All Questions
2,494 questions
17
votes
1
answer
2k
views
How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?
In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived ...
- 3,309
3
votes
1
answer
301
views
Given a smooth hyperplane section Y of a variety X there exists a Lefschetz pencil of hyperplane sections of X containing Y
Let $X$ be a variety contained in $\mathbb{P}^N$ and let $Y$ be a smooth hyperplane section of $X$. I have read in page 54 of Voisin's book "Hodge theory and complex algebraic geometry II" ...
- 519
1
vote
0
answers
120
views
Large Tate-Shafarevich group of an elliptic curve with the form $E_{p,n}:y^2=x^3+p^nx$
Let $p$ be a prime number and $n$ be positive integer.
Let $E_{p,n}:y^2=x^3+p^nx$ be an elliptic curve.
LMFDB reads in the case $(p,n)=(73,3)$ , $\#Sha(E_{p,n})=64$.
This is the biggest size of $Sha(...
- 1,541
2
votes
1
answer
160
views
Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$
Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
- 412
32
votes
4
answers
5k
views
Why is this "the first elliptic curve in nature"?
The LMFDB describes the elliptic curve 11a3 (or 11.a3) as "The first elliptic curve in nature". It has minimal Weierstraß equation
$$
y^2 + y = x^3 - x^2.
$$
My guess is that there is some ...
- 35.5k
6
votes
0
answers
528
views
Simple motivation for mixed characteristic algebraic geometry?
Can anyone give a road map for how Bhatt–Scholze's fancy recent p-adic work applies to questions in more general algebraic geometry and commutative algebra? I'm aware that it does, following Andre - ...
- 1,645
89
votes
9
answers
13k
views
Why should I believe the Mordell Conjecture?
It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points.
I am interested to know why Mordell and ...
- 2,827
6
votes
1
answer
771
views
A regular, geometrically reduced but non-smooth curve
Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book Algebraic ...
- 620
3
votes
1
answer
332
views
Finite subschemes of projective bundles
$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Proj{\mathbf{Proj}}$Let $S$ be a Noetherian scheme and $X$ finite $S$-scheme. The finite morphism $X \to S$ is projective in the sense of the ...
user505967
1
vote
0
answers
79
views
A general theory of pairings
Bilinear forms and bilinear maps for vector spaces over a field are standard material for an introductory course in linear algebra.
There are also text books for bilinear forms and related quadratic ...
- 231
5
votes
0
answers
348
views
Are there toy models of full poly-isomorphism from Inter-universal Teichmüller theory?
I am learning Inter-universal Teichmüller theory and I am interested in the concept of full poly-isomorphism. According to "INTER-UNIVERSAL TEICHMÜLLER THEORY I: CONSTRUCTION OF HODGE THEATERS&...
4
votes
1
answer
240
views
Cycles contained in ample enough hypersurfaces
Let $X$ be an irreducible smooth projective variety of pure dimension $d$ over the complex numbers and $Z\subset X\times X$ a codimension $d$ irreducible smooth closed subvariety.
Is there a smooth ...
user497064
5
votes
1
answer
312
views
Conductor at 2 of abelian surfaces with real multiplication
Let $A/\mathbb{Q}$ be an abelian surface such that $\text{End}_{\mathbb{Q}}(A)\otimes \mathbb{Q}$ is a real quadratic field $E$. I am interested in bounding the conductor of $A$ at $2$.
Let $\mathfrak{...
- 984
2
votes
1
answer
191
views
Cohen-Macaulay fiber products
Let $R$ be a regular local ring, $X$ and $Y$ smooth $R$-schemes, $T\to Y$ a regular closed immersion over $R$ with $T$ smooth over $R$, and $f: X\to Y$ an $R$-morphism.
Is the fiber product scheme $...
user501361
1
vote
0
answers
70
views
Simplicity of abelian varieties and localization
Let $A$ be an abelian variety defined over a number field $K$. Let $v$ be a place of $K$ and denote by $K_v$ the $v$-adic completion of $K$ with respect to $||\cdot||_v$.
Assume $A$ is simple, is it ...
- 2,907
0
votes
1
answer
159
views
Zariski dense in abelian scheme
Let $A \to S$ be an abelian scheme over an irreducible curve $S$ over complex numbers of relative dimension $g \geq 1$. Let $s: S \to A$ be a non-constant section and denote by $X:=s(S)$. Is it true ...
- 247
3
votes
0
answers
161
views
Redefining connected Shimura datum
Firstly, let us fix a semisimple reductive linear algebriac group $G$ over $\mathbb{Q}$.
I am interested in seeing if I can bring the definition of connected Shimura datum (which is defined using some ...
- 831
4
votes
0
answers
284
views
Intermediate arithmetic results in F_1 geometry
Much is made of the search for a proof of the Riemann hypothesis via the field with one element. Are there any lesser classic arithmetic results that have been proved with F_1 geometry, such as the ...
- 447
15
votes
1
answer
911
views
A possible gap in Faltings note to prove the Tate conjecture for finitely generated field over $\mathbb{Q}$
Qeustion:
Given a Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_\ell$ with an ideal $\mathfrak{g}^O$ and a subalgebra $\mathfrak{h}$,
such that $\mathfrak{g}=\mathfrak{g}^O+\mathfrak{h}$.
Now given a ...
- 178
2
votes
1
answer
290
views
Flat scheme-theoretic closure
Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$.
Let $C_R$ be a flat ...
user501361
10
votes
1
answer
3k
views
Roadmap to understand the Scholze's proof of the local Langlands correspondence for $\text{GL}_n$ over $p$-adic fields
I would like to know which books I should read to understand the paper "The local Langlands correspondence for $\mathrm{GL}_n$ over $p$-adic fields" written by Peter Scholze.
I only know ...
- 111
8
votes
1
answer
339
views
On actions of finite groups on adic spaces
Let $K$ be an algebraically closed complete non-archimedean field and consider the unit ball $\mathbb{B}^{1}_{K}=Sp(K\langle t\rangle)$. We have an action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{B}^{1}...
- 541
1
vote
1
answer
146
views
Roots of local zeta function of curves
For a smooth proper curve $C$ over a finite field $\mathbb{F}_q$, local zeta function can be shown that
$$
Z(C,t)=\frac{P(t)}{(1-t)(1-qt)}
$$
with a polynomial of degree $2g$, where $g$ is the genus ...
- 519
0
votes
1
answer
204
views
Are degrees and ramification degrees preserved upon passing to the smooth compactification?
Let $\phi ：C_1\to C_2$ be morphism of projective singular curve. Let $\tilde{C}_1$ and $\tilde{C}_2$ be their smooth compactification.
Then $\phi$ extends to $\tilde{\phi} : \tilde{C}_1\to \tilde{C}...
- 1,541
3
votes
0
answers
170
views
Smoothness of height in Manin conjecture
Set up: Let $K$ be a number field. Let $M_K$ be the places of $K$, and define the standard height on $\mathbb{P}^n(K)$ as
$$H([x_0, \cdots, x_n]) = \prod_{v \in M_K} \max\{|x_0|_v, \cdots, |x_n|_v\}$$
...
- 267
5
votes
2
answers
314
views
Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension
Let $E/ \Bbb{C}$ be an elliptic curve which has complex multiplication over a number field $K$.
Then it is widely known that $j(E) \in \overline { \Bbb{Z}}$.
What is the known generalization of this ...
- 1,541
1
vote
0
answers
162
views
Exercise in Cassels's book
I am trying to prove the following theorem:
Theorem. Let $d=q_1q_2$ where $q_1> 0$, $q_2>0$ are rational primes, with $q_1\equiv 2\mod 9$ and $q_2 \equiv 5 \mod 9$. Then the only rational point ...
- 11
4
votes
1
answer
248
views
Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$
There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the lattice $E_7$ and $\mathbb{F}_2^6 \setminus \{0\}$ (where $\...
- 3,779
1
vote
0
answers
47
views
Absolute irreducibility implies free action on framed universal deformation ring
Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...
- 2,907
59
votes
1
answer
9k
views
IMO 2017/6 via arithmetic geometry
The (very nice) final problem of IMO 2017 asked contestants to show:
If $S$ is a finite set of lattice points $(x,y)$ with $\gcd(x,y)=1$, then there is a nonconstant homogeneous polyonmial $f \in \...
- 1,207
36
votes
2
answers
3k
views
How to visualize Dirichlet’s unit theorem?
As the question title asks for, how do others "visualize" Dirichlet’s unit theorem? I just think of it as a result in algebraic number theory and not one in algebraic geometry. Bonus points for ...
63
votes
11
answers
8k
views
Why certain diophantine equations are interesting (and others are not) ?
It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" ...
- 23.4k
0
votes
1
answer
107
views
Is there a method to check if two sections of an elliptic surface are dependent over the endomorphism ring or not?
Let $E\!: y^2 = x^3 + a(t)x + b(t)$ be an elliptic curve over the function field $\mathbb{F}_{q}(t)$ over a finite field $\mathbb{F}_{q}$ of characteristic $5$ or greater. For simplicity, let $E$ be ...
- 1,833
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 ...
- 455
3
votes
0
answers
132
views
How does the number of connected components of the Néron model change in a family of abelian varieties?
Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other ...
- 333
3
votes
0
answers
177
views
Does every (ordinary) elliptic curve over a quite large prime field $\mathbb{F}_{p}$ have a lift to the function field with huge Mordell-Weil rank?
Ulmer (and then other authors) showed existence of elliptic curves $E$ over the function field $\mathbb{F}_{p}(t)$ (where $p$ is a prime) with huge Mordell-Weil ranks (i.e., depending on $p$). The ...
- 1,833
2
votes
1
answer
243
views
Finite flat pullback of the diagonal
Let $X, Y$ be smooth projective connected complex varieties of the same pure dimension $d$ and $f : X\to Y$ a finite flat surjective morphism.
Let $\Delta_X$ be the closed subscheme of $X\times X$ ...
user497064
2
votes
1
answer
438
views
Sheaf cohomology in number theory
I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of ...
- 186
4
votes
1
answer
915
views
What is the idea behind the proof of the Isogeny theorem and Theorem III.7.9 (Serre) in Silverman's book?
Let $E_1$ and $E_2$ be Elliptic curves over the field $K$ and $\ell\neq\mathrm{char}(K)$ be a prime number. Let $T_\ell(E_i)$ is the Tate module of $E_i$, $i=1,2$ and $\mathrm{Hom}_K(T_\ell(E_1),T_\...
user102974
0
votes
0
answers
162
views
Norm map of Tate-Shafarevich group $\mathrm{Sha}(E/K)\to \mathrm{Sha}(E/\Bbb{Q})$
Let $K$ be a quadratic number field. Let $\sigma$ be a generator of Galois group of $K/\Bbb{Q}$. Let $E$ be an elliptic curve defined over $\mathbb{Q}$.
Let $\mathrm{Sha}(E/K)$ denote the Tate-...
- 1,541
150
votes
2
answers
22k
views
What is a Frobenioid?
Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion ...
- 13.6k
2
votes
1
answer
94
views
Base change for fundamental group prime to p in mixed characteristic?
I found the answer to this question while typing it up, but since I've already written it, it is probably worthwhile to post-and-answer in case someone finds it useful.
Let $S=\operatorname{Spec}\...
- 371
5
votes
0
answers
192
views
Image of $\gamma-1$ on etale $(\varphi,\Gamma)$-modules
Let $p\geq 3$ be a prime, $D$ be an etale $(\varphi,\Gamma)$-modules over the classical period ring $A_{\mathbb{Q}_p}=\mathbb{Z}_p[\![T]\!][1/T]^{\widehat{\phantom{xx}}}_p$ and $\gamma$ be a ...
- 549
6
votes
2
answers
656
views
Finite flat group schemes over $\mathbb{Z}$ that are extensions of $\mu_p$ by $\mathbb{Z}/p\mathbb{Z}$
Suppose $X$ is a finite flat group scheme over $\mathbb Z$, killed by a prime number $p$ and such that there exists an extension as finite flat group schemes defined over $\mathbb Z$:
$$0\to \mathbb{Z}...
- 608
5
votes
0
answers
168
views
Generalization of Deuring's theorem
Deuring has the following important theorem on elliptic curves over a finite field: Let $p$ be a prime at least $5$, and $N=p+1-a$ an integer with $|a|\leq 2\sqrt{p}$, then the number of elliptic ...
- 213
2
votes
1
answer
272
views
System of two linear Diophantine equations
Let $n\in\mathbb{N}$ be a positive integer. Denote by $f(n)$ the number of integral solutions of the following system
$$
\left\lbrace\begin{array}{l}
\sum_{i=1}^nx_i = 3n; \\
\sum_{i=1}^n (2i-1)x_i = ...
- 8,998
6
votes
1
answer
445
views
Why do Chern forms show up in Arakelov geometry?
Let $X$ be a regular, projective flat scheme over $\Bbb{Z}$, let $\bar{L}$ be a hermitian line bundle on $X$. In order to define the height of an integral closed subset $Y$ we define it on closed ...
- 193
6
votes
1
answer
877
views
Symmetric powers, localisation and Frobenius
I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and ...
- 1,093
1
vote
1
answer
150
views
Is Krämer's local model for ramified unitary groups isomorphic to the blow-up of Pappas' flat model at the singular point?
I am reading the following two papers:
Pappas, On the arithmetic moduli schemes of PEL Shimura varieties, 1999 (it seems to be difficult to find online nowadays - only a .ps file remains available),
...
- 769
0
votes
1
answer
240
views
Can I calculate congruent zeta function of given hyperelliptic curve by hand?
How can I calculate the numerator of congruent zeta function of given hyperelliptic curve ?
For example, let $C:y^2=(x^2+1)(x^4-8x^3+2x^2+8x+1)$.
numerator of congruent zeta function mod$23$ of this ...
- 1,541