All Questions
2,494 questions
14
votes
3
answers
1k
views
$\ell$- vs. $p$-adic and de Rham vs. Betti in the geometric Langlands correspondence
In Emerton–Gee–Hellmann’s IHÉS notes on the categorical $p$-adic local Langlands programme, one finds the following remark:
The differences between the $\ell$-adic and $p$-adic settings are ...
- 617
21
votes
3
answers
808
views
Examples when quantum $q$ equals to arithmetic $q$
First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor.
In the world of quantum mathematics, the letter $q$ is a standard ...
- 82.7k
2
votes
1
answer
154
views
Extending line bundle to regular model
Let $S$ be the spectrum of an excellent discrete valuation ring with field of fractions $K$ and $C$ be a proper integral regular curve over $K$. Assume, $C$ admits a proper regular flat model $\...
- 39.3k
4
votes
0
answers
178
views
Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
- 5,986
4
votes
0
answers
100
views
Structure of points of elliptic curves in field with restricted ramification
Let $k$ be a finite field of characteristic $p$ and let $C$ be a curve over $k$. Let $E$ be a non-constant elliptic curve over $k(C)$. Taking the Néron model of $E$ and removing the singular fibers ...
- 177
2
votes
0
answers
280
views
Why is the weight monodromy hard in mixed characteristics?
I know very little about the conjecture, beyond Grothendieck's monodromy theorem perhaps (a dense open subgroup of inertia acting unipotently on pure motives). But I heard that it was completely ...
- 63.9k
3
votes
1
answer
187
views
Reference Request: Preservation of étale maps under rigid analytic GAGA
Let $K$ be a finite extension of $\mathbb{Q}_p$. As the title says, I am looking for a reference in which it is shown that given an étale map $f:X\rightarrow Y$ between smooth algebraic $K$-varieties, ...
- 998
3
votes
1
answer
241
views
Could I get an interpretation for application of Euler characteristics in number theory？
As a beginner who just get in touch with Euler characteristics in this field, could I get some intuition for the arithmetic meaning of Euler characteristics of bounded complexes, for example Selmer ...
- 10.9k
7
votes
1
answer
233
views
Proven results for the refined Birch Swinnerton-Dyer conjecture over rationals when rank at most $1$
For an elliptic curve, $E$, defined over $\mathbb{Q}$ and with the Mordell-Weil group, $E(\mathbb{Q})$, having rank, $r$, the Birch Swinnerton-Dyer (BSD) conjecture states that
$$
c_{E} = \lim_{s \...
- 10.9k
1
vote
0
answers
85
views
Action of Atkin--Lehner involution on CM points
In their first paper on Heegner points and derivatives of $L$-series, Gross and Zagier describe the action of Atkin--Lehner involutions on certain CM $\mathbf{C}$-points of the modular curve $X_0(N)$. ...
- 265
1
vote
0
answers
137
views
Crystalline at $\ell=p$ implies unramified at $\ell\neq p$
Let $X$ be a smooth, projective variety defined over some $p$-adic field $K$. Is it true that if the etale cohomology $H^i_{et}(X_{\overline{K}},\mathbb{Q}_\ell)$ is crystalline at $\ell=p$, then $H^...
- 141
2
votes
0
answers
96
views
On the root numbers of quadruples of quadratic twists of elliptic curves
We got strong numerical evidence for the root numbers and analytic ranks
of quadruples of elliptic curves over the rationals.
Related to this question.
Let $k,k_1,k_2$ be squarefree pairwise coprime ...
- 25.4k
3
votes
0
answers
241
views
Generating algebraic points on elliptic curves
Let $E$ be an elliptic curve over $\mathbf{Q}$. One has the modular parameterisation
\begin{align*}
\mathbf{H} \to X_0(N)(\mathbf{C}) \to E(\mathbf{C})
\end{align*}
where $X_0(N)$ is the modular curve ...
- 265
1
vote
0
answers
219
views
What is the relation between two abelian varieties that have the same formal group?
Consider two abelian varieties $A$ and $B$ over the $p$-adic number field $\mathbb{Q}_p$. Let $\hat{A}$ and $\hat{B}$ be the associated formal groups of $A$ and $B$, respectively. Assume that $\hat{A}=...
- 195
4
votes
0
answers
125
views
Absolute purity in p-adic case
Suppose that $X$ is a regular proper scheme over $\mathbb{Z}_2$ with smooth generic fiber $X_{\eta}$ and singular special fiber $X_s$. I want to understand $H_{\textrm{et}}^*(X_{\eta},\mathbb{Z}/2)$. ...
- 918
44
votes
2
answers
6k
views
Clausen's modified Hodge Conjecture
In a recent talk at the University of Geneve, Dustin Clausen presented a "modified Hodge Conjecture". I found the abstract intriguing but couldn't find videos or notes available online.
If I'...
- 40.3k
5
votes
0
answers
236
views
Triviality of $\unicode{1064}(T_pE ⊗ T_pE)$ for elliptic curves and Bogomolov's lemma
Consider the case of an elliptic curve $E$ over Q, and let $S$ be a finite set of primes including all places of bad reduction and a place $p$ of good reduction.
Bogomolov's Lemma says that when $p$ ...
- 2,907
4
votes
1
answer
291
views
Effective version for Silverman’s specialization theorem
In his paper, Silverman proves Theorem C (page 208, the indexes of the pages don't match the file), which says that the set $$\{t\in C^0(\bar{K})\mid \sigma_t \text{ is not injective}\}$$ is a set of ...
- 47.4k
3
votes
1
answer
364
views
Height pairing and the Néron model of an elliptic curve
I have a question on Joseph Silverman's book ``Advanced topics in the arithmetic of elliptic curves’’ (1999 printing). I asked him; he answered that he doesn't know off-hand and suggested that I put ...
CommunityBot
- 1
13
votes
2
answers
1k
views
Finiteness of the Brauer group for flat proper schemes over $\operatorname{Spec} \mathbf{Z}$
One fundamental conjecture on the Brauer group is that $\operatorname{Br}(X)$ is finite for $X/\operatorname{Spec} \mathbf{Z}$ proper. By class field theory (the theorem of Albert–Brauer–Hasse–Noether)...
- 12.9k
1
vote
1
answer
143
views
Algorithm for computing isogeny class of elliptic curve
Is there an algorithm for computing the entire isogeny class of a given elliptic curve $E/\mathbb{Q}$?
References/ideas are welcome. Thanks!
- 8,945
17
votes
2
answers
2k
views
How to think of algebraic geometry in characteristic p?
How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
- 4,713
11
votes
1
answer
406
views
Closed image of curves under $p$-adic logarithm, Coleman integrals and Bogomolov
Disclaimer: my knowledge of $p$-adic analysis/geometry is minimal.
Consider a smooth, complete curve $C$ of genus $g$ over $\mathbb{C}_{p}$, denote by $J$ its Jacobian and consider the embedding $C\...
- 1,000
3
votes
0
answers
192
views
How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
1
vote
0
answers
85
views
Companions for positive characteristic arithmetic representations viewed as representations of the topological fundamental group?
Suppose $X / K$ is a variety over a finitely generated field over $\mathbb{Q}$. Fix an embedding $K \subset \mathbb{C}$ and let $\pi := \pi_1(X(\mathbb{C}), x)$ be the topological fundamental group. ...
- 3,730
4
votes
0
answers
167
views
Is the group of homologically trivial cycles in a variety over a finite field torsion?
Let $X$ be a smooth projective variety over $\mathbb{F}_q$. Is any cycle in the Chow group $CH^i(X)$ which is trivial in $\ell$-adic cohomology automatically torsion? For abelian varieties I believe ...
- 531
9
votes
3
answers
2k
views
Characterisation for separable extension of a field
Can someone verify this for me.. or tell me what reference shows me this... is this true:
Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \...
- 11
2
votes
0
answers
234
views
Elliptic curve with rank at least $6$
I was going through a research paper which proves the existence of an infinite family of rank $6$ elliptic curves over $\mathbb{Q}$ with invariant equal to $0$.
Let $k$ be a field of characteristic ...
- 12.9k
2
votes
1
answer
345
views
Why this genus one curve over $\mathbb{F}_5$ appear to violate Hasse-Weil bound?
Working over $\mathbb{F}_5$, the affine curve $x^4+2=y^2$ has no points.
The projective curve $x^4+2y^4=z^2y^2$ has only one point $(0 : 0 : 1)$.
Both curves appear to violate Hasse-Weil bound of $4....
- 79.9k
3
votes
0
answers
174
views
On the sheaves-functions dictionary
Let $X$ be a variety over a finite field $k$. Let $\pi_{1}(X)$ be the arithmetic etale fundamental group of $X$, and $\rho:\pi_{1}(X)\to k^{\times}$ a continuous character. If $x: \text{Spec}(k)\to X$ ...
- 667
7
votes
1
answer
361
views
Shouldn't we expect analytic (in the Berkovich sense) étale cohomology of a number field to be the cohomology of the Artin–Verdier site?
Let $K$ be a number field. Consider $X=\mathcal{M}(\mathcal O_K)$ the global Berkovich analytic space associated to $\mathcal O_K$ endowed with the norm $\|\cdot\|=\max\limits_{\sigma:K \...
- 12.9k
2
votes
0
answers
54
views
Conductor of hyperelliptic curve after adding a rational root
Let $r\geq 5$ be a prime. Suppose I have a specified hyperelliptic curve $C: y^2=f(x)$ defined over $\mathbb{Q}$, where $f\in\mathbb{Q}[x]$ has degree $r$.
Note: the roots of $f$ are not rational but ...
- 83
8
votes
1
answer
365
views
Evidence for the equivariant BSD conjecture with higher multiplicity
Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho$ be an irreducible Artin representation. Let $K_\rho/\mathbb{Q}$ be the smallest Galois extension such that $\rho$ factors through $\mathrm{Gal}(...
- 83
1
vote
0
answers
98
views
Are there known effective bounds on the number of semisimple Galois representations?
In continuation to my question here, are there known effective bounds on the total number of semisimple $p$-adic Galois representations unramified outside a finite set of primes $S$, of dimension $d$, ...
- 2,907
0
votes
1
answer
223
views
Reference for Faltings' proof on finiteness of semisimple $d$-dimensional $p$-adic Galois representations
I'm looking for a reference to Faltings' proof concerning the finiteness of $d$-dimensional semisimple $p$-adic Galois representations. Specifically, the result states that there are only finitely ...
- 1,561
4
votes
0
answers
228
views
The definition of complex multiplication on K3 surfaces
I am reading this paper on the complex multiplication of K3 surfaces. It seems that this is only defined for complex K3 surfaces, or K3 surfaces over number fields. Is there a more general defintion ...
- 141
2
votes
0
answers
103
views
Existence of supersingular abelian variety over $\mathbb F_p$ with $\mathcal O_E$-action where $E/\mathbb Q$ is quadratic imaginary ramified at $p$
Let $E/\mathbb Q$ be a quadratic totally imaginary number field with ring of integers $\mathcal O_E$. Let $p > 2$ be an odd prime number which ramifies in $\mathcal O_E$. Let $\mathcal P$ be the ...
- 769
8
votes
0
answers
333
views
Do automorphisms actually prevent the formation of fine moduli spaces?
I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...
- 12.9k
7
votes
1
answer
613
views
Are all representations of the geometric étale fundamental group subquotients of representations of the arithmetic étale fundamental group?
Let $X$ be a variety over a field $k$. The étale fundamental group of $X$ fits into the exact sequence:
$$1 \to \pi_1^{\text{geom}}(X) \to \pi_1^{\text{arith}}(X) \to \text{Gal}(\overline{k}/k) \to 1,$...
- 28.8k
1
vote
0
answers
115
views
Compactifications of product of universal elliptic curves
Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving ...
- 12.1k
9
votes
2
answers
413
views
Arithmetic schemes with the same zeta function
Suppose $X$ and $Y$ are $n$-dimensional regular separated schemes of finite type over $\mathbb{Z}$ such that number of $\mathbb{F}$-points of $X$ and $Y$ are equal for all finite fields $\mathbb{F}$.
...
- 1,553
4
votes
0
answers
184
views
Étale- or fppf-crystalline sites
I have a straightforward question. Let (say) $X/\mathbb{F}_p$ be a smooth proper scheme. On the big crystalline category over $\mathbb{Z}/p^n$ one can take the Zariski or étale topology, and one can ...
- 371
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 28.8k
16
votes
2
answers
687
views
Deligne's theorem on finite flat group schemes and generalizations
Recall Deligne's theorem that for a finite flat commutative group scheme $G$ of order $n$, the multiplication by $n$ map $[n]: G \to G$ is the zero map.
I have seen the proof a few times but I can't ...
- 21
3
votes
1
answer
131
views
Depth of the filtration of higher ramification groups in the ramified case in Serre's modularity conjecture
I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I have some questions about Serre's definition of "peu ...
- 385
5
votes
1
answer
253
views
Rational isogenies of prime degree $p\in\{11,17,19,37,43,67,163\}$
Let $p\in\{11,17,19,37,43,67,163\}$ be a prime number. In [1], B. Mazur proves that there are only finite number of elliptic curves $E$ [over $\mathbb{Q}$] having an isogeny of degree $p$.
Here is my ...
- 79.9k
3
votes
0
answers
201
views
Cup product structure on Galois cohomology
Let $G_S$ denote the Galois group of the maximal extension of $\mathbb{Q}$ unramified outside a finite, non-empty set of primes, $S$. Let $p\in S$ and let $V, W$ be a pair of finite dimensional $p$-...
CommunityBot
- 1
4
votes
1
answer
458
views
Bad prime of torsor and original elliptic curve ; Definition of Tate–Shafarevich group $Ш(E/K)$
Let $E/K$ be an elliptic curve over number field $K$. Let $M_K$ be a set of all places of $K$.
My question is, Does there exist a finite set $S\subset M_K$ such that
$\forall C$: $E/K$-torsor, $\...
- 12.9k
3
votes
0
answers
84
views
Practical way of computing bitangent lines of a quartic (using computers)
Are there known practical algorithms or methods to calculate the bitangent lines of a quartic defined by $f(u,v,t)=0$ in terms of the 15 coefficients? Theoretically you can set up $f(u,v,-au-bv)=(k_0u^...
- 101
3
votes
1
answer
370
views
Bloch–Beilinson conjecture for varieties over function fields of positive characteristic
Is there a version of the Bloch–Beilinson conjecture for smooth projective varieties over global fields of positive characteristic? The conjecture I’m referring to is the “recurring fantasy” on page 1 ...
- 1,404