All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
2
votes
0
answers
269
views
Is there any relation between Berkovich spaces over $\Bbb Z$ and Arakelov theory?
As I understand it, both Arakelov geometry and Berkovich geometry over $\Bbb Z$ (or $\mathcal O_K$) consider geometric objects that contain in some sense information about both archimdean and ...
- 804
3
votes
1
answer
230
views
Cubic polynomial over $\mathbb{Z}_p$
Let
$$
f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b,
$$
where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all pairs $(a,b)\in\mathbb{Z}_p^2$ such that $f_{a,b}(x)$ factor linearly. Then what ...
user126352
7
votes
1
answer
310
views
Faithful representations of integral models
I am reposting a question that I had asked on stackexachage three weeks ago.
Let $G/\mathbb{Q}$ be a connected reductive group, and $\mathcal{G}/\mathbb{Z}$ be an integral model (i.e. flat affine ...
- 821
12
votes
3
answers
911
views
Does there exist some $p(x) \in \mathbb{Q}[x]$, deg$(p) > 1$, which maps $\mathbb{Q}$ onto itself surjectively?
Clearly this is impossible for $p$ of even degree, and I imagine that Cardano’s formula quickly reveals it to be impossible in the cubic case, although I have not checked in detail. My guess is that ...
- 531
0
votes
1
answer
160
views
Embedding of symmetric square in Jacobian
Let $C$ be a projective curve defined over a field $K$, and let $C^{(2)}$ and $J$ be its symmetric square and Jacobian, respectively.
There is a natural map $C^{(2)}\hookrightarrow J$, defined as ...
- 2,907
0
votes
1
answer
205
views
Rational points on genus 3 curves defined by short equations
(a) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3-y=x^4-x.
$$
(b) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3+y=x^4+x.
$$
If not a complete answer, I would be happy to ...
- 7,182
2
votes
1
answer
242
views
Varieties whose residue fields do not generate the algebraic closure of the ground field
Let $K$ be a number field, $\overline{K}$ an algebraic closure, and $X$ be a positive dimensional finite type $K$-scheme.
Could there exist a proper subfield $L\subset\overline{K}$ such that the ...
- 7,527
2
votes
0
answers
184
views
Will Coppersmith's method work for this bivariate modular polynomial shape?
I have a bivariate modular polynomial of shape
$$f(x,y)=x^2y-g(x)\equiv 0\bmod q$$
where
$q=(2p-1)(2p+1)$ is a product of two primes $2p-1$ and $2p+1$,
$g(x)\in\mathbb Z[x]$ is of degree four and
$f(...
- 13.9k
2
votes
0
answers
259
views
The group of the modular automorphisms of the Shimura curves
Let $B$ be a rational indefinite division quaternion algebra, $(X,G)$ the Shimura datum associated with $B$ (i.e., $X$ is the upper half plane and $G(R) = (B \otimes_\mathbb{Q} R)^*$ for a ring $R/\...
- 1,364
2
votes
0
answers
113
views
Selmer groups associated with Drinfeld modules
Given an elliptic curve $E_{/\mathbb{Q}}$ (or more generally, a number field) and a prime $p$, there is a standard short exact sequence
$$0\rightarrow E(\mathbb{Q})\otimes \mathbb{Q}_p/\mathbb{Z}_p\...
- 685
1
vote
1
answer
238
views
When $E_D:y^2=x^3+17D^2x$ has even rank?
Let $E:y^2=x^3+17x$ be an elliptic curve.
In this MO page(Infinitely many elliptic curve with twist rank more than $1$ in specific case), Nulhomologous's and other's comment reads from parity ...
- 1,541
2
votes
1
answer
154
views
On the estimate for the mixed 3-dimensional hyper-Kloosterman sum
There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum:
For any positive integer $n$ not divisible by $p$, how to prove
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p}
\...
- 563
4
votes
1
answer
291
views
Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB
The result of LMFDB claims (https://www.lmfdb.org/EllipticCurve/Q/1640/c/1 )
that (2-part of) Tate-Shafarevich group $\mathrm{Sha}(E/\Bbb{Q})$ of elliptic curve $y^2=x^3-8747x-314874$ has order $16$. ...
- 1,541
5
votes
1
answer
513
views
Learning Inverse Galois Theory
Can someone give me a roadmap for learning Inverse Galois theory?
I am a PhD student in the representation theory of finite groups. I studied Galois theory when I was an undergraduate student. The ...
- 195
2
votes
1
answer
214
views
Cohomology of $\mathcal{O}_{F^S}[\frac{1}{S}]^\times$
$\quad$Let $F$ be a number field, $\ell$ a prime, and $S$ a finite set of places of $F$ including all Archimedean places and places over $\ell$.
$\quad$Then we have
$$\mathrm{H}^1\left(G_{F,S},\...
user508433
10
votes
1
answer
550
views
Igusa's $\chi_{10}$ and Borcherds products
Igusa defined a genus 2 Siegel modular form $\chi_{10}$, which vanishes on the Humbert surface $G_{1}$ (the image of a "degenerate" Hilbert modular surface, the product of modular curves, ...
- 3,309
4
votes
1
answer
408
views
The notion of morphisms between two moduli problems in Katz-Mazur
I am reading Katz-Mazur Arithmetic Moduli of Elliptic Curves, and have some questions about the notion of morphisms between two moduli problems.
What is the proper definition of morphisms between two ...
- 161
1
vote
1
answer
198
views
Crystalline fibre of a morphism of Galois cohomology stacks
Let $K = \mathbb{Q}_p$, $G = G_K$ its absolute Galois group. Let
$$1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$$
be a split exact sequence of (not necessarily abelian) group ...
- 2,907
0
votes
1
answer
91
views
Construct next polynomial from predecessor and resulting GCD
I have a sequence of polynomials built from an interpolation derived in a combinatorial problem. For each integer value of a parameter $n$ there is a different polynomial.
After trying to find a way ...
- 47
2
votes
0
answers
220
views
Zero dimensional varieties and the L-function $1/(1-p^{-n})$
I am interested in positive characteristic varieties which produce an L-function of the form $\frac{1}{1-χ} = \frac{1}{1-p^{-s}} = \sum_{n = 0}^\infty p^{-ns}$. It seems related to the positive ...
user30211
2
votes
1
answer
410
views
Galois cohomology of Tate modules
Let $E,E'$ be a pair of elliptic curves defined over $\mathbb{Z}$. Let $T_p[E], T_p[E']$ be their associated ($p$-adic) Tate modules. These are Galois representations for the absolute Galois group of $...
- 2,907
17
votes
1
answer
2k
views
How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?
In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived ...
- 3,309
2
votes
0
answers
118
views
polynomials with no repeated factors
Assume that $F(x_1,\ldots, x_n)$ is a polynomial with integer coefficients that is "square-free" over $\mathbb Q$, i.e. it does not have repeated polynomial factors whose coefficients are in ...
- 3,062
0
votes
0
answers
92
views
A question on the evaluations of certain three-dimensional hyper-Kloostermans
There is a basic question regrading the 3-dimensional hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,h \in \mathbb{N}$, how to estimate the sum:
$$\sideset{_{}^{...
- 563
4
votes
0
answers
307
views
Equations involving sum of fourth powers
Do there exist rational numbers $x,y,z$ such that
$$
\quad \quad z^3 - 1 = x^4+y^4 \neq 0 \tag{$a$} \quad ?
$$
Also, do there exist rational numbers $x,y,z$ such that
$$
\quad \quad z^3 - z = x^4+y^4 \...
- 7,182
6
votes
0
answers
525
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 - ...
- 1,635
2
votes
0
answers
82
views
Is there any work on the intersection loci of the universal theta divisor with torsion sections?
Let $Y$ be a Siegel modular variety of some non-stacky level and genus $g$, carrying over it a universal principally polarized family of dimension-$g$ abelian varieties $A\to Y$. Inside $A$, with fine ...
- 2,044
1
vote
0
answers
140
views
Kernel of restriction map in Galois cohomology
Let $S$ be the algebraic group $SL_2/\mathbb{Q}_p$ with a $G=G_{\mathbb{Q}}$ action, (acts by conjugation with a representation $\rho: G\longrightarrow GL_2$.)
Let $G_p$ be the decomposition group at ...
- 2,907
1
vote
0
answers
128
views
Representability of twists of projective schemes
Let $K$ be a perfect field, and let $S$ be a projective $K$-scheme. Denote by $\text{Twist}(S/K)$ the set of twists of $S$ up to $K$-isomorphism. These are (apriori) sheaves $\mathcal{F}$ on the ...
- 2,907
3
votes
1
answer
225
views
Deformations of Galois cohomology
Let $M = (\mathbb{Z}_p)^2$ be a Galois representation, with Galois action given by $\rho: G\longrightarrow SL_2(\mathbb{Z}_p)$. I am trying to understand how sensitive the Galois cohomology group $H^1(...
- 2,907
2
votes
1
answer
132
views
Zeros of a sequence in $\overline{\mathbb F_q(T)}$
Let $\beta$ be an element of $\overline{\mathbb F_q(T)}\setminus\overline{\mathbb F_q}$.
Is it true that the sequence $(\beta^{q^n}-T)_n$ admits infinitely many zeros, that is there exist infinitely ...
- 3,998
3
votes
0
answers
160
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 ...
- 821
0
votes
0
answers
319
views
Percent of rational coordinates that is a multiple of another point on the elliptic curve
Consider elliptic curves $E:= y^2=x^3+Ax+B $ (A, B are integers) which have points $P, Q$ with rational coordinates and satisfy $P=[n]Q, n>1$. Now consider the below problem:
Input: Rational ...
5
votes
0
answers
237
views
Hyperelliptic curve with prescribed rational points?
Given a set of rational points $S$, does there always exist a hyperelliptic curve $C$ such that $C(\mathbb{Q})=S$?
Namely, which sets could arise as the set of rational points of a hyperelliptic curve?...
- 203
1
vote
0
answers
152
views
How difficult is to find rational points on these genus 3 curves:
How difficult is to find all rational points on these genus 3 curves:
$$
(a) \quad \quad x^3 + y^3 x +y^2 - y = 0
$$
$$
(b) \quad \quad x^4 - y^3 + x y + x = 0
$$
Short motivation. Consider the ...
- 7,182
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
3
votes
1
answer
510
views
p-adic period map in Lawrence and Venkatesh
In Lawrence and Venkatesh's paper on the Mordell conjecture, they prove that there are finitely many $K$-rational points on a hyperbolic curve $X$, where $K$ is a number field, by showing that there ...
- 2,907
2
votes
0
answers
107
views
Extensions of groups with a $G$-action
Let $1\longrightarrow A\longrightarrow \mathcal{G}\longrightarrow R\longrightarrow 1$ be an exact sequence of algebraic group schemes, with $\mathcal{G}$ being an extension of $R$, an affine reductive ...
- 2,907
12
votes
1
answer
984
views
Jouanolou thesis on l-adic cohomology
Does someone have a copy of the Jean-Pierre Jouanolou's thesis:
Catégories dérivées et cohomologie l-adique
or has the ability to make a digitalization? The thesis was done at Université de Paris 1969....
user339690
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
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 ...
3
votes
0
answers
375
views
On the analogy between $p$-derivations and derivations
$\DeclareMathOperator\Spec{Spec}$Let $p$ be a prime number, and $A$ a commutative ring. Recall that a $p$-derivation on $A$, or a $\delta$-ring structure on $A$ is a set map $\delta : A \to A$ such ...
- 63.9k
8
votes
6
answers
2k
views
How many solutions are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$?
Let $p$ be a prime. How many solutions $(x, y)$ are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$? Here $x, y \in \{0, 1, \ldots p-1\}$. This paper (https://arxiv.org/abs/1404.4214) seems like ...
- 1,703
1
vote
0
answers
146
views
Is the functor $\mathrm{Hom}(\mathrm{spec}\,k[x^{1/{p^\infty}}]/(x), -)$ from the category of finite commutative group schemes exact?
Question. Let $B \twoheadrightarrow C$ be a fully faithful homomorphism of finite connected commutative group schemes over a perfect field $k$. Let $T = k[x^{1/p^\infty}]/(x) = \varinjlim k[t]/(t^p)$. ...
- 199
2
votes
1
answer
437
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
12
votes
1
answer
1k
views
An omission in K. Conrad's notes on the conductor ideal
I am referring to the very useful K. Conrad's notes on the conductor ideal of an order in a Dedekind domain: https://kconrad.math.uconn.edu/blurbs/gradnumthy/conductor.pdf
$\DeclareMathOperator\Cl{Cl}$...
- 675
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
7
votes
1
answer
315
views
Rational points on regular curves over global fields
Let $k$ be a global field and $C$ a smooth projective curve over $k$ which is not isotrivial. Then there is a well-known trichotomy:
If $g(C) = 0$ and $C(k) \neq \emptyset$, then $C \cong \mathbb{P}^...
- 21.3k
2
votes
0
answers
226
views
Number of roots of a multivariate polynomial
What could be the best known asymptotic for the number of solution of the following polynomial in $(F_p)^s$:
$$
(1-x_1)(1-x_2)\cdots(1-x_s)(1-x_1x_2...x_s)=ux_1x_2...x_s
$$
where $u$ is a non-zero ...
user126352
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