All Questions
2,494 questions
5
votes
2
answers
572
views
Birational geometry over finite fields
I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ...
- 3,730
5
votes
0
answers
278
views
Torsion points of an elliptic curve over number fields. Another proof of Silverman AEC theorem
I am studying the following theorem from Silverman's AEC:
I am wondering whether there exists another proof that doesn't make use of formal groups and is still valid for a number field $K$. Could you ...
- 233
7
votes
0
answers
262
views
Are unramified simple Rapoport-Zink spaces smooth?
I have read on different occasions that unramified simple Rapoport-Zink spaces are formally smooth, eg. stated in Remark 4.13 of Rapoport and Viehman's survey article. These spaces are formal schemes $...
- 769
2
votes
0
answers
287
views
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 ...
- 63.9k
3
votes
0
answers
173
views
Smooth proper varieties over the integers that are not toric
Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric?
By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by ...
- 7,746
5
votes
2
answers
725
views
Embedding torsors of elliptic curves into projective space
Suppose I have a genus 1 curve $C$ over a field $k$. If $C$ has a point, then we can embed it into the projective plane by a Weierstrass equation. Now let us suppose that $C$ does not have a point (so ...
- 21.4k
16
votes
2
answers
1k
views
Is the tangent space functor from PD formal groups to Lie algebras an equivalence?
The previous version of this question was rather badly broken, and I hope this version makes some sense.
There have been a few questions on MathOverflow about how much representation-theoretic ...
- 35.5k
7
votes
2
answers
788
views
Reference request: the geometry of vanishing cycle
I’m currently studying basics on étale cohomology, by Fu’s and Milne’s book. The formalization of vanishing cycle and nearby cycle particularly interests me. I realized it may relate with reduction ...
- 1,606
4
votes
0
answers
231
views
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
5
votes
0
answers
139
views
Liouville property in the Bost theorem on foliations
Let $X$ be a smooth algebraic variety over a number field $K$, and let $\mathcal{F}$ be an involutive coherent subsheaf of the tangent bundle. After a pullback to $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
- 485
7
votes
1
answer
1k
views
L-functions and Galois representations: What’s the explicit relation?
It was mentioned in a lecture on Faltings’s proof of Mordell Conjecture that there’s some kind of correspondence between Galois representation (of cohomology, or some complex of constructible sheaves?)...
- 149k
3
votes
1
answer
769
views
Explicit defining equations for del Pezzo surfaces
Are there known explicit parametrizations of smooth del Pezzo surfaces, say of degree $5$ or higher, as surfaces cut out by equations in some projective space?
The closest I've been able to find is on ...
- 1,306
15
votes
2
answers
1k
views
Lang's conjecture beyond the curve case
An algebraic variety $V$ is said to be of general type if it is of maximal Kodaira dimension. If $V$ is defined over a number field $K$, then one has the following conjecture due to Lang (Bombieri had ...
- 9,497
2
votes
0
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
6
votes
0
answers
173
views
Orlik-Solomon algebra and hyperplane complements in positive characteristic
Let $k$ be an algebraically closed field of characteristic $p\geq 0$, $\underline H:=\{H_1,\dots, H_m\}$ a set of hyperplanes in $\mathbb A_k^n$ and $X:=\mathbb A^n-(\bigcup H_i)$.
Given a ring $R$ ...
- 728
3
votes
0
answers
188
views
Does the construction of arithmetic toroidal compactification of $A_{g}$ depend on semistable reduction theorem?
If there is a good theory of arithmetic toroidal compactification over $\mathbb{Z}_{p}$ of the Siegel modular variety with deep enough level structure, then it seems like semistable reduction theorem ...
- 1,024
11
votes
4
answers
3k
views
What does ramification have to do with separability?
Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely ...
- 411
4
votes
0
answers
166
views
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 $\...
- 63.9k
6
votes
0
answers
113
views
$S_n$-invariant polynomials on the dual of reflection representation
Let $W=S_n$ (the symmetric group) acting on $V=k^n$ via permutation of the indices, where $k$ is an algebraically closed field. Closely related to this are the reflection (sub-)representation $V_0=\{ (...
- 1,336
21
votes
5
answers
5k
views
Mirror symmetry mod p?! ... Physics mod p?!
In his answer to this question, Scott Carnahan mentions "mirror symmetry mod p". What is that?
(Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than $\mathbb{C}$...
- 35.5k
14
votes
2
answers
568
views
Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?
Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
- 149k
4
votes
0
answers
205
views
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
7
votes
0
answers
308
views
Number of rational points over finite fields mod $q$ is birational invariant
I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) \equiv \#(Y(\mathbf F_q)) \pmod q$, and I heard that the proof ...
- 988
2
votes
1
answer
433
views
Why geometric generic point (in abstract algebraic geometry) replace general points in the unit disk?
In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
On $\mathbb{C}$ Lefshietz local results are as follows. Let $X$ be a ...
- 16.5k
6
votes
2
answers
664
views
Could the Weil zeroes of curves be evenly distributed?
If $X$ is a smooth, geometrically connected, projective curve of genus $g$
over $\mathbb{F}_q$, then the zeta function of $X$ is of the form $P(s)/(1 - s)(1 - qs)$, where $P(s)$ is a polynomial of ...
- 149k
3
votes
0
answers
377
views
Meaning of "the" general fiber in the paper "La conjecture de Weil : I"
In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
Let $X$ be a non singular analytic space and purely of dimension $n+1$....
- 35.5k
4
votes
0
answers
141
views
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
1
vote
0
answers
246
views
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
6
votes
1
answer
418
views
Number of points of parabolic Springer fibres
Let $P$ be a parabolic subgroup of $\mathrm{GL}_n$ and $u\in P$ a unipotent element. The parabolic Springer fibre associated to $(P,u)$ can be defined by
$$
\mathcal{P}_u:=\{gP\in G/P \mathrel\vert g^{...
- 2,751
6
votes
0
answers
449
views
Proof of Lemma 6.5 in Scholze's Perfectoid Spaces
In the proof of Lemma 6.5(approximation lemma) in Scholze's Perfectoid Spaces,
I have the following three questions about $h = f - g^\sharp_c$ and $g^\sharp_{c'}$.
(Maybe it's something you'll find ...
- 61
5
votes
1
answer
361
views
diagonal cubic hypersurfaces
At the end of
https://encyclopediaofmath.org/index.php?title=Cubic_hypersurface#References
it is stated that the diagonal cubic hypersurface
$$
\sum_{i=0}^{2m+1} a_i x_i^3 = 0, m\ge 2
$$
(and ...
- 38k
56
votes
2
answers
10k
views
What is prismatic cohomology?
Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...
- 63.9k
6
votes
0
answers
439
views
Cohomology theories for algebraic varieties over number fields
There is a standard line which is repeated by anyone writing/talking about motives and cohomology of algebraic varieties over number fields: namely, there are many such cohomologies and then the ...
- 2,751
1
vote
0
answers
172
views
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
4
votes
0
answers
266
views
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
7
votes
0
answers
550
views
Mistake in Silverman's book in proof of Neron-Ogg-Shafarevich criterion?
In Silverman's The arithmetic of elliptic curves, p. 201, theorem $7.1$ (Criterion of Neron-Ogg-Shafarevich),
he applies the theorem "When $K$ is complete with respect to it's discrete value, ...
- 37k
1
vote
0
answers
208
views
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
4
votes
1
answer
309
views
Does the modular form associated to cubic twist of a elliptic curve $E$ corresponds to some twist of $f_E$?
Let $E$ be an elliptic curve defined over $\Bbb Q$ and $f_E$ be the modular form associated with the elliptic curve $E$.
Suppose the elliptic curve $E^D$ is a quadratic twist of $E$.
I understand that ...
- 8,945
6
votes
1
answer
242
views
Contraction of some surfaces over a ring of algebraic integers
The situation:
Let $X$ be a 2 dimensional normal quasi-projective $\mathcal{O}_K$-scheme, where $K$ is an algebraic number field. Assume the following conditions on $X$:
$X$ is integral.
$X_K$ is ...
- 19.6k
19
votes
1
answer
419
views
Varieties with the same number of $\mathbb{F}_p$-points for all but finitely many primes
If two varieties over $\mathbb{Q}$ have the same number of $\mathbb{F}_p$-points for all but finitely many primes do they have the same number of $\mathbb{F}_{p^n}$-points for all $n>1$ and for all ...
- 325
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
15
votes
0
answers
2k
views
A question on Fargues-Scholze
As far as I understand it, the main goal of the recent work of Fargues and Scholze on the geometrization conjecture is to show that the local Langlands conjecture of a local field is equivalent to the ...
- 4,428
4
votes
0
answers
134
views
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 $E_{\mathbb{Q}_\infty}[l^\infty]=0$. ...
- 9,501
5
votes
1
answer
172
views
Reductive groups over number rings
Let $F$ be a number field.
If $G$ is a reductive group over $\mathcal{O}_F$ then we can look where $G\otimes \mathbb{C}$ fits in the classification of complex reductive groups and get a "standard ...
- 13k
1
vote
0
answers
167
views
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
19
votes
1
answer
2k
views
Examples of solid abelian groups
I am reading through Clausen's and Scholze's Lectures on condensed mathematics. I am struggling to understand the concept of solid abelian groups so I am looking for some examples.
Is the underlying ...
- 21.3k
19
votes
1
answer
2k
views
The Infinitesimal topos in positive characteristic
This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie ...
- 19.4k
12
votes
3
answers
2k
views
Order of the Tate-Shafarevich group
I thought that the order of the Tate-Shafarevich group should always be a square (it's also supposed to be finite, but for the purposes of this question let's assume we know this) but I don't seem to ...
- 19.6k
2
votes
2
answers
270
views
Why is $P^n(K)$ compact, when $K$ is a local field?
In Silverman's book AEC, question 7.6 asks to prove $E_0(K)$ has finite index in $E(K)$ for $K$ a local field. For part (a), I know the topology on $P^{n}(K)$ is the quotient topology on $K^{n+1}$, ...
- 63.9k
14
votes
0
answers
933
views
Relation between Igusa tower and $p$-adic modular forms
As the title suggests, my question is devoted to understand (and maybe get some good references) the relation between the Igusa tower for a modular curves and $p$, or maybe $T$-adic modular forms. I ...
- 28.2k