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Let $p$ be an odd prime. Suppose an elliptic curve $E/\mathbb{Q}$ has reduction of type $I_n$ at $p$. Let $\mathcal{E}$ be its Néron model, and assume that the component group $\tilde{\mathcal{E}}/\...
elliptic's user avatar
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3 votes
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228 views

Let $A/K$ be an abelian variety where $K = \operatorname{Frac}(R)$ for some discrete valuation ring $R$. Let $\mathcal{A}$ be its Neron model. Assume that $A$ has potential good reduction, so that the ...
user47437's user avatar
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Let $S$ be a henselian DVR and $X/S$ be a flat and proper curve with $X$ being regular. Under what conditions the specialization map $Pic^0_{X/S}(S)\to Pic^0_{X/S}(Spec(k(s)))$ is surjective? Here $s\...
Nanjun Yang's user avatar
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206 views

Let $A$ be an abelian variety of dimension $g$ over the quotient field $K$ of a DVR $R$ which is a subfield of the complex field $\mathbb{C}$. Then by a result of Grothendieck, we know that there is a ...
divergent's user avatar
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Let $S$ be the spectrum of an excellent discrete valuation ring with field of fractions $K$ and $C$ be a proper integral regular curve over $K$. Assume, $C$ admits a proper regular flat model $\...
user267839's user avatar
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Let $K$ be a complete (but think Henselian suffice for purposes of this question) local field of characteristic $0$ with residue field $k$ of characteristic $p>0$ and ring of integers $R=\mathcal{O}...
user267839's user avatar
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Let $S$ be the spectrum of an excellent discrete valuation ring with field of fractions $K$ and $C$ be a proper integral regular curve over $K$. Assume, $C$ admits a proper regular flat model $\...
user267839's user avatar
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1 vote
2 answers
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Let $S$ be a Dedekind scheme with function field $K=K(S)$ and $C$ a projective regular curve over $K$, so we can fix certain closed embedding $e:C \subset \mathbb{P}^n_K$. Let compose this embedding ...
user267839's user avatar
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Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other ...
Nathan Lowry's user avatar
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Let $K$ be a field which is complete with respect to a discrete valuation $v$ with ring of integers $R$ and residue field $k$. Let $A$ an abelian variety over $K$ and let $A^t$ be the dual abelian ...
Hans's user avatar
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Let $B$ a discrete valuation ring, say for simplicity with residue field of characteristic $0$, and $K$ its quotient field. Assume that I have an abelian variety $A$ over $K$ and let $A'$ be its Néron ...
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Let $f:T_1 \to T_2$ be an isogeny of tori over a number field $K$. Does $f$ induce an isomorphism of the Lie algebras of the lft Néron models of $T_1$ and $T_2$ ? Are there some interesting properties ...
Adrien MORIN's user avatar
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I have a question about proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite. Could you please tell (correct) me any strange parts about my understanding of the outline of the proof ? My ...
Duality's user avatar
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156 views

Let $\cal A$ be a smooth commutative group scheme over $S$, where $S$ is the spectrum of a discrete valuation ring with fraction field $K$ and residue field $k$. Suppose that $A:={\cal A}_K$ is an ...
Damian Rössler's user avatar
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Let $A$ be an abelian variety over $\mathbb{Q}_p$. Does there exist a proper flat morphism $X\to \mathrm{Spec}\:\mathbb{Z}_p$ such that the generic fiber is isomorphic to $A$, the special fiber is ...
user avatar
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395 views

Let $(S,\eta,s)$ be spectrum of a discrete valuation ring $R$. Let $E$ be an elliptic curve over $\eta$. Let $\mathcal{E}$ be the Neron model of $E$. Is there a concrete example of an $E$-torsor (...
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Let $A$ be an abelian variety over a discretely valued field $K$ and $\mathcal A$ its Neron model over $R$ (the ring of integers of $K$) and $\mathcal A^0$ the identity component of $\mathcal A$. ...
kkgz's user avatar
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Let there be an isogeny $f:A_1 \rightarrow A_2$ between two abelian varieties over a $p$-adic field $F$ and assume $f$ has degree $p^n$. By the universal property we get a moprhism $f_0: \mathcal{A}_1 ...
Zhiyu's user avatar
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7 votes
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675 views

Let $R$ be a D.V.R. with the fraction field $K$, $A$ a $K$- abelian variety, $\mathfrak{A} \to \operatorname{Spec}R$ the Neron model of $A$. We say $A$ has good reduction if there exists a smooth ...
k.j.'s user avatar
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Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$, and let $A$ be an abelian variety over $K$. Suppose that there is a smooth proper scheme $\mathcal{X}$ over $\mathcal{O}_K$ whose ...
Kriss's user avatar
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10 votes
2 answers
706 views

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$. Let $A$ be an abelian variety over $K$. Does $A$ have a Neron model? If $\mathcal{O}_K$ is a discrete valuation ring, then this is ...
Kriss's user avatar
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1 answer
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Let $X$ be a smooth projective $k$-scheme, $k$ being a number field. Let $\mathcal{O}_k$ be the ring of integers of $k$. Fix a large enough category of schemes $\text{Sch}/k$ containing $X$, and ...
user avatar
3 votes
3 answers
377 views

Please provide some references on Voronoi and Delaunay decompositions which is mathematically written. I mean I can find several texts or links on this written for computer science students without ...
user avatar
5 votes
0 answers
622 views

Let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $R$ and residue field $k$. Let $A/K$ be an abelian variety with Neron model $\mathcal{A}/R$. We denote by $\tilde{\mathcal{A}}/k$ the ...
Jędrzej Garnek's user avatar
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1 answer
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My question is related to a previous question on the Mordell-Weil rank of the elliptic curve $E/\mathbf{Q} : y^2 = x^3- 2$ asked here. More precisely, I want to understand the following. Let $E'/\...
David Benjamin Lim's user avatar
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9 votes
2 answers
656 views

I encountered this result while reading a few things, and it was stated without reference. I am having a hard time finding a reference for it (or a simple proof), so maybe you can help me: let $E$ be ...
dbluesk's user avatar
  • 203
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1 answer
520 views

Let $R$ be a complete DVR with fraction field $K$, $X$ be a regular scheme flat over $R$. Let $L$ be a finite field extension of $K$ and $Q$ be the integral closure of $R$ in $L$. Denote by $Y:=X \...
Ron's user avatar
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1 vote
1 answer
235 views

Let $X$ be a normal Noetherian algebraic space and $\mathscr{L}$ a line bundle on $X$. If $X$ is a scheme, then there is locally principal Weil divisor on $X$ that gives rise to $\mathscr{L}$. Is the ...
Question Mark's user avatar
  • 1,123
9 votes
1 answer
544 views

Let $f\colon X \rightarrow S$ be a semistable curve of genus $g \ge 0$. Being a semistable curve means that $f$ is a morphism of schemes such that $f$ is proper, flat, and of finite presentation; The ...
Question Mark's user avatar
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1 vote
1 answer
438 views

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise ...
Question Mark's user avatar
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4 votes
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My question is about the proof of Example 8 in section 9.2 of the book "Neron models." There we have a semistable curve $X$ over an algebraically closed field $K$ and we let $\pi\colon \widetilde{X} \...
Question Mark's user avatar
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1 vote
1 answer
714 views

I am trying to understand the first sentence of the proof of 9.1/5 in "Neron models." There we have a proper curve $X$ over a field $K$ and a line bundle $\mathscr{L}$ on $X$. Our ultimate goal is to ...
Question Mark's user avatar
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3 votes
1 answer
707 views

My question is about the proof of 8.4/2 in "Neron models." The claim is that if $f\colon X \rightarrow S$ is a proper flat morphism of finite presentation such that $H^2(X_s, \mathscr{O}_{X_s}) = 0$ ...
Question Mark's user avatar
  • 1,123
2 votes
1 answer
436 views

Let $X$ be a scheme and let $\mathscr{I} \subset \mathscr{O}_X$ be a quasi-coherent sheaf of ideals. Suppose that for each $x \in X$, the stalk $\mathscr{I}_x$ is generated by an element $f_x \in \...
Question Mark's user avatar
  • 1,123
4 votes
0 answers
395 views

Let $f\colon X \rightarrow S$ be a proper morphism of schemes. Is the cohomology group $H^2(S, f_* \mathbb{G}_m)$ the same regardless of whether it is computed in the etale or the fppf topology? And ...
Question Mark's user avatar
  • 1,123
3 votes
1 answer
231 views

I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...
Question Mark's user avatar
  • 1,123
4 votes
0 answers
249 views

Let $R$ be a discrete valuation ring, $K$ its field of fractions, and $$ 0 \rightarrow G_K' \rightarrow G_K \rightarrow G_K'' \rightarrow 0$$ a short exact sequence of smooth $K$-group schemes of ...
Question Mark's user avatar
  • 1,123
3 votes
2 answers
688 views

Let $f\colon X \rightarrow Y$ be a proper morphism with $Y$ Noetherian (and even affine, if you wish), and let $\mathscr{A} = \bigoplus_{n \ge 0} \mathscr{A}_n$ be a quasi-coherent graded $\mathscr{...
Question Mark's user avatar
  • 1,123
2 votes
1 answer
441 views

Let $S$ be a locally noetherian scheme and let $G$ be a separated and smooth $S$-group algebraic space of finite presentation. Does there exist an open sub-(group algebraic space) $G^0 \subset G$ ...
Question Mark's user avatar
  • 1,123
2 votes
0 answers
178 views

Let $R$ be a Noetherian ring and let $X$ be a finite type, separated $R$-scheme that is normal and integral. Let $x_1, \dotsc, x_n \in X$ be points of height $1$. Does there exist an open affine $U \...
Question Mark's user avatar
  • 1,123
3 votes
1 answer
252 views

My question is about the proof of Lemma D.3 on p. 147 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud. Namely, towards the end of that proof there is the sentence "That $\varphi$ ...
Question Mark's user avatar
  • 1,123
4 votes
0 answers
192 views

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...
Question Mark's user avatar
  • 1,123
1 vote
0 answers
1k views

Let $R$ be a discrete valuation ring with a uniformizer $\pi$ and $(A, \mathfrak{m}_A)$ a local $R$-algebra that is essentially of finite type (i.e., is a localization of a finite type $R$-algebra), ...
Question Mark's user avatar
  • 1,123
3 votes
1 answer
570 views

Recall that a discrete valuation ring $R$ is excellent if the extension $\widehat{K}/K$ is separable, where $\widehat{R}$ is the completion of $R$ (with respect to the maximal ideal), $K = \mathrm{...
Question Mark's user avatar
  • 1,123
1 vote
1 answer
360 views

Suppose I have Neron model over some discrete valuation ring. Is there a result such that the number of components of the fiber over the closed point cannot decrease after some based change? In ...
Maxim's user avatar
  • 444
6 votes
1 answer
1k views

One's can find following definition of tamagawa numbers in Dino Lorenzini paper "Torsion and Tamagawa numbers": Let $K$ be any discrete valuation field with ring of integers $O_K$ , uniformizer $\...
Dr.van's user avatar
  • 63
7 votes
2 answers
2k views

Suppose I have a hyperelliptic curve of genus $2$ over $\mathbb Q$. I want to get information about its Jacobian reduction at prime $p$ (especially, in case $p=2$). Also I'm interesting in the group ...
Maxim's user avatar
  • 444
4 votes
1 answer
833 views

The modular curve $X_0(15)$ has a canonical model over $\mathbf{Q}$, and it has genus $1$. As the cusp $\infty$ is rational, it is an elliptic curve. Roughly, my question is whether we can find all ...
Bertie Wooster's user avatar
  • 295
2 votes
1 answer
538 views

Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice). Let $\eta$ be the generic point of $S$, $K = S(\...
None's user avatar
  • 33
1 vote
0 answers
294 views

How does the line bundles look like on a proper model (or Néron model) of an abelian variety? Who knows references about this? In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ ...
Heer's user avatar
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