All Questions
2,494 questions
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
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
5
votes
0
answers
235
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
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
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
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!
- 2,907
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
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....
- 25.4k
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
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
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 ...
- 2,907
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
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
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 ...
- 831
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
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:
$$1 \to \pi_1^{\text{geom}}(X) \to \pi_1^{\text{arith}}(X) \to \text{Gal}(\overline{k}/k) \to 1,$...
- 2,907
8
votes
1
answer
365
views
Evidence for the equivariant BSD conjecture with higher multiplicity
Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho$ be an irreducible Artin representation. Let $K_\rho/\mathbb{Q}$ be the smallest Galois extension such that $\rho$ factors through $\mathrm{Gal}(...
- 83
1
vote
0
answers
115
views
Compactifications of product of universal elliptic curves
Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving ...
- 12.1k
2
votes
0
answers
234
views
Elliptic curve with rank at least $6$
I was going through a research paper which proves the existence of an infinite family of rank $6$ elliptic curves over $\mathbb{Q}$ with invariant equal to $0$.
Let $k$ be a field of characteristic ...
- 127
4
votes
0
answers
184
views
Étale- or fppf-crystalline sites
I have a straightforward question. Let (say) $X/\mathbb{F}_p$ be a smooth proper scheme. On the big crystalline category over $\mathbb{Z}/p^n$ one can take the Zariski or étale topology, and one can ...
- 371
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53
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 ...
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 ...
- 622
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 $p$-...
- 2,907
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, $\...
- 1,541
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
2
votes
1
answer
319
views
Bounding $H^4_{\text{ėt}}$ of a surface
Let $X\longrightarrow X'$ be a smooth proper map of smooth proper schemes defined over $\mathbb{Z}[1/S]$, where $S$ is a finite set of primes. Assume $X'$ is a curve of positive genus, and $X$ is a ...
- 2,907
4
votes
0
answers
129
views
How does one compute the group action of the automorphism group on integral cohomology?
Suppose I have a curve $X$ (for concreteness, we can take $X$ to be a smooth, projective curve over a finite field $\mathbb F_q$, and even more concretely consider the family of curves described by ...
- 7,746
3
votes
1
answer
250
views
Action of complex conjugation on etale cohomology
Let $X$ be a genus $g$ smooth projective curve, defined over $\mathbb{Q}$, and let $\overline{X}$ denote the base change of $X$ to $\overline{\mathbb{Q}}$.
It is well known that $H^1_{\text{ét}}(\...
- 2,907
3
votes
1
answer
187
views
Difficulties in the proof of finiteness of n-Selmer group using cohomology
I was reading the proof of finiteness of n-Selmer group $S^n(E/\mathbb{Q})$ from Milne's Elliptic curve book(1st Edition). While reading the proof I had some difficulties in some arguments.
1st ...
- 127
19
votes
1
answer
1k
views
What are $L$-functions?
I am coming at this question from the point of view of someone who is working in arithmetic geometry around the Langlands program.
We have $L$-functions associated to many different structures that we ...
- 831
1
vote
1
answer
159
views
Zeta function of variety over positive characteristic function field vs. local zeta factor of variety over $\mathbb{F}_p$
Let $X = Y \times_{\mathbb{F}_q} C$, with $Y, C / \mathbb{F}_q$ smooth projective varieties, $C$ a curve. Let $d = \dim_{\mathbb{F}_q} X$. We can consider the local zeta function $Z(X, t) = \prod\...
- 658
0
votes
1
answer
208
views
Finiteness of Selmer group
I was reading the proof of finiteness of $S^n(E/\mathbb{Q})$ but I am unable to understand from the following lemma how it follows that $S^n(E/L)$ finite.
LEMMA 3.13 For any finte subset $T$ of $\...
- 127
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 \...
- 804
6
votes
1
answer
918
views
Understanding the Hodge filtration
Let $X$ be a smooth quasiprojective scheme defined over $\mathbb{C}$, and let $\Omega^{\bullet}_X$ denote its cotangent complex, explicitly, we have:
$\Omega^{\bullet}_X:=\mathcal{O}_X\longrightarrow \...
- 2,907
0
votes
0
answers
100
views
Hodge filtration vs Hodge structure on algebraic de Rham cohomology
I have a basic question on the relation between the definitions of the Hodge structure on the algebraic de Rham of a smooth proper scheme defined over a subfield of $\mathbb{C}$ and the Hodge ...
- 2,907
7
votes
1
answer
1k
views
What's the relation between analytic stacks and higher complex/non-archimedean analytic stacks?
Recently Clausen and Scholze have developed a theory of analytic stack to unify different analytic geometries. Actually I do not know many details about it. It says an analytic stack is a sheaf $\...
- 83
2
votes
0
answers
101
views
Action of monodromy on the $p$-adic period domain in Lawrence-Venkatesh
In here, I asked various questions related to Lawrence and Venkatesh's work on the Mordell-Weil conjecture, which failed to receive any answers.
This is my attempt to try and focus the question.
In ...
- 2,907
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 ...
- 531
1
vote
1
answer
141
views
Grössencharakter or Galois representation associated to a CM elliptic curve in characteristic $p$
When $E$ is an elliptic curve over a number field $L$, with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic ...
- 531
3
votes
0
answers
141
views
Monodromy action on the period domain in the Lawrence-Venkatesh paper
Let $Y\longrightarrow X$ be a smooth proper map of smooth quasiprojective schemes over $\mathbb{Z}_p$.
In Lawrence and Venkatesh's paper on Mordell-Weil, the authors consider the $p$-adic period map: $...
- 2,907
7
votes
0
answers
148
views
Is the $\ell$-adic cohomology ring of a cubic threefold a complete invariant?
The only interesting $\ell$-adic cohomology of a smooth cubic threefold $X$ is $H^3(X,\mathbb{Z}_{\ell}(2))$, which is isomorphic as a $\mathrm{Gal}_k$-module to $H^1(JX,\mathbb{Z}_{\ell}(1))^{\vee}$ ...
- 679
0
votes
0
answers
78
views
Bounding the dimension of $H^1(G, V\otimes V^{\vee})$
Let $G_S$ denote the Galois group of the maximal extension of $\mathbb{Q}$ unramified away from a finite set of primes, $S$. Let $V$ be a finite dimensional, $G_S$-representation over $\mathbb{F}_p$ (...
- 2,907
7
votes
0
answers
249
views
What justifies the following isomorphism in Cassels' proof of the Cassels–Tate pairing?
In Cassels' paper Arithmetic on curves of genus 1. IV introducing the Cassels–Tate pairing the following lemma is stated.
Lemma 5.1: Let $q$ be a rational prime and $\Gamma$ the Galois group of the ...
- 221
0
votes
1
answer
111
views
Kernel of restriction in étale cohomology of curves over number fields
Let $X$ be a smooth projective curve defined over a number field $K$. Let $\overline{K}$ denote the algebraic closure of $K$, and set $\overline{X} := X\otimes \overline{K}$. Denote by $\iota: \...
- 2,907
2
votes
2
answers
272
views
Finding rational points on intersection of quadrics in affine 3-space
Consider the subvariety of Spec $\mathbb{Q}[x,y,z]$ cut out by the equations
\begin{eqnarray*} f_1: a_1x^2 - y^2 - b_1^2 & = & 0 \\
f_2 : a_2x^2 - z^2 - b_2^2 & = & 0
\end{eqnarray*}
...
- 7,547
2
votes
1
answer
176
views
Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$
$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...
- 185