All Questions
975 questions with no upvoted or accepted answers
2
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221
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What is the p-adic Plancherel measure?
What I know as the Plancherel measure for a group is a measure on the spectrum of $G$, aka the set of irreducible representations - at least for finite groups, this makes perfect sense.
Now, this ...
- 7,746
2
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162
views
Faltings' height theorem for isogenies over finite fields
For an Abelian scheme over a ring of integers in a number field, Faltings has a theorem that describes how the Faltings' height changes through an isogeny. There are multiple references for this ...
- 7,746
2
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245
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What unramified Galois representations come from geometry?
I think we don't know what crystalline representations come from geometry. What about the unramified ones? Specifically let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}...
user158636
2
votes
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answers
170
views
Complex Geometric Interpretation of Mordell conjecture
The Mordell conjecture/Falting's Theorem says that any smooth projective curve $X$ of genus $g\geq 2$ over $\mathbb{Q}$ has finitely many integer points (using the valuatlive criterion).
We can of ...
- 4,428
2
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109
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arithmetic del Pezzo surfaces in comparison with del Pezzo surfaces over a field
A del Pezzo surface is a smooth, 2-dimensional projective variety $X$ with ample anticanonical divisor, i.e. a 2-dimensional Fano variety.
I am interested in the arithmetic analogue, a 2-dimensional ...
- 1,338
2
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175
views
A normal proper model of an abelian variety with geometrically integral special fiber smooth at the reduction of the origin
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 ...
user158636
2
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answers
131
views
versal deformation ring of a p-divisible group with some tensors
I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with ...
- 1,093
2
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100
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Composing equal characteristic and mixed characteristic deformations
Suppose $k$ is an algebraically closed field of characteristic $p>0$ and $A$ is a $k$-point on a moduli space of certain objects over $k$. Suppose, moreover, that $A$ deforms to a $k[[t]]$-point $B$...
- 245
2
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145
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Hasse invariant of abelian varieties with complex multiplication
Is there a good way to compute Hasse invariants of elliptic curves or higher dimensional Abelian varieties with complex multiplication?
For example, if $E$ is an elliptic curve with CM by an ...
- 949
2
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122
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A Lefschetz style formula for the $\ell^\infty$ torsion of an Abelian variety over a finite field
Let $A/\mathbb F_q$ be an abelian variety over a finite field. Define $A_\ell = A[\ell^\infty](\mathbb F_q)$, the $\ell^n$ ($n\geq 0)$ torsion points defined over the base field. I can assume $\ell \...
- 7,746
2
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111
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Inseparable field extensions of degree p and linear independence
Let $F$ be a field of characteristic $p$; let $\alpha \in F$ such that $\alpha \neq \beta^p$ for any $\beta \in F$, and let $K := F(x)$ where $x=\sqrt[p]{\alpha}$.
Is it true that the elements $1,(x-...
- 21
2
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answers
243
views
Some questions regarding computation of the Mordell-Weil group
I was reading the theory relevant to Selmer and Shafarevich-Tate groups from Silverman's AEC. And I have a lot of doubts related to these topics:
First, I don't understand the reasoning behind the ...
- 401
2
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58
views
Number of representations by the norm in a division algebra corresponding to endomorphism rings of elliptic curves
Let $E$ be a supersingular curve over a field of characteristic $p$ with endomorphism ring $\mathcal O_D$ which is a maximal order in a division ring $D$ over $Q$ ramified at $p$ and $\infty$.
The ...
- 7,746
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189
views
References for heights of algebraic or projective variety
In my research I am using the notion of a $\textbf{rational function f on a domain}\: U\subset \mathbb C^{n}$. By this I mean that the graph of $f$ is an analytic component of an affine variety $X$ ...
- 478
2
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257
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Rigid analytic geometry and Tate curve
I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-...
user141691
2
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295
views
Fiber of normalisation morphism
Let $X$ be an integral, excellent scheme and $\eta: \widetilde{X} \to X$ be its normalization. If $x \in X$ is a closed point there is the following powerful tool (from EGA, Ch IV, 7.8.3, vii) to find ...
- 727
2
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answers
252
views
Which fields and schemes "have enough finite residue fields"?
I am looking for assumptions on the spectrum $S$ of a field $K$ that ensure the following: there exists an excellent noetherian finite dimensional (integral) scheme $S'$ such that $S$ is its generic ...
- 16.9k
2
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answers
275
views
Étale group scheme exact sequence
Consider the exact sequence of finite flat group schemes over the $2$-adic integers ring $\mathbb{Z}_2$:
$$0\longrightarrow\mathbb{Z}/2\mathbb{Z}\longrightarrow A\longrightarrow\mathbb{Z}/2\mathbb{Z}\...
user141691
2
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answers
244
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Standard application of Oort-Tate classification theorem
$\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...
user141691
2
votes
0
answers
226
views
Lift the relative Frobenius automorphism to zero characteristic
Let $X$ be a algebraic variety of finite type over $\mathbb{Z}$. Let $\mathcal{F}$ be a foliation in codimension one over $X$. Let $X_p$ and $\mathcal{F}_p$ be the reductions modulo $p$ of $X$ and $\...
- 527
2
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550
views
Fontaine - Wintenberger field of norms and imperfect case
Let $K$ be a complete discrete valued field whose residue field $k_K$ has characteristic $p$ and has the property that $[k_K:k_K^p]=p^d$ for some $d$. Let $t_{\alpha}, 1 \leq \alpha \leq d$ be a set ...
- 313
2
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177
views
vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves
I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:
$H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
- 1,142
2
votes
0
answers
167
views
Weighted projective lines and elliptic curves
The modular curves of low level can sometimes be describes as weighted projective lines. For example, over $\mathbb{Z}[1/2]$ the compactified stack of elliptic curves with full level 2 structure is ...
user144105
2
votes
0
answers
277
views
Which endomorphisms of the Tate module of an abelian variety are "algebraic"?
For an abelian variety $A$ over a field $k$ with characteristic different from $\ell$ and Galois group $G = Gal(\overline k/k)$, there is always an injective map of the form:
$$\mathbb Q_\ell\otimes ...
- 7,746
2
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answers
162
views
Can Berthelot-Ogus be used to prove that varieties do not lift?
Given a "nice" scheme $X$ over $F_p$, Berthelot-Ogus comparison relates its crystalline cohomology to algebraic de Rham cohomology of its lift to $\mathbb{Z}_p$. The nice thing about crystalline ...
user140765
2
votes
0
answers
79
views
Integral lifts of families of varieties over a finite field
Let $X\rightarrow \mathrm{Spec}\:F_q[[t]]$ be a flat morphism with smooth proper geometrically connected fibers. Suppose the central fiber lifts to a scheme $X'_0$ smooth proper over $W(F_q)$. Is our ...
user140761
2
votes
0
answers
182
views
Local-global compatibility and modular curves
I have been told by some people that local-global Langlands compatibility for $GL_2$ (the vanilla version, not the one being developed in this decade by Emerton and others) implies Shimura conjecture ...
user138661
2
votes
0
answers
209
views
Is there literature on a de Rham analogue of the Mumford-Tate group or ell-adic monodromy group?
Let $X$ be a smooth projective variety over $\mathbb{Q}$. The theory of motives predicts that for each cohomology theory, there should be a distinguished Zariski closed subgroup of $GL(H^k_{\bullet}(X)...
- 9,061
2
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219
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Igusa curve at infinite level
In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent....
- 149
2
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232
views
Berthelot’s comparison theorem and functoriality
Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$.
Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...
- 181
2
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answers
80
views
Points of low height and low degree
Let $K$ be a number field, and let $C$ be an algebraic curve of genus $g \geq 2$ defined over $K$. We define $d_C$ to be the positive integer with the following property: there exists at most finitely ...
- 27k
2
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198
views
Point of smallest height on an algebraic curve
Let $C$ be an algebraic curve of genus $g \geq 2$ defined over a number field $K$, and we suppose that $C(K) \ne \emptyset$ ($C$ may very well be defined over a proper subfield of $K$, but perhaps ...
- 27k
2
votes
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answers
152
views
Moduli interpretation of the integral anticanonical tower
This question is related to my reading of On torsion in the cohomology of locally symmetric varieties by Scholze.
In chapter $3$, using the theory of canonical subgroup, he produces Frobenius maps ...
- 149
2
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0
answers
114
views
Algebraic theta-functions of level $2$ on an elliptic curve
Consider an elliptic curve $E\!: y^2 = x(x-1)(x-\lambda)$, where $\lambda \in k \setminus \{0, 1\}$ for some field $k$ of $\mathrm{char}(k) \neq 2$. Also consider the ample divisor $D = 2P_0$, where $...
- 1,833
2
votes
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answers
60
views
A conjectural formula for the "minimal degree function", $k\rightarrow d\min(k)$, attached to recursion, $f\rightarrow A(f)$, in char $3$
THE RECURSION: $f\rightarrow A(f)$
$A: t(\mathbb{Z}/3)[t^3]\rightarrow t(\mathbb{Z}/3)[t^3]$ is the $\mathbb{Z}/3$-linear map with $A(t)=0, A(t^4)=t, A(t^7)=t^4$, and $A((t^9)f)=(2t^9)A(f)+(t^3)A((t^...
- 5,422
2
votes
0
answers
183
views
Solving solutions to systems of polynomial equations over $\mathbb Z$
Macaulay Resultants help identify common root in $\mathbb P^{n-1}(\mathbb K)$ of $n$ homogeneous polynomials in $n$ variables when $\mathbb K$ is algebraically closed. Is it possible that some type of ...
- 13.9k
2
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138
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The growth of class number in $\mathbb{Z}_p$-extensions of function fields
Let $X$ be a curve (proper, smooth, ...) over a finite field $\mathbb F_q$ where $q$. Suppose also that $\mathbb F_q$ contains the $p$-th roots of unity, in this case we have the following (unique) ...
- 7,746
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answers
154
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Categorical representations of absolute Galois groups
I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.
user127738
2
votes
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answers
294
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Specialization map on geometric points
Let $\mathcal{X}$ be a proper and smooth scheme over $\text{Spec}(\mathbf{Z}_p)$, and let’s call $X$ the geometric generic fiber of $\mathcal{X}$, and $X_0$ the geometric special fiber of $\mathcal{X}$...
user125985
2
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110
views
What is the canonical divisor on an arithemetic curve?
Let an arithemetic curve be an integral scheme $X$ whose structure morphism $\pi:X\rightarrow B=Spec(O_{K})$ is projective, flat and of pure dimension $0$, and whose generic fiber is regular. I am ...
- 91
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answers
63
views
Product formula over arithmetic surface
Let $X$ be an arithmetic surface and $f\in K(X)$ be a function in the function field. Is there any analagous "product formula" showing $\deg(f)=0$? This is motivated by the number field case, where $X=...
- 91
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answers
46
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Counting intersections of rectilinear lattices
The following proposition
Let $g>0$ be an integer and let $\Lambda \subset \mathbb{R}^g$ be a rectilinear lattice (possibly shifted) with mesh $d$ at most $D$. Then we have
$$
\left| \#(\Lambda \...
- 389
2
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answers
290
views
Elliptic fibrations on the Fermat quartic surface
Consider the Fermat quartic surface
$$
x^4 + y^4 + z^4 + t^4 = 0
$$
over an algebraically closed field $k$ of characteristics $p$, where $p \equiv 3$ ($\mathrm{mod}$ $4$).
Is there the full ...
- 1,833
2
votes
0
answers
208
views
Is the Fermat quartic surface a generalized Zariski surface?
Consider the Fermat quartic surface $$F\!: x^4 + y^4 + z^4 + t^4 = 0$$ over an algebraically closed field $k$ of odd characterstics $p$. Shioda proved that for $p=3$ this surface is a generalized ...
- 1,833
2
votes
0
answers
257
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Absolute approximation of formal schemes
Let $\mathfrak{X}_j$ be an inverse system of qcqs $p$-adic formal scheme, flat over $\mathbf{Z}_p$, with affine transition maps, and assume $\mathcal{O}_{\mathfrak{X}_j}$ is a coherent sheaf of ...
user95222
2
votes
0
answers
219
views
Liftability of varieties, after fpqc base change
Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable.
Suppose there exists an fpqc cover $S'\to S$, such ...
user124171
2
votes
0
answers
112
views
affine Lefschetz and Poincaré duality for syntomic cohomology
Let $X$ be a smooth variety over an algebraically closed field $k$ of characteristic $p > 0$. Is there an affine Lefschetz theorem and Poincaré duality for sheaves represented by finite flat ...
user19475
2
votes
0
answers
242
views
Is there any generalization of Weil conjecture for non-smooth variety?
Is there any generalization of Weil conjecture for any non-smooth geometric-connected variety? For example, for more general curve, or at least some numerical evindence (example)?
- 806
2
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answers
325
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A question on direct limits of rings, and descent of ideals
Motivated by an étale cohomology calculation I am going to do, here is a question that should have positive and not too hard answer.
Let $A$ be a ring, $A_0$ a ring such that $A_0$ is equipped with ...
user92332
2
votes
0
answers
165
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Galois action on posets of number fields and $p$-adic fields
In the theory of group actions on posets one studies the action of a group $G$ on a poset $\mathcal{P}$ (via order-preserving permutations) partly through its derived action on the order complex of ...
user122285