All Questions
974 questions with no upvoted or accepted answers
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etale cohomology and algebric K theory for algebraic stack
Let $X$ be a smooth variety over a perfect field $k$. Fix a prime $p$ which is invertible in $k$.
Thomason proved that there is Atiyah-Hirzebruch type spectral sequence that computes $K(1)$-local $K$ ...
- 349
2
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76
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Galois weight of a Serre twisted pure Galois representation
Let $V$ be a pure, 2-dimensional weight $k$ $p$-adic global Galois representation for $G$, the absolute Galois group of the rationals.
Let $c\in Z^1(G,\text{SL}_2)$ be a cocycle and consider the Serre ...
- 2,907
2
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153
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Order $4$ element of Tate-Shafarevich group
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Tate-Shafarevich group $\mathit{Sha}(E/\Bbb{Q})$ is defined as follows.
$$\mathit{Sha}(E/\Bbb{Q})\stackrel{\text{def}}{=} \operatorname{Ker}...
- 1,541
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142
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$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ and Selmer group
This question is essentially related to the theory of elliptic curves (Selmer group), but this question itself is just a field theoretic one.
To calculate the Selmer group of given elliptic curve, we ...
- 1,541
2
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119
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Resolution of singularities of the resultant locus
We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
- 7,746
2
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241
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References to let me know about current directions of research in arithmetic geometry
I have knowledge of basic algebraic geometry and good deal of number theory. I have studied roth theorem and I am currently studying proof of Mordell-Weil theorem. These two topics come under ...
- 793
2
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197
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Mumford's computation of the determinant of cohomology of a relative curve
In Integral Grothendieck-Riemann-Roch theorem, Pappas mentions that Mumford computed the determinant of cohomology of $f:X\to S$ a relative curve integrally, and thus proved an integral version of GRR ...
- 2,054
2
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220
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Is the ring of power series with $p$-adic coefficients Huber?
I have been reading the Berkeley lectures and got stuck with this question. Let $\mathbb{Q}_p [[t]]$ denote the ring of power series with $p$-adic coefficients. Is there a natural topology (e.g. the ...
- 103
2
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90
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Reconciling two notions of finite descent obstructions
Let $k$ be a number field and $X$ a smooth geometrically connected variety over $k$. We denote by $H(k,X)$ the set of sections $G_k \rightarrow \pi_1(X)$, where $G_k$ is the absolute Galois group of $...
- 895
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146
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How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?
It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
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47
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Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
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147
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Can we say anything about the zeros and Galois group of the polynomial $(x^p-a)^{p^2}-p^{p^2+1}x+p^{p^2} a=0$?
Let $p$ be an odd prime number and $\mathbb Q_p$ be the $p$-adic number field. Let $K=\mathbb Q_p(a)$ be the extension by $a=p^{\frac{p^2+1}{p^3-1}}$.
Consider the polynomial $f(x)=(x^p-a)^{p^2}-p^{p^...
- 930
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177
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How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 13k
2
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103
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Selmer ranks unbounded?
Is it known if the Selmer ranks of quadratic twist families are unbounded?
Suppose that $E/K$ is an elliptic curve defined over a number field. For each quadratic extension $F/K$ I can form the twist $...
- 3,730
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136
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Is there the specialisation map of etale K theory?
Take a smooth proper morphism of schemes $X\to S$. Fix a point $t\in S$ and a point $s\in \overline\{s\}$. For a prime $l$ which is invertible in $S$, is there the natural specialization map of etale ...
- 519
2
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101
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Number of points of parabolic Springer fibres for general reductive groups
My question is the same as this post but for an arbitrary reductive $G$ instead of just $\mathrm{GL}_n$.
Let $G$ be a connected split reductive group over a finite field $k$.
Let $P$ be a parabolic ...
- 2,751
2
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253
views
Künneth formula for algebraic de Rham cohomology
Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
- 333
2
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212
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Intermediate extensions of pure perverse sheaves (BBD 5.4.3)
I am working my way through "Faisceaux pervers" by Beilinson, Bernstein and Deligne and can't wrap my head around Corollary 5.4.3. To me it seems that one of the hypotheses is extraneous, ...
- 1,071
2
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190
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Relation between division point of elliptic curve and formal group of elliptic curve, $\Bbb{Q}_p(E[p])＝\Bbb{Q}_p(\hat{E}[p])$
Let $E/ \Bbb{Q}_p$ be an elliptic curve over $ \Bbb{Q}_p$. $\hat{E}$ denote the corresponding formal group of $E$.
I want to prove
$\Bbb{Q}_p(E[p])＝\Bbb{Q}_p(\hat{E}[p])$.
$ \hat{E}[p]$ denotes $p$ ...
- 1,541
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98
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Control on the locus of bad reduction for divisors
Let $X$ be a smooth variety over a number field $K$ and let $\mathcal X$ be a normal, projective model of $X$ over the ring of integers $O_K$.
Now assume that $D\subset X$ is an irreducible divisor ...
- 321
2
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218
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Borel-Weil-Bott theorem for wonderful compactification in characteristic p
Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
- 41
2
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92
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Linear system of a relative effective divisor on an arithmetic surface contains vertical divisors
I am puzzled by the behavior of some divisors in my attempt to understand the relative Picard functor $\mathrm{Pic}_{X/S}$ of an arithmetic surface $\pi:X\to S$. This is defined by relative divisors $...
- 1,142
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147
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Automorphism groups of "reductive" Lie algebras in positive characteristic
I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras.
Let $G$ be a reductive group ...
- 13k
2
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458
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Connected-étale sequence $ 0 \to G^0 \to G \to G^{\text{ét}} \to 0$ for affine finite group scheme $G$
Let $G$ be an affine finite commutative group scheme over a complete (or at least Henselian; thanks to Jason Starr for calling attention to it) local ring $R$, and assume the residue field $\kappa=R/m$...
- 6,016
2
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166
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Theorem on formal functions when the initial data is a proper map of formal schemes
Let $\pi: X \to S:=\mathrm{Spf}\text{ } A$ be a proper morphism of $\mathbb{Z}_p$-admissible formal schemes and $\mathcal{F}$ be a coherent sheaf on $X$.
Set $S_0=\{x\}$ be a closed point of $S$ and $...
- 1,066
2
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0
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97
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Non-noetherian Cartier Isomorphism
A result in positive characteristic is that if $R/\mathbb{F}_p$ is a smooth ring, then we have a Cartier isomorphism
$$\Omega_{R}^\bullet\cong H^\bullet(\Omega_R^\bullet)$$
which is essentially ...
- 4,428
2
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182
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Image of the Kummer map for abelian varieties over $p$-adic local fields
The following statement might be well-known to the community: let $K$ be a finite extension of $\mathbb{Q}_p$ for some prime $p$. Let $A$ be an abelian variety over $K$. Then the image of the Kummer ...
- 1,428
2
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176
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Path spaces vs arc spaces
Let $X$ be a smooth projective variety over $\mathbf{C}$ and denote by $\mathcal{L}_m(X)$ the $m$-th jet space, a smooth $\mathbf{C}$-scheme representing the functor on $\mathbf{C}$-algebras
$$A\...
user178246
2
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171
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Monogenic function fields
Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can ...
- 103
2
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467
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Confusion regarding Proposition 1.1 in Wiles's Fermat paper
This is from p. 459 of Wiles's Fermat paper.
Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of ...
- 1,805
2
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99
views
Are tamely ramified representations $\widehat{\mathbb{Q}_p^\text{tr}}$-admissible?
Let $K$ be a finite field extension of $\mathbb{Q}_p$. Let $G_K$ denote the absolute Galois group of $K$, $I_K$ the inertia subgroup and $I_K^{(p)}$ the $p$-Sylow subgroup of $I_K$, i.e. the wild ...
- 455
2
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answers
287
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Frobenius endomorphism is not flat
I am actually going through "Twenty four hours of local cohomology". They discuss the Frobenius endomorphism in Chapter 21. Here is the Exercise 21.6 which I am finding hard to solve:
Find a ...
- 183
2
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answers
505
views
Relative homology in Fargues-Scholze paper
if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they ...
- 1,093
2
votes
0
answers
131
views
Base change of Hodge-Witt cohomology
Let $k$ be a perfect field of characteristic $p$, and $L$ be a finite extension of $k$.
For a smooth projective variety $X$ defined over $k$, we denote the base change $X \times_k L$ by $X_L$. In this ...
- 349
2
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answers
184
views
Points on Galois conjugate curves
Let $C$ be a curve defined over a number field $F$ and let $C^{\sigma}$ be its Galois conjugate obtained by applying $\sigma \in Gal(F/\mathbb{Q})$ to the coefficients defining $C$. What can we say ...
- 81
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180
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Reference request for an English translation of a book of Tate
In this ongoing program, Professor Mahesh Kakde said that the best reference for learning about Stark and Gross-Stark conjecture is this book of John Tate. But this book is in French. Is there any ...
- 333
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187
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Does the map $\theta[1/p]: A_{\mathrm{inf}} \otimes \mathbb Q_p \to \mathbb C_p$ split?
This question might be very elementary to someone who knows p-adic hodge theory/perfectoid stuff etc.
Recall that $\mathbb C_p = \hat{\overline{\mathbb Q_p}}$ and $\mathbb C_p^\flat$ is it's tilt. We ...
- 7,746
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85
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Are there non-trivial $\mathbb{F}_q$-covers of the j-invariant 0 elliptic curve by a hyperelliptic or cyclic trigonal curve?
Consider the ordinary elliptic curve $E\!: y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_q$ such that $\sqrt{b}, \sqrt[3]{b} \not\in \mathbb{F}_q$. Also, for any $n \in \mathbb{...
- 1,833
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209
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Is there a smooth proper family whose fibers are not Mazur-Ogus?
Set $K$ to be a number field, denote by $\mathcal{O}_K$ the integer ring of $K$. My question is the following:
Is there a smooth proper family $X \to \mathcal{O}_K$ whose fibers are not Mazur-Ogus?
- 519
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243
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Cartier operator and logarithmic differentials
Let $k$ be an algebraically closed field of characteristic $p$, let $C$ be a curve over $k$ and let $\omega$ be a meromorphic differential form on $C$. If $\omega$ gets mapped to itself by the Cartier ...
- 75
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381
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A computation of the rank of the Jacobian of a hyperelliptic curve over a number field using MAGMA
In this paper,
the authors says that, in order to show the rank of a Jacobian over $\mathbb{Q}$ is 0, they use the L function.
In the section 3.3, the authors compute the rank of the Jacobian of $X_1(...
- 1,364
2
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0
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411
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Equidimensional Morphism
I am reading the paper "Relative Cycles and Chow Sheaves" due to Suslin and Voevodsky. Here we have the following definition:
Definition 2.1.2.
A morphism of schemes $p:X\rightarrow S$ is ...
- 519
2
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answers
286
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Moduli interpretation and Ogg's notation for the cusps on modular curves
In Ogg's paper "rational points on certain elliptic modular curves", the author says, using Ogg's notation for cusps,
that for fiexed $d$, if $(y, N) = d$, then for any $x$ satisfying $(x, y,...
- 1,364
2
votes
0
answers
198
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Finding rational points via birational map
Let $C$ be an affine curve given by $p_C(x,y)=0$ where $$ p_C=2x^3y + 2xy^3 +x^3 + y^3 + 5x^2y + 5xy^2 + 2x^2 + 2y^2 + 2x^2y^2 + 2xy $$
and let $\overline{C}$ denote the projective closure of $C$. For ...
- 21
2
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answers
161
views
Abelian variety corresponding to a vector space
I would like to know what the following statement means:
"Let $B_t$ be the Abelian subvariety in $J_t$ corresponding to the $\mathbb{Q}$-vector subspace $H^1(C_t,\mathbb{Q})_{van}$ in the space $...
- 519
2
votes
0
answers
130
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Iwasawa theory over function fields - How do eigenvalues vary in $\mathbb Z_\ell$ towers?
Consider a tower of curves $\dots \to C_n \to C_{n-1} \dots \to C_1$ over $\mathbb F_q$ where $C_n: f(x^{\ell^n},y) = 0$.We can look at the multiset $S_n$ of eigenvalues of the Frobenius $\sigma_q$ on ...
- 7,746
2
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128
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How do elliptic units generate the module of Euler systems over abelian extensions of imaginary quadratic fields?
I am trying to undesrtand the analogy between the Euler systems over abelian extensions of the rationals and the Euler systems over abelian extensions of imaginary quadratic fields.
As Soogil Seo ...
- 99
2
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answers
190
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Computing monodromy groups of curves over function fields
Suppose I consider a hyperelliptic curve given by an equation such as $y^2 = x^{n} + tx + 1$ or some variation on this (where $t$ is a parameter on $\mathbb P^1$ and this curve is really a surface ...
- 7,746
2
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216
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Have the following summations been studied before?
Suppose $q^2-4pr<0$, and consider the set of integral points
$$\mathcal Z=\{(X,Y)\in\mathbb Z^2:$$
$$px^2+qxy+ry^2+sx+ty+u=0\}$$
which lie on an ellipse. Then define $$M=\sum_{(X,Y)\in\mathcal Z}\...
- 13.9k
2
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0
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165
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Is the cohomology of rigid varieties semisimple?
Let $X$ be a smooth projective geometrically connected scheme over $\mathbb{Q}_p$. Assume that $H^1(X, T_{X/\mathbb{Q}_p})=0$.
Is the Galois representation $H^*(X_{\overline{\mathbb{Q}_p}}, \mathbb{Q}...
- 21