$. Let $s$ be a rational point of the base. Then $E_s$ is an elliptic curve over $\mathbb Q$. What does it mean to "reduce $E_s$" in this case? And do you mean by "glueing everything to a curve over $S$"? Isn't $E\to S$ already the object you need?
May
15
comment Some questions about reduction of elliptic curves
I still don't understand 4. Is $E\to S$ an elliptic curve or is only its generic fibre an elliptic curve? Also, what do you mean by finding the Neron model of $E_s \to Spec(k(s))$? To speak of a model for $X/k$ over a scheme $S$, we usually ask $k$ to be function field of this scheme $S$. Are you asking about whether an elliptic curve $E\to S$ is uniquely determined by its generic fibre? In that case, the answer is positive if $S$ is integral noetherian and regular.
May
15
comment Some questions about reduction of elliptic curves
Have a look at Proposition 1.2.4 in the book "Neron models" by Bosch et al. This proposition implies that, if $N$ is the Neron model of the elliptic curve $E$ over $\mathbb Q$, then $N_{\mathbb Z_{(p)}}$ is the Neron model of $E$ over $\mathbb Z_{(p)}$. I agree that the unicity is not all you need here. To prove that Proposition you need to use limit arguments and reduce to the statement that "the Neron model commutes with etale base-change".
May
12
comment Some questions about reduction of elliptic curves
Yes to 1). The Neron model is unique. In 2), the isomorphisms you write are only true over the algebraic closure. Have a look at Chapter 10 of Liu's book for more "scheme-theoretic" proofs. The same chapter also answers 3). Indeed, split multiplicative is the same as split semi-stable (+ singular). I don't completely understand 4). If $E\to S$ is an elliptic curve (so flat in particular) over a scheme $S$, then for all $s$ in $S$ the fibre $E_s$ is an elliptic curve. This is the reduction of $E/S$ at $s$.
May
10
comment $R^nf_*\mathbb{Z}$ trivial for a morphism with hypersurface fibers.
@floflo You say that $f$ has smooth fibres. In particular, assuming $X$ is integral and $Y$ is a smooth curve, then $f$ is flat so that $f$ is a smooth morphism. Therefore $U=Y$ in your second paragraph. Also, a local system being trivial usually means it's zero. In any case, what you claim is not true if $n$ is odd. I tried explaining this in my answer below.
May
10
answered $R^nf_*\mathbb{Z}$ trivial for a morphism with hypersurface fibers.
Apr
28
comment Fibers of a scheme over $\text{Spec}\,\mathbb{Z}$
@miguels The affine line and the projective line over $\mathbb Z$ answered your first two questions. To construct schemes with less points just remove points from these schemes. For instance, you can remove the zero section of $\mathbb A^1_{\mathbb Z}$ to get a scheme with $p-1$ rational points in its fiber over $(p)$.
Apr
14
comment Surface constructed using curves
@dario Compute $E^2$ by writing it as $p^{-1}(y) \cdot p^{-1}(y)$ with $y$ some point in $F$ and $p$ the projection $E\times F\to F$.
Mar
26
comment Higher dimensional analogue for Riemann Hurwitz formula
The setting in the higher-dimensional case is similar to the case of curves: you start with a reasonable map $f:X\to Y$ (say proper with finite fibres) and you want to relate topological invariants of $X$ to topological invariants of $Y$. Unfortunately, one of the points made in the answers to the questions you link to is that you should forget about "numerical relations between topological invariants" and rather work with equalities of line bundles (or divisors). So maybe try to learn a bit about canonical bundles and understand the meaning of Riemann-Hurwitz: $K_X = f*K_Y + R$.
Mar
17
comment Zariski dense implies classically dense?
Dear @DustanLevenstein , I will to translate the proof for you if I find the time today.
Mar
17
revised Zariski dense implies classically dense?
added 32 characters in body
Mar
17
answered Zariski dense implies classically dense?
Feb
20
revised (Reference Request) Desingularization of Fibrations
deleted 142 characters in body
Feb
20
revised (Reference Request) Desingularization of Fibrations
Fixed a typo
Feb
20
comment (Reference Request) Desingularization of Fibrations
Dear @Paul you are completely right. You need an $n$th root of $t$. I was probably thinking of $X_2$ when writing this.
Feb
7
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Jan
28
comment Properties of fibers of a morphism of varieties
@Dubious I miswrote some things in my last comment as well. If $X$ is irreducible and generically non-reduced, then the locus of smoothness is empty. Of course, you can have an irreducible variety with only one non-reduced local ring. Then the locus of smoothness is non-empty (if $X$ has more than one point).
Jan
24
comment Number of elements in fiber
@PeterWalker What might bother you is the definition of $\deg f$ is $f$ is not finite but only quasi-finite. In general, by $\deg f$ one denotes the degree of the corresponding extension of function fields. This makes sense for etale morphisms of integral schemes $X\to Y$ as etale morphisms are dominant and quasi-finite.