All Questions
974 questions with no upvoted or accepted answers
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Implicit function theorem and compactification of algebraic curve
Let $C$ be a singular curve defined over a local field $K$.
Let $\tilde{C}$ be its smooth compactification(maybe this is not normalization).
Why $\tilde{C}(K)\neq \emptyset$ implies ${C}(K)\neq \...
- 1,541
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108
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Does an open immersion "cut out" points surviving finite descent?
Let $X$ be a smooth affine curve over a number field $k$, and let $C$ be its smooth compactification. Let $i:X(\mathbb{A}_k) \rightarrow C(\mathbb{A}_k)$ be the induced morphism of adelic points by ...
- 895
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96
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Are the irreducible components appearing in the resolution of singularities of a Hilbert modular surface defined over $\mathbb{Q}$?
It seems to me that this is claimed in van der Geer's "Hilbert modular surfaces" on p. 245 at the beginning of XI.2 (without justification).
My current state of belief/knowledge:
The ...
- 93
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192
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Vanishing of the local étale cohomology sheaf (?)
Let $X$ be a locally noetherian regular scheme, and let $Z$ be a closed subscheme of $X$ whose codimension $d > 0$ at every point.
Let $U$ be the complement of $Z$ in $X$.
For a sheaf $\mathscr{F}$ ...
- 185
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83
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Are there known situations where this weaker form of the section conjecture holds?
Let $k$ be a number field. The section conjecture predicts that for a (smooth geometrically connected) hyperbolic curve over $k$, the profinite Kummer map $\kappa :X(k) \rightarrow \mathscr{J}_{\pi_1(...
- 895
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180
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Maximal unramified extension and algebraic closure of $\operatorname{Frac}(\widehat{A_{\mathfrak{m}_A}})$
$\DeclareMathOperator\trdeg{trdeg} \DeclareMathOperator\Frac{Frac} $
Let $k$ be an algebraically closed field of characteristic $0$ and $K$ a function field over $k$. let $(A, \mathfrak{m}_A)$ a ...
- 6,006
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205
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Projective scheme over the integers
Let $X$ be a projective scheme over $Spec(\mathbb{Z})$. Let $X_{p}$ be the reduction at $p$ of $X$. If for any prime $p$, $X_{p}$ is normal, can we deduce $X$ is normal? Or any counterexamples?
- 653
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186
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Moduli interpretation for integral models of PEL Shimura variety at parahoric level?
Kottwitz has built canonical integral models for a large family of PEL Shimura varieties, associated to a certain reductive group $G$ over $\mathbb Q$, when the structure level has the form $K = K_pK^...
- 769
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98
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$K$-ranks of some algebraic groups in Lubotzky's "Discrete groups, expanding graphs and invariant measures"
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Let $G$ be a semisimple algebraic group and $K$ any field. Then
the $K$-rank of $G$ is the maximal rank $r$ of a $K$-splitting
torus $T \cong (K^...
- 6,006
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180
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Homomorphism of formal group of elliptic curve corresponding to its endomorphism
Let $E$ be an elliptic curve and $ \hat{E}$ be its formal group.
Rubin's lemma $3.7$ in 'Elliptic curves with complex multiplication' reads
For arbitrary $φ∈End(E)$, there exists unique $φ(t)∈End( \...
- 1,541
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117
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Arithmetic analogues in Liouville quantum gravity
I recently discovered about Minhyong Kim's work on what can be coined "Arithmetic Gauge Theory/Arithmetic Chern Simmons Theory". Since Liouville quantum gravity is fully understood, I was ...
- 115
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89
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Image of higher displays in isocrystals
There is a functor from the category of higher displays over $k$ of type $\mu$ to the category of isocrystals over $k$ where $k$ is an algebraically closed field where you forget about the filtration ...
- 1,093
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102
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About Definition 2 in Roĭtman's Paper
Let $A(X)$ be the Chow group of $0$-cycles on a nonsingular irreducible projective variety $X$ over an uncountable algebraically closed field of characteristic zero.
In Definition 2 of Roĭtman's paper ...
- 519
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246
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To justify the intuition about #$E(\Bbb Q_p)$＝$∞$
Let $E$ be an elliptic curve on $\Bbb Q_p$.
$E_0(\Bbb Q_p)$ is points of $E(\Bbb Q_p)$ reduced to nonsingular points.
How to prove #$E(\Bbb Q_p)$＝$∞$ directly ?
According to Silverman's book 'the ...
- 1,541
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172
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Does Lemma 5.4 in Deligne's Ramanujan paper generalize to Shimura varieties of PEL type?
It is generally not known if a smooth variety over a perfect field embeds into a smooth proper variety.
Lemma 5.4 in Formes modulaires et représentations $\ell$-adiques provides such an embedding for ...
- 31
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208
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Outline of the proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite
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 ...
- 1,541
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167
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The existence of two $p$-isogenies implies the existence of one $p^2$-cyclic isogeny
Let $E$ be an elliptic curve over $\mathbb{Q}$.
(or over a number field $K$.)
If $E$ has two $p$-isogenies over $\mathbb{Q}$, then $E$ has $p^2$ cyclic isogeny over $\mathbb{Q}$.
I want to show it ...
- 185
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194
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Brauer-Manin obstruction and affine curves
I'm looking for references that can justify to what extent is the following statement true:
Statement. Let $X$ be a smooth geometrically integral curve over a number field $k$. Then the Brauer-Manin ...
- 895
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193
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How to construct a sheaf on the infinitesimal site from a stratified module
Let $X\to S$ be a morphism of schemes.
Proposition 2.11 of the book "Notes on crystalline cohomology" by Berthelot and Ogus states that a stratified $\mathcal{O}_X$-module $(E,\{\...
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143
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A specific Diophantine equation related to the congruent number question
Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
- 159
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223
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Cohomological dimension of continuous étale cohomology of finitely generated fields
Given a finitely generated field $F$ with prime field $k$, we assume $k$ is finite, of characteristic $p$. Fix a prime $\ell$ invertible in $k$.
In the discussion right after [K, Lemma 2.3], the ...
- 5,901
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73
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Would the iterated finite abelian descent obstruction equality hold for curves?
Let $X$ be a smooth projective geometrically integral variety over a number field $k$. We begin with some established notions in the theory of descent obstruction to the local-global principle, ...
- 895
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377
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Bad primes for algebraic curves
I am confused with general notion of integral models of algebraic varieties. Let us focus on, say, algebraic curves.
If $X$ is a not necessarily projective algebraic curve over a number field $K$, is ...
- 1,428
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192
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p-adically completed Hodge-completed de Rham algebra
Let's look at this paper on page 40.
Let $K$ be a finite extension of $\mathbb Q_p$. One can define the derived de Rham algebra $L\Omega_{\mathcal{O}_{\overline{K}}/\mathcal{O}_K}^{\bullet}$. There is ...
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152
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Image of pullback for Brauer groups
If a have a dominant morphism $\pi:X \rightarrow \mathbb{P}^{1}$ where $X$ is a projective, geometrically integral $k$-scheme. Then this gives rise to a pullback map
\begin{align*}
\pi^{*}:\text{Br}(k(...
- 481
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255
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Construction of the Hilbert Scheme
I am reading the book "Rational Curves on Algebraic Varieties" of János Kollár. Definition-Proposition 1.2, begin like this:
Let $g:Y\rightarrow Z$ be a projective morphism and $\mathcal{O}(...
- 519
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152
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Rational points on towers of surfaces
Take infinitely many 2-variable polynomials $p_k(X,Y)\in \mathbb{Q}[X]$ ($k\in \mathbb{N}$) and let $S_n$ be the surface given by $p_1(X,Y)=Z_1^2,\dots, p_n(X,Y)=Z_n^2$
Assume that no $p_k$ equals the ...
- 2,959
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293
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Shimura varieties which are not of abelian type but has a good modular description
Deligne's idea was that Shimura varieties should be understood as moduli space of motives(with extra structures). lot's of Shimura varieties of abelian type can be understood as moduli space of ...
- 1,093
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104
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A subgroup of the $n$-Selmer group
Let $p$ be an odd prime and for the purpose of this question let $n$ be an integer which is a power of $p$.
Let $E$ be an elliptic curve over a number field $F$.
The $n$-Selmer group, denoted by $S_n(...
- 1,283
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62
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Polynomial sized arithmetic map from circle to ellipse preserving integral points
Let $n$ be a square free integer and a product of $O(m/\log m)$ number of primes $1\bmod 4$ where $m$ is $\log_2n$.
Take the circle around origin of radius $n^2$. It has ${\exp}(m/\log m)$ number of ...
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158
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Is this an explicit construction of a Hurwitz space with Galois group Z/p, p distinct branch points, and inertia group Z/(p-1)?
I am desperately confused and would like a sanity check that the following moduli space/stack is a Hurwitz space/stack. I would also appreciate any references on the topic of the explicit construction ...
- 3,539
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78
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Simultaneous square values of binary quadratic forms
Let $f_1, \cdots f_k$ be binary quadratic forms with $\mathbb{Z}$-coefficients, such that no two of them share a common linear factor. I want to count the number of pairs of integers $(x,y)$ such that
...
- 27k
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155
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Cubic surface in $\mathbb{P}^3$ singular along a line
Maybe it is a stupid question but I'm not able to find the answer anywhere else.
My goal is to prove in an "algebraic geometry fashion" that $\sqrt{n}$ is not a rational number for $n$ not a ...
- 1,343
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Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?
Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...
- 1,479
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88
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Distribution of number of integer solutions in box to bivariate polynomials?
Take a bivariate polynomial of degree $d_x+d_y>\max(d_x,d_y)>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree).
What is the ...
- 13.9k
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80
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Number of integers in relation to Pythagorean triples with some quadratic relations
Given an integer $m>0$ possibly composite we can find non-negative integers which are not equivalent to $0\bmod m$ with
$$ab+cd\equiv0\bmod m$$.
Is there any integer quadruples bounded in $[0,m-1]^...
- 1,836
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89
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Sifted sets can't accumulate on a curve
Let $f,g,h$ be elements in $\mathbb{Z}[x,y]$, each geometrically integral and at least two of them are distinct. Without loss of generality, suppose that $f$ is not proportional to $g$ over $\mathbb{C}...
- 27k
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Does the Galois groups $G_1$ and $G_2$ are isomorphic under `some suitable assumptions`?
Let $E_1,~E_2$ be two elliptic curve on the $p$-adic field $K \supseteq \mathbb{Q}_p$.
Consider the $p$-power torsion points and adjoin them with $K$.
Denote $E_i[p^n], ~i=1,2$ to be the set of $p^n$-...
- 930
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$n$-variable polynomials modulo $p$
The Hasse-Weil bound implies that for any 2-variable polynomial $P(x,y)$, there exists approximately $p$ solutions in $\mathbb{F}_p$ of $P(x,y) \equiv a \pmod p$ for sufficiently large $p$, and any ...
- 141
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galois deformation ring with type is union of irreducible components
Notation:
$K$ finite extension of $\mathbb{Q}_p$, $G_K$ absolute Galois group of $K$,
$E$ is finite extension of $\mathbb{Q}_p$ (coefficient field), $O_E$ is ring of integer in $E$.
In this paper of ...
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191
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Group cohomology of sheaves under closed immersion
Suppose $X$ is a scheme over Spec $\mathbb{Z}$, and $p$ is a non-zero prime in $\mathbb{Z}$. Then we have a closed immersion from the special fibre $i_p: X_p \rightarrow X$. If $\mathscr{F}$ is a ...
- 11
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178
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L-function in p-adic spaces
I've been learning more about different $p$-adic geometries, namely Berkovich spaces, Huber's Adic spaces and ridgid analytic spaces. In arithmetic geometry, it is often very interesting to assoicate ...
- 4,428
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59
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Over derivations with an inusual property
Let $A$ be a ring finitely generated over $\mathbb{Z}$. Let $D$ be a derivation on $A$. Let $D^p$ be the composition of $D$ with itself, $p$ times. We suppose that $D^p(x)$ belongs to the ideal $pA$, ...
- 527
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83
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Can someone suggest some references for rank discussion of elliptic curves (bonus if it systematic)?
First, I apologise if such a question has been asked before. Please feel free to refer me to the previous question, if it answers my current query then I will delete this post.
I am reading the ...
- 401
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246
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Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
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88
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Algebraic definition of the "pseudo complement" of algebraic curve
Not sure if this makes sense.
Let $K$ be field and $C : f(x,y)=0$ algebraic curve curve over $K$.
Define the "pseudo complement" $\hat{C}$ to be the rational surface
$z f(x,y) - 1=0$ with ...
- 25.4k
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111
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An irrational complete intersection surface with good reduction everywhere
Do there exist 3 absolutely irreducible homogeneous polynomials in $\mathbb{Z}[a, b, c, d, e, f]$ such that
each one defines a hypersurface in $\mathbb{P}^5_{\mathbb{Z}}$ smooth over $Spec(\mathbb{Z})...
- 11
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272
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Is a birational morphism between normal projective varieties residually separable?
My goal is to use a "normal Bertini" theorem (see https://link.springer.com/article/10.1007%2Fs000130050213)
More specifically, let $k$ be a field (you may assume that k is infinite but it should be ...
- 121
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56
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Can someone help in understanding this isomorphism helpful in proving finiteness of n-Selmer group?
This is from J.S.Milne's Elliptic Curves book.
We have $H^1(\mathbb{Q}, E_2) \cong (\mathbb{Q}^× / \mathbb{Q}^{×2})^2$ because $Gal(\mathbb{Q}^{al} / \mathbb{Q})$ acts trivially on $E_2(\mathbb{Q}^{...
- 401
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132
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Having difficulty in understanding a result that'll help in proving the finiteness of Selmer group
I'm reading and trying to understand the proof of the finiteness of n-Selmer group from J.S.Milne's Elliptic Curves book but having difficulty in understanding it. Here's a screenshot from the book-
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