All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
3
votes
1
answer
330
views
Generalization of torsion points on Jacobian of genus 2 over finite fields (with respect to the theta divisor)
Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \...
- 125
58
votes
3
answers
4k
views
What is the geometry of an undecidable diophantine equation?
As an arithmetic algebraic geometer of the highest moral fiber, I am trained to look at Diophantine equations in terms of the geometry of the corresponding scheme. For instance, if the Diophantine ...
- 148k
8
votes
1
answer
855
views
What is the motivation for excellent rings?
First of all I am not formally educated in mathematics so pardon my ignorance if this is obvious and I am skipping something vital, but I am interested nonetheless in what the original motivation and ...
- 81
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
9
votes
2
answers
744
views
Number of solutions mod p and Betti numbers
Suppose $X$ is proper flat scheme over Sepc$\mathbb{Z}$. If $p$ is a large prime, then by Weil conjectures one can recover the Betti numbers of $X$ from the size of $X(\mathbb{F}_{p^n})$ for all $n$. ...
- 123
3
votes
1
answer
184
views
Non-abelian isomorphic absolute Galois groups of fields of different characteristic
Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.
Assume the absolute Galois groups of $K$ and $L$ are non-abelian and isomorphic as profinite groups.
Can $L$ ...
- 55
11
votes
1
answer
3k
views
Books with exercises to learn Langlands program, Galois representations, modular forms
I want to develop a basic background in number theory with a goal towards contributing to some part of the Langlands program (which I know is vast, and has many different aspects; I want to do ...
- 785
3
votes
1
answer
335
views
Comparison of two definitions of the modular sheaf $\omega$
I have seen two equivalent definitions of the modular sheaf $\omega$. Let $S$ be some base scheme. If $p \colon \mathcal{E} \to X$ is the universal generalized elliptic curve over the modular curve $X$...
- 949
8
votes
1
answer
692
views
Why can Euler systems constructed from algebraic cycles only be anticyclotomic?
In a footnote to the 2018 Zerbes-Loeffler lecture notes from the Arizona Winter School, it's stated that Euler systems constructed from algebraic cycles "cannot give a full Euler system, only an ...
- 2,044
11
votes
5
answers
4k
views
How much do I need to learn algebraic geometry to understand arithmetics over number fields
I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to ...
- 1,613
25
votes
3
answers
2k
views
Least number of non-zero coefficients to describe a degree n polynomial
I'd be grateful for a good reference on this, it feels like a classic subject yet I couldn't find much about it.
Polynomials in one variable of the form $x^n+a_{n-1}x^{n-1}+\dots +a_1 x+a_0$ can be ...
- 2,901
7
votes
3
answers
553
views
Two queries on triangles, the sides of which have rational lengths
Let us define a "rational triangle" as one in the Euclidean plane, with lengths of all sides rational.
We are aware that a positive integer is called "congruent" only if it is the area of a right ...
3
votes
0
answers
377
views
Meaning of "the" general fiber in the paper "La conjecture de Weil : I"
In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
Let $X$ be a non singular analytic space and purely of dimension $n+1$....
- 519
6
votes
2
answers
926
views
Motivating the coefficient field of $\ell$-adic cohomology
It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over $\mathbb{F}_p$ would allow for a possible solution to the Weil conjectures.
It was also ...
- 317
4
votes
1
answer
264
views
Reference request: Long exact sequence in profinite Galois cohomology up through $H^2$
I'm looking for a reference of the following statement. Let $G$ be the Galois group of a Galois extension $L/K$, not necessarily finite. Let $A,B,C$ be groups with a continuous $G$-action, and let
$$1\...
- 7,527
6
votes
0
answers
198
views
$\mathbb{Z}$-points in a given $\widehat{\mathbb{Z}}$-isomorphism class
Given a finite type $\mathbb{Z}$-scheme $X$ with $X(\widehat{\mathbb{Z}})\neq\emptyset$ can we find a finite type $\mathbb{Z}$-scheme $Y$ with $X\times \widehat{\mathbb{Z}}\cong Y\times\widehat{\...
user409739
5
votes
0
answers
524
views
Generalization of Weil Conjectures
is there a reference in English, besides Deligne's original publication: "La conjecture de Weil: II", not synthetic but complete that deals with the original argument of the generalization ...
- 51
10
votes
2
answers
1k
views
periodic cyclic homology and tilting in the sense of Scholze
Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost ...
- 7,952
10
votes
0
answers
481
views
What is the precise definition of "Hypergeometric motives over $\mathbb{Q}$"?
The question is as in the title, but here is some background:
Section 4 of this paper by Beukers, Cohen and Mellit is called "Hypergeometric motive over $\mathbb{Q}$" but no actual (pure) ...
- 10.5k
4
votes
1
answer
418
views
Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$)
In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n \geq 3$ integer):
''The scheme $\overline{M}_n - M_n$" over $\mathbb{Z}[1/n]$ is finite and étale, and over $\mathbb{Z}[1/n,...
- 65
20
votes
5
answers
4k
views
Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 671
13
votes
1
answer
760
views
Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
We found infinitely many integer solutions to
$$X^4+Y^4-18Z^4= -16 \qquad (1)$$.
The interesting part in this diophantine equation is the sum of
the reciprocals of the degrees is $3/4 < 1$, which ...
- 25.4k
24
votes
1
answer
3k
views
What do Hecke eigensheaves actually look like?
Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands ...
- 148k
2
votes
0
answers
161
views
Nearby cycle is tamely ramified?
Let $S$ be a henselization of a closed point $s$ in a smooth algebraic curve $C$ over some finite field $\mathbb{F}_q$. Then we can consider nearby cycles over $S$. Let $s$ be the closed point of $S$ ...
- 71
4
votes
0
answers
205
views
Grothendieck group of admissible $p$-adic representations
Let $K$ be a $p$-adic local field; $G = \mathop{\mathrm{Gal}}(\overline K | K)$; $\tau \in \{\text{HT}, \text{dR}, \text{crys}\}$, $B_\tau$ the corresponding period ring; $\mathop{\mathrm{Rep}}_{\...
- 988
6
votes
1
answer
298
views
Weil cohomologies with given field of definition and coefficient field
Fix a perfect field $k$. Fix a field $K$ of characteristic $0$.
A Weil cohomology induces a functor from the category of smooth projective geometrically connected $k$-schemes to the category of $\...
- 61
24
votes
3
answers
3k
views
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in\mathbb Z$ be unbounded as $n$ varies?
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in\mathbb Z$ be unbounded as $n$ varies?
$x,y$ are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of $n$ (...
- 454
31
votes
2
answers
15k
views
A road to inter-universal Teichmuller theory
What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...
- 1,099
3
votes
0
answers
188
views
Does the construction of arithmetic toroidal compactification of $A_{g}$ depend on semistable reduction theorem?
If there is a good theory of arithmetic toroidal compactification over $\mathbb{Z}_{p}$ of the Siegel modular variety with deep enough level structure, then it seems like semistable reduction theorem ...
- 1,024
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
25
votes
1
answer
687
views
Geometry of algebraic curve determined by point counts over all number fields?
Let $C$ be a smooth (geometrically irreducible) projective curve of genus $g>1$ over a number field $K$. The Mordell conjecture (first proved by Faltings) says that for any finite field extension $...
- 2,665
9
votes
1
answer
458
views
Prehomogeneous varieties
A prehomogeneous vector space is a pair $(G,V)$ where $V$ is a finite dimensional $\mathbb{C}$-vector space of dimension $n$ and $G$ is a reductive group of complex dimension $n$, such that $G$ admits ...
- 26.9k
9
votes
1
answer
860
views
Complex manifold defined over $\mathbb{Q}$
If we consider complex projective varieties, to be defined over $\mathbb{Q}$ means that there is a projective embedding whose image is the vanishing locus of a polynomial system with coefficients in $\...
user164740
12
votes
9
answers
6k
views
Proofs of Mordell-Weil theorem
I would like to ask if there exist pedagogical expositions of the Mordell-Weil theorem (wikipedia). What parts of number theory (algebraic geometry) one should better learn first before starting to ...
- 14.3k
31
votes
2
answers
1k
views
The Sylvester-Gallai theorem over $p$-adic fields
The famous Sylvester-Gallai theorem states that for any finite set $X$ of points in the plane $\mathbf{R}^2$, not all on a line, there is a line passing through exactly two points of $X$.
What ...
- 20.8k
10
votes
2
answers
1k
views
Are the ideles literally a Picard group?
I understand that in the number field / function field analogy, the ideles $\mathbb I_K$ of a number field $K$ are supposed to be analogous to the Picard group of a function field.
Question: Is this ...
- 64k
17
votes
4
answers
4k
views
Did Grothendieck write about modular forms?
This question might be astoundingly naive, because my understanding of modular forms is so meek. It occurred to me that the reason I was never able to penetrate into the field of modular forms, ...
- 6,348
0
votes
1
answer
325
views
On the elliptic curve $y^2 = x^3 + z^{4k}$
Are there any rational numbers $x, y, z$ with $xyz \neq 0$ such that $y^2 = x^3 + z^{4k}$ for some $k \in \mathbb{Z}_{>1}$ ?
- 1,019
26
votes
4
answers
1k
views
Variety acquiring rational point over any quadratic extension
Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$?
If ...
- 463
13
votes
1
answer
1k
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Is there a substitution that relates every Fermat curve to an elliptic curve?
I asked this question on MSE but didn't get any response, so I'm asking here. I apologize in advance if this question is not research level.
A Fermat Curve of degree $n$ is the set of solutions to $x^...
- 236
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
19
votes
0
answers
1k
views
Mumford-Tate conjecture for mixed Tate motives
Let $X$ be a (not necessarily smooth or proper) variety over a number field $k$. Suppose we are given
A subquotient $V_{dR}$ of the algebraic de Rham cohomology $H_{dR}^i(X)$ (defined in the non-...
- 23k
4
votes
1
answer
415
views
3-, 6-, 12-descent for Z2xZ6 elliptic curves
We are trying to write a snippet of Magma code to clarify the steps in the simplified procedure of applying $3$-, $6$-, $12$-descent and hopefully resolve the missing generator of the following $\...
- 913
19
votes
1
answer
3k
views
Mazur secret Bourbaki report "Analyse p-adique"
Does anyone happen to know if a scan of Mazur's report exists, and, if so, where to find it? It appears in the references for Katz's "Higher congruences" and "Eisenstein measure" papers.
4
votes
1
answer
282
views
Mean square estimate for the Kloosterman sums
For $m,n\in \mathbb{N}$, denote the Kloosterman sum
$$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$
denotes the multiplicative inverse of $a\bmod c$.
Does ...
- 353
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
13
votes
2
answers
527
views
Naive point count underestimates the number of mod $p$ points of an elliptic curve for infinitely many primes
Let $E$ be an elliptic curve over $\mathbb{Z}[1/N]$ where $N$ is some non-zero integer. Can one show that that the integer $n_p-p-1$ (where $n_p$ is the number of points of $E$ mod $p$) is positive ...
user145520
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
2
votes
2
answers
541
views
A new simple formula is needed
The following question is related to the families of high rank elliptic curves with torsion subgroup $\mathbb{Z}/6\mathbb{Z}$.
The SageMath/Python code below produces a list of small fractions $a$ for ...
- 913
50
votes
5
answers
10k
views
Definition and meaning of the conductor of an elliptic curve
I never really understood the definition of the conductor of an elliptic curve.
What I understand is that for an elliptic curve E over ℚ, End(E) is going to be (isomorphic to) ℤ or an ...
- 5,530