All Questions
974 questions with no upvoted or accepted answers
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Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence
My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
- 485
4
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236
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Number of homomorphisms from a group to $\mathrm{GL}_n(\mathbb{F}_q)$
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms ...
- 2,751
4
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195
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Rational points on ramified coverings of abelian varieties
Let $K$ be a number field $A$ an abelian variety over $K$ and $f: X \to A$ a possibly ramified covering of degree $d$ with $X$ a proper variety. My question is:
Suppose that $f(X(K)) \neq A(K)$, can ...
- 2,224
4
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231
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How big are small inverse powers of 2 mod powers of 3?
Let $a \bmod b$ take value as an integer in $[0, b)$. For any $T \ge 1$, for what $R \in [0, 3^n)$ is
$$\min \{2^{-t}\bmod 3^n: t =1, \dotsc, T\} := \min A_T > R?$$
When $T$ is fixed as $n$ ...
- 459
4
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166
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Constructing motivic representations through extensions of $\mathrm{SL}(2, \mathbb{Z})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the Esquisse, Grothendieck explains how to find the representations of $G_{\mathbb{Q}}$ on Tate modules of Jacobians through its action on $\...
- 216
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205
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Grothendieck group of admissible $p$-adic representations
Let $K$ be a $p$-adic local field; $G = \mathop{\mathrm{Gal}}(\overline K | K)$; $\tau \in \{\text{HT}, \text{dR}, \text{crys}\}$, $B_\tau$ the corresponding period ring; $\mathop{\mathrm{Rep}}_{\...
- 988
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141
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Explicit toroidal compactification of Hilbert modular varieties
Hirzebruch's construction of toroidal compactification of Hilbert modular surfaces is explicit, namely one can explicitly choose rational polyhedral cone decomposition in a sort of optimal way using ...
- 1,024
4
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266
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Defining the h-topology via v-covers
I have two questions about v-covers and the h-topology (as defined by Voevodsky) which arose when reading Bhatt-Scholze's "Projectivity of the Witt vector affine Grassmannian" available here ...
- 313
4
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Clarification of argument in "Elliptic curves over $\mathbb{Q}_{\infty}$ are modular"
In https://arxiv.org/abs/1505.04769 in the proof of Theorem 5 it is asserted that since $\rho_{E, l}:G_\mathbb{Q}\to\mathrm{GL}_2(\mathbb{Z}_l)$ is surjective then $E_{\mathbb{Q}_\infty}[l^\infty]=0$. ...
- 41
4
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112
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An example of a projective surface of general type where we know all of the rational points
I am looking for an example of a surface $X \subset \mathbb{P}^3$ defined over $\mathbb{Q}$ with the following properties: 1) $X(\mathbb{Q}) \ne \emptyset$; and 2) we know all of the elements $X(\...
- 27k
4
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189
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If the Frobenius endomorphism of a characteristic $p$ ring is epimorphic, is it surjective?
MO question 19282 is about integral epimorphisms of commutative rings, and a counterexample is given to surjectivity. What about the case of the Frobenius endomorphism of a commutative, characteristic ...
- 4,492
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233
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Structure of $A(L)/A(K)$
Let $L/K$ be an extension of number fields and $A/K$ an abelian variety (or an elliptic curve, or a modular abelian variety).
Then what can we say about the structure of $A(L)/A(K)$ (or of $A(L)_{\...
- 1,364
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144
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How often is the rank of J_0(p)^- zero
As mentioned in this answer there is a conjecture by
Kimball Martin that, formulated slightly informally, has the following special case.
Conjecture:
On average $J_0(p)$ has 2 simple components when ...
- 2,224
4
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262
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de Rham Bloch-Ogus theory in positive characteristic
In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
- 2,054
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117
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Laurent polynomials of the form $p(x)\cdot p(x^{-1})$
Let $R$ be a commutative, associative ring with $1$ and let $\tau: R[x^{\pm 1}]\to R[x^{\pm 1}]$ be the $R$-algebra involution $\tau(x)=x^{-1}.$ Then it is natural to ask which elements of the ...
- 2,390
4
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277
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Explicit computations of the fundamental groups of perfectoid spaces
If $X$ is a perfectoid space then it has the same étale site as its tilt $X^\flat$. This means that the fundamental groups (suitably defined) of $X$ and $X^\flat$ are isomorphic.
Can you give ...
- 41
4
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289
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Formal integration (?) in Chabauty’s method
In Mccallum, Poonen’s paper “The method of Chabauty and Coleman”,
the authors define, for the Jacobian $J$ of a geometrically connected smooth proper curve over the rational field and for $\omega \in ...
- 1,364
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130
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Castelnuovo–Mumford regularity and wedge powers in positive characterisitc
A vector bundle on $\mathbb{P}^n$ is said to be $r$-regular if
$$H^i(\mathbb{P}^n,F(r-i))=0$$ for all $i>0$.
It is always true that if $F$ is $r$-regular and $G$ is $s$-regular (both vector bundles)...
- 2,356
4
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211
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Traces vs. Determinants in Artin's $L$-functions
Loosely put, my question is:
What happens if we swap determinant by the trace in an Artin $L$-function?
This question is not very precise and can be a little misleading, so I explain the specific ...
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4
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155
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Can we attach (formal) abelian varieties to $p$-adic modular forms?
The Jacobian of the modular curve $X_1(N)$ over $\mathbb Q$ $J_1(N)$ can be decomposed up to isogeny, as a product of abelian subvarieties $A_f$ corresponding to Galois conjugacy classes of Hecke ...
- 461
4
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181
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Sato-Tate over function fields
Suppose we have an elliptic surface $\pi: \mathscr E \to C$ over a curve over a finite field $\mathbb F_q$. We consider only the places on $C$ over which we have good reduction.
Over any point $x\in C$...
- 7,746
4
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410
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What is a Shimura variety?
It so look like that Shimura variety is a modular space that contains classes of zeros of modular functions that have certain symmetry that have structure of Hodge type.
Is this true?
4
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108
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Algorithmically recover the $l'$-adic Galois representation from the $l$-adic one (assuming the Tate conjecture)
Let $E$ be a number field. For any finite Galois extension $E\subset F$ there is a continuous homomorphism $\pi_F:\mathrm{Gal}(\overline{E}/E)\to \mathrm{Gal}(F/E)$.
Let $X$ be a smooth projective ...
user158636
4
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265
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Explicit linear object underlying $l$-adic cohomology for almost all $l$
If you are working with closed manifolds you can consider cohomology with any coefficients you like but ultimately everything is determined by the singular cohomology with $\mathbb{Z}$-coefficients.
...
user158636
4
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173
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Tannakian group of Galois representations coming from geometry
Let $K$ be a number field. Let $G_K$ be its absolute Galois group.
Let $p$ be a rational prime.
Let $\mathcal{R}_{K,p}^g$ be the category of finite-dimensional continuous $p$-adic representations of $...
- 41
4
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221
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Galois action of Weil restriction
Let $K/\mathbb{Q}$ be a quadratic field. Let $E$ be an elliptic curve defined over $K$ but not over $\mathbb{Q}$, and let $\bar{E}$ be the Galois conjugate of $E$. Then by the descent theory (for ...
- 461
4
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130
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Honda-Tate theorem and prescribing roots of $L$-functions
I'm currently working with Artin $L$-functions in function field extensions (on one variable, over finite fields). I have heard about a theorem of Honda-Tate which vaguely states that a polynomial $P \...
- 1,322
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126
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The differential topology of varieties with good reduction
Choose a prime $p$. Let $X$ be a smooth proper scheme over $\mathbb{Q}$. Does there exist a smooth proper scheme $X'$ over $\mathbb{Z}_{(p)}$ such that $X\otimes_{\mathbb{Q}} \mathbb{C}$ is ...
user145520
4
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129
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What does hyperbolicity of curves buy us in the arithmetic context?
This is going to be a fairly vague question but hopefully it will have concrete answers:
There is this recurrent phenomenon in the arithmetic of curves (possibly stacky,affine) where there is a "...
- 7,746
4
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130
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Existence of a "p-adic Mahler measure" or alternatively, the converge of a p-adic sequence
Let $f \in \mathbb Z_p[[t]]^\times$ be an invertible power series and let $\log_p$ be the p-adic logarithm with the normalization that $\log p = 0$. Consider the sequence:
$$a_n = \frac{1}{p^{n-1}}\...
- 7,746
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100
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The profinite topology on the Mordell Weil group
In this lecture of Serre on his open image theorem, around 6 minutes, Serre mentions the following theorem of Tate:
Let $A/k$ be an abelian variety over a number field and consider the Mordell-Weil ...
- 7,746
4
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339
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Beilinson regulator: a road map
I'm approaching to the Beilinson Conjecture and after studying some properties of the Deligne-Beilinson cohomology, I want to understand the regulator maps. But I don't know anything about K-theory ...
4
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179
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Is having no rational point always witnessed over a place?
Let $K$ be a finitely generated extension of $\mathbb{Q}$ of transcendence degree at least 1.
Recall that a valuation ring of $K/\mathbb{Q}$ is a sub-$\mathbb{Q}$-algebra $V\subset K$ such that for ...
4
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389
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Kottwitz global gerbes
I've been trying to understand global gerbes constructed by Kottwitz in $B(G)$ for all local and global fields (arXiv:1401.5728). Scholze explained in $p$-adic geometry (arXiv:1712.03708) why I would ...
- 4,428
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169
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Extensions of fraction field and residue field
Let $A \subset B$ be integrally closed local domains, $K(A), K(B)$ be fields of fractions, and $k(A),k(B)$ be residue fields. How to prove $[K(B):K(A)] \ge [k(B):k(A)]$?
This question should be easy ...
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119
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Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)?
I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist ...
- 16.9k
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419
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Is the Fargues–Fontaine curve proper?
It is well known that Fontaine's curve $X=\bigoplus_{k\geq0}B_{\text{cris}}^{+,\varphi=p^k}$ is a Noetherian irreducible complete scheme of dimension $1$. And completeness means that the degree ...
user141691
4
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163
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References for the computation of the Mordell-Weil group of an elliptic curve
I am reading about the Mordell-Weil group of an elliptic curve over a number field using primarily Silverman's AEC. While the book is excellent in discussing materials prior to Chapter 8, I think ...
- 401
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267
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Generalized Tate Conjecture
I have seen a statement of the generalized Tate conjecture over finite fields (see for example page 4 of Milne's "The Tate conjecture over finite field" (https://jmilne.org/math/articles/2007e.pdf)).
...
- 726
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319
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Keep blowing up all $k$-rational points
In the construction of Drinfeld space, one uses the following construction of formal schemes (see appendix B of "Period mappings and differential equations. From $\mathbb C$ to $\mathbb C_p$" by Yves ...
- 6,622
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234
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Why does $\theta: \mathbb{B}^+_{dr} \rightarrow \mathbb{C}_p$ have no continuous or equivariant section?
Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathbb{C}_K$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathbb{C}_K/p,$$ where the transition maps in ...
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Next step in studying arithmetic geometry
This relates to this post.
I want to study arithmetic, such as Fermat's last theorem, Faltings' theorem, Mazur's torsion points theorem, Weil conjecture and so on.
For understanding these theorems (...
- 1,364
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100
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Eta quotient and order of a cuspidal divisor
Let $X_0(N)$ be the modular curve associated to a congruence subgroup $\Gamma_0(N)$. If $N=p$ is a prime, then there are two cusps $0$ and $\infty$ on $X_0(N)$. Suppose that $p>7$ so that the genus ...
- 136
4
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133
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Non-cyclic Galois groups over the field of formal Laurent series in positive characteristic
This should be an easy question, but I am unfortunately not able to give an answer, so I am sorry if this is not the appropriate level for the site.
Let $C$ be an algebraically closed field of ...
- 41
4
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169
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Fibered surfaces degenerating to Frobenius
Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
- 599
4
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189
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Weil cohomology theories "genuinely" of positive characteristic
One of the reasons why Weil cohomology theories are required to have coefficients in a field of characteristic 0 is that they are supposed to be robust enough to solve Weil conjectures, i.e. to count ...
user140971
4
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0
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105
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When does a morphism of schemes induce a morphism of crystalline sites (not topoi)?
Here it is stated that the crystalline site of a scheme is not functorial in general. Is there a non-tautological characterization of morphisms of schemes which in fact do induce morphisms of ...
user140658
4
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0
answers
206
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Higher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?
In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an interesting identity (Proposition 6.4.11.):
$\sum_{k}...
- 6,622
4
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349
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Regular functions vs holomorphic functions
Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.
Is ...
- 181
4
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195
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lemma II.2.4 in Harris-Taylor (about drinfeld-katz-mazur level structure on 1-dimensional $p$-divisible groups)
Lemma II.2.4 on page 82 in Harris and Taylor's "The Geometry and Cohomology of Some Simple Shimura Varieties" (or lemma 3.2 here), says that given a Drinfeld(-Katz-Mazur) level structure $\alpha:(p^{-...
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