All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
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
3
votes
0
answers
529
views
The cycle class map with values in crystalline cohomology
Let $ k = \mathbb{F}_q $ be a finite field of characteristic $ p > 0 $.
Let $ X $ be a smooth proper scheme of dimension $ d $ over $ k $.
Consider the associated $ K $ - linear cycle class map ...
- 595
9
votes
0
answers
380
views
How can I "see" that a map is birational?
This came up with the Euler brick.
Let $T=(p,q,r)$ be a Randall triple, i.e. $$(p^2-1)(q^2-1)(r^2-1)=8pqr\ \qquad\text{[eq.1]}.$$ There are tons of maps that map a triple $T$ to another $T'=(p',q',r')$...
- 4,829
5
votes
0
answers
215
views
Integer points of rational function in 2 variables
Is there an algorithm that, given polynomials $P(x)$ and $Q(y)$ with integer coefficients, decides whether there exists integers $x$ and $y$ such that $\frac{P(x)}{Q(y)}$ is an integer?
This is a ...
- 7,182
6
votes
2
answers
268
views
Curve with a rational point but no new points in number fields of low degree
Given an integer $d\geq 2$ is there an algebraic curve $C/\mathbb{Q}$ with $C(\mathbb{Q})\neq\emptyset$ and the natural map $C(\mathbb{Q})\to C(F)$ bijective for all number fields of degree at most $d$...
- 61
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
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
1
vote
1
answer
150
views
Counting $\mathrm{mod}\:p$ solutions of Diophantine equation in two variables taking $O(p^2)$ time
Are there Diophantine equations in two variables such that counting solutions $\mathrm{mod}\:p$ requires $O(p^2)$ time?
Geometrically this means we have to sort through a positive proportion of the ...
- 21
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 ...
- 4,418
2
votes
1
answer
215
views
The growth of the number of Fano complete intersection families
I recently calculated the number (possible multidegrees) of Fano complete intersections of dimension $n$ , because I wanted to make the remark that it grows "very rapidly" as $n \rightarrow \...
- 6,995
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 ...
- 455
4
votes
0
answers
236
views
Is this property of polynomials generic?
Let $n \geq 2$, and consider a polynomial $f$ in $n$ variables, say over a field $K$ of characteristic 0. Recall that $f$ is geometrically irreducible if $f$ is irreducible over the algebraic closure ...
- 26.9k
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
3
votes
0
answers
150
views
When is the Fermat Catalan surface a rational surface?
Related to Fermat Catalan conjecture and scholar.google.com didn't return any results.
Define the Fermat Catalan surface
$$ S_{m,n,k}: x^m+y^n=z^k$$
Where $\frac1m+\frac1n+\frac1k < 1$.
Q1 When is ...
- 25.4k
4
votes
0
answers
202
views
Local global principle for a system of polynomial equations
Suppose $T$ be a system of polynomials homogenous of degree 2 solvable over $\mathbb{R}$ and $\mathbb{Q}_p$ for all primes $p$. So, can we claim that $T$ is solvable over $\mathbb{Q}$? I think as of ...
- 159
0
votes
0
answers
96
views
Polynomial sparsity of conductors of elliptic curves
Let $f:\mathbb{N}\to\mathbb{N}$ be the map sending $n$ to the number of isomorphism classes of elliptic curves over $\mathbb{Q}$ with conductor $n$.
Is there a polynomial $P$ such that $P(f(n))>n$ ...
- 183
1
vote
0
answers
67
views
Curves covering elliptic curves with polynomially bounded genus
Let $S$ be the set of isomorphism classes of elliptic curves over $\mathbb{Q}$. Consider the following claim.
There is a map $f:S\to \mathbb{N}$ such that $|f^{-1}(n)|$ is finite for all $n\in \...
- 183
15
votes
1
answer
340
views
Are there only finitely many $m$ such that $m$ is the number of elliptic curves with a given conductor?
Let $f:\mathbb{N}\to\mathbb{N}$ be the map sending $n$ to the number of isomorphism classes of elliptic curves over $\mathbb{Q}$ with conductor $n$.
Is $f(\mathbb{N})$ finite?
- 183
0
votes
0
answers
67
views
Neccessary and sufficient condition for trivial rational solution of rational homogeneous cubic polynomials
If we consider a cubic homogeneous polynomial in $ 5 $ variables , $ ax_{1}^{3} + bx_{2}^{3} + cx_{3}^{3} + dx_{4}^{3} + ex_{5}^{3} + \sum_{i < j<k =1}^{5} f_{ijk} x_{i}x_{j}x_{k} $ where a,b,c,...
- 923
5
votes
2
answers
391
views
Abelian variety with CM defined over real numbers
Is there an abelian variety $A/\mathbb R$ of dimension $n$ such that $End_{\mathbb R}(A)\otimes \mathbb Q$ contains a field $K$ of degree $[K:\mathbb Q]=2n$? ($End_{\mathbb R}(A)$ is the ring of $\...
- 73
2
votes
0
answers
184
views
Points on Galois conjugate curves
Let $C$ be a curve defined over a number field $F$ and let $C^{\sigma}$ be its Galois conjugate obtained by applying $\sigma \in Gal(F/\mathbb{Q})$ to the coefficients defining $C$. What can we say ...
- 81
2
votes
1
answer
474
views
Chevalley–Warning theorem for rational field $\mathbb{Q} $
At 1st we consider some weak statement of Chevalley–Warning theorem for any finite field: If $f$ is a homogeneous polynomial of degree $d$ with $n$ independent variables over a finite field $F$. Then ...
- 923
2
votes
1
answer
106
views
Common roots to "independent" equations $P_1(x) = Q_1(y)$ and $P_2(x) = Q_2(y)$ in $\mathbb{F}_p \times \mathbb{F}_p$
Problem:
let $P_1(x), P_2(x), Q_1(y), Q_2(y)$ be some polynomials of degree $d$ in $\mathbb{F}_p$. Let
\begin{equation}
A := \{ (x, y) \in \mathbb{F}_p^2 : P_1(x) = Q_1(y) \},\\
B := \{ (x, y) \in \...
- 23
4
votes
0
answers
204
views
$\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the poles of $\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts ...
- 563
2
votes
0
answers
480
views
About derived divided power envelope
Assume $A$ is a $\mathbb{Z}_{(p)}$-algebra with ideal $I$ and $A,A/I$ are $p$-torsionfree.
In this survey, Akhil Mathew defines the derived divided power envelope $LD_I(A)$ in Construction 7.15, after ...
- 121
5
votes
1
answer
467
views
Weak Mordell-Weil for EC using Chevalley-Weil theorem
I am reading the book Applications of Diophantine Approximation to Integral Points and Transcendence by Zannier and Corvaja and, after their proof of the Chevalley-Weil theorem, in Example 3.8 they ...
- 233
3
votes
0
answers
311
views
Eichler orders in a certain quaternion algebra
Let us consider a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We shall consider the quaternion algebra $D$ over $K$ such that $D$ splits everywhere at finite places ...
- 781
3
votes
2
answers
382
views
Localization at multivariate monic polynomials
Let $R$ be a ring and consider a monomial order $<$ on $R[X_1,\ldots,X_n]$. A nonzero polynomial $f \in R[X_1,\ldots,X_n]$ is said to be monic if its leading coefficient with respect to $<$ is $...
5
votes
1
answer
377
views
When $p(x)^2 \mid f(g(x))$?
Let $f(x),g(x),p(x)$ be non-constant polynomials with rational coefficients.
Is it true that for all $f$ exist $g,p$ such that $p(x)^2 \mid f(g(x))$?
Partial results:
$f(g(x))$ is divisible by square ...
- 25.4k
4
votes
0
answers
233
views
Structure of $A(L)/A(K)$
Let $L/K$ be an extension of number fields and $A/K$ an abelian variety (or an elliptic curve, or a modular abelian variety).
Then what can we say about the structure of $A(L)/A(K)$ (or of $A(L)_{\...
- 1,364
0
votes
0
answers
326
views
Field extension generated by the roots of multivariate-polynomials
Let us consider a field $K$ of characteristic $0$. Then we know that any finite extension $L$ of $K$, which is a Galois extension as well, is produced the roots of a separable polynomial $f(x) \in K[x]...
- 930
0
votes
1
answer
131
views
Primes of the form $p=u^2+1$ and number of points on the elliptic curve $x^3+a x z^2=y^2 z$
Let $p$ be prime of the form $p=u^2+1$. For $a \in \mathbb{F}_p,a \ne 0$,
define
$E_a : x^3+a x z^2=y^2 z$
Let $B= \lfloor 2 \sqrt{p}\rfloor$
Must we have $(\#E_a(\mathbb{F}_p) -p - 1) \in \{2,-2,B,-B\...
- 25.4k
2
votes
1
answer
323
views
Coprime multivariate polynomials
Let ${\bf R}$ be a gcd domain, $n \geq 2$, $k \in \mathbb{N}^*$, and $f,g \in
{\bf R}[X_1,\ldots,X_n]$. Supposing that $f$ and $g$ are coprime in ${\bf R}[X_1,\ldots,X_n]$, that is, $\gcd(f,g)=1$, ...
5
votes
0
answers
524
views
Generalization of Weil Conjectures
is there a reference in English, besides Deligne's original publication: "La conjecture de Weil: II", not synthetic but complete that deals with the original argument of the generalization ...
- 51
4
votes
0
answers
262
views
de Rham Bloch-Ogus theory in positive characteristic
In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
- 2,044
10
votes
3
answers
1k
views
What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?
I want to calculate the number of solutions to the quadratic equation $$x_1^2+\dots+x_m^2=0$$ where $m$ is odd (a given number) and $x_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given ...
user178596
10
votes
0
answers
481
views
What is the precise definition of "Hypergeometric motives over $\mathbb{Q}$"?
The question is as in the title, but here is some background:
Section 4 of this paper by Beukers, Cohen and Mellit is called "Hypergeometric motive over $\mathbb{Q}$" but no actual (pure) ...
- 10.5k
5
votes
0
answers
225
views
Belyi functions with prescribed image of a given point
$\newcommand{\bP}{\mathbb{P}}\newcommand{\bQ}{\mathbb{Q}}$Definition. A Belyi function is a non-constant rational function $f:\bP_{\bQ}^1\to \bP^1_{\bQ}$ such that the image of any of its critical ...
- 7,377
3
votes
1
answer
312
views
Geometric line bundles on the Tate curve
Let $E_q$ be the rigid analytic Tate elliptic curve over a complete algebraically closed non-archimedean field $K$ of mixed characteristic $(0,p)$, with parameter $q\in K^{\times}$ with $|q|<1$.
...
user178246
1
vote
0
answers
176
views
Motivic cohomology of Weil restriction
hopefully this isn't too obvious or well-known, but I couldn't find it by searching. The motivic cohomology of $\mathbb{G}_m$ and its powers over any base with known motivic cohomology can be computed ...
- 2,044
3
votes
1
answer
152
views
Are there any results on an upper bound for the number of secondary invariants needed to generate the invariant ring of a finite group?
If $ G $ is a finite cyclic group, $ \beta: G \to \operatorname{GL}(\mathbf{V}) $ is a linear $ n $ dimensional representation of $ G $, and $ \{x_{1},\dots,x_{n}\} $ is a basis of $ \mathbf{V}^{\ast} ...
- 782
2
votes
2
answers
399
views
What fraction of polynomials with integer coefficients are indecomposable?
It is well-known that "most" integers are composite: the Prime Number Theorem tells us that only about $1/\log(N)$ of the integers in the interval $1 \ldots N$ are prime. For polynomials, ...
- 1,703
3
votes
0
answers
151
views
Computing the group structure of $J(\mathbb{F}_q)$
Let $k$ be a finite field, $X/k$ a smooth curve, $f$ a polynomial of 2 variables which gives an affine model of $X$ and $J$ its Jacobian.
Then how can I compute $J(k)$?
If $X$ is a hyperelliptic curve,...
- 1,364
2
votes
1
answer
348
views
What is a generic pencil?
In Voisin book "Hodge theory and Complex Algebraic Geometry 2". There is the following corollary
Corollary 2.10. If $X\subset \mathbb{P}^N$ is a smooth projective complex variety, then a ...
- 519
1
vote
0
answers
254
views
Construction of the Hilbert Scheme
I am reading the book "Rational Curves on Algebraic Varieties" of János Kollár. Definition-Proposition 1.2, begin like this:
Let $g:Y\rightarrow Z$ be a projective morphism and $\mathcal{O}(...
- 519
8
votes
2
answers
703
views
Isomorphic endomorphism algebras implies isogenous (for abelian varieties over finite fields)?
$\newcommand{\F}{\mathbb{F}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}$
I would like to know if the following is true:
Proposition A : Let $A_1, A_2$ ...
- 1,742
14
votes
0
answers
821
views
What goes wrong with this alternate proof of Dirichlet's Theorem?
I had an idea for an alternate proof of Dirichlet's theorem, but something goes wrong. Dirichlet's theorem on primes in arithmetic progression says that for $ m,a \in \mathbb{N} $ which are ...
- 782
4
votes
1
answer
309
views
Torsion points on $E/\mathbb{Q}$ with large coordinates
Let $E/\mathbb{Q}$ be an elliptic curve with finitely many rational points.
What are some examples where at least one rational point has large coordinates (compared to the height of $E$)?
- 41
4
votes
0
answers
402
views
Is every Dedekind domain the integral closure of some principal ideal domain?
I mean that $B$ is a Dedekind domain with fraction field $L$, which is a finite separable extension of a field $K$ that is the fraction field of a PID $A$ such that $B$ is the integral closure of $A$ ...
- 1,053