All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
4
votes
1
answer
444
views
Surfaces of general type
What methods do we know about proving-disproving existence of rational points on surfaces of general type?
I was recently asked. My gut answer was- 'nothing'.
user95222
4
votes
1
answer
765
views
Conic bundles on quartic del Pezzo surfaces
I'm trying to understand better conic bundles on quartic del Pezzo surfaces over non-algebraically closed fields.
Let $k$ be a field. A conic bundle surface is a smooth projective surface $S$ over $k$...
- 21.3k
11
votes
1
answer
1k
views
Is Proposition 2.6 in J. Silverman's book Arithmetic of Elliptic Curves correct?
In J. Silverman's book "Arithmetic of Elliptic Curves" Chapter 2 Proposition 2.6 (a) it is considered a non constant morphism $\Phi:C_{1}→C_{2}$ between two smooth curves defined over a perfect field $...
- 675
14
votes
0
answers
1k
views
Status of the "anabelian dream" ($\mathrm{dim} \leq 1$)
The anabelian conjectures for small dimensions have been known for quite some time. In full generality the results are:
Dimension 0. Finitely generated fields are anabelian (Pop)
Dimension 1. ...
- 17.6k
9
votes
0
answers
262
views
Injectivity of map in Beilinson's conjectures
In Beilinson's conjectures on special values of L-functions, he uses the image of the motivic cohomology of a a regular proper model in the motivic cohomology of the generic fiber to state the ...
- 1,553
29
votes
1
answer
4k
views
What was achieved on IUT summit, RIMS workshop? [closed]
I would like to know what was achieved in the workshop towards the verification of abc conjecture's proof and the advance of understanding of IUT in general.
A comment from a participant:
C ...
3
votes
1
answer
527
views
Selmer and free rank of Elliptic Curves
If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal.
This is one of the many implications of the Birch and Swinnerton-Dyer ...
- 1,805
1
vote
1
answer
329
views
Kummer extension of Galois modules
Let $k$ be a field of characteristic $p \geq 0$, $n$ an integer prime to $p$, and $x$ an element of $k \setminus \{0, 1\}$. I have read that the $n^{th}$ root of $1-x$ gives rise to a Galois module $E$...
- 11
2
votes
0
answers
87
views
Rank growth in ray class fields of primes that are inert in an imaginary quadratic extension
If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that:
(1.) $\...
- 1,805
1
vote
0
answers
86
views
Counting points in a certain 4-dimensional region
Let $(a,b,c)$ be a fixed tuple of co-prime integers, with $a \ne 0$ and at least one of $b,c$ non-zero. Define
$$L = -\frac{a p_1 q_1 - b p_2 q_1 - b p_1 q_2 + 4 c p_2q_2}{a},$$
$$Q = \frac{(b^2 - ...
- 26.9k
5
votes
1
answer
702
views
How to construct an abelian variety with CM by a given CM field?
Let $F$ be a totally real number field, and let $K$ be a quadratic extension of $F$ which cannot be embedded into $\mathbb{R}$.
Then $K$ is a so called CM field.
For instance, take $F = \mathbb{Q}(\...
- 11.3k
17
votes
1
answer
852
views
Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?
Fix $N\ge4$. Let $Y_1(N)$ and $X_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (...
- 734
1
vote
0
answers
179
views
Spivakovski-Popescu-Neron desingularisation
For $A \colon= {\Bbb F}_p[[X_1,...,X_d]]$, by generalising Popescu-Neron's method, Spivakovski proved that $A$ is written by smooth sub-algebras. That is,
$A \cong \underset{\lambda \in \Lambda}{\...
- 781
8
votes
1
answer
1k
views
Semisimplicity of Frobenius on *integral* Tate module
Let $K$ be a number field and $A/K$ an Abelian variety; let $l$ be a (rational) prime. Do there exist infinitely many primes $\mathfrak{p}$ of $K$ such that the Frobenius at $\mathfrak{p}$ acts ...
- 23k
14
votes
1
answer
1k
views
Rational curves on the Fermat quartic surface
Let $X$ be the Fermat quartic $x^4+y^4+z^4+w^4=0$ in $\mathbb P^3$. It is known that $X$ contains infinitely many $(-2)$-curves, that is, smooth rational curves. (One way to obtain in infinitely many ...
- 666
10
votes
1
answer
529
views
Polylogarithm sheaves
In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{...
- 1,428
2
votes
2
answers
640
views
An example of "all non-torsion rational points on an elliptic curve are integral points''?
For an elliptic curve $E$ over $\mathbb{Q}$, it is well-known that the torsion points on $E$ are integral points.
Then, is it possible that there exists an example whose all of non-torsion rational ...
- 342
15
votes
2
answers
1k
views
Can you use Chevalley‒Warning to prove existence of a solution?
Recall the Chevalley‒Warning theorem:
Theorem. Let $f_1, \ldots, f_r \in \mathbb F_q[x_1,\ldots,x_n]$ be polynomials of degrees $d_1, \ldots, d_r$. If
$$d_1 + \ldots + d_r < n,$$
then the ...
- 28.7k
6
votes
0
answers
463
views
Lifting points via étale morphism of adic spaces
This question was suggested to me during the reading of Huber's book about Etale Cohomology of Adic Spaces. I formulate this question here in the context of adic spaces, but I think, since a morphism ...
- 181
7
votes
1
answer
1k
views
Ordinary primes vs supersingular primes
Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. As shown by Serre, the set of supersingular primes for $E$ has density zero.
Is the analytic rank of $L(E,1)$ determined only by the ...
- 342
1
vote
0
answers
65
views
Sieving for points not on a conic and quadratic residues
Let
$$\displaystyle F(x,y,z) = \sum_{\substack{0 \leq i_1, i_2, i_3 \leq 2 \\ i_1 + i_2 + i_3 = 2}} a_{i_1, i_2, i_3} x_1^{i_1} x_2^{i_2} x_3^{i_3}$$
be a ternary quadratic form with integer ...
- 26.9k
2
votes
1
answer
733
views
Are there analogies between $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\Bbb Z$?
Much progress in understanding $\Bbb Z$ is made from analogies between $\Bbb F_q[x]$ and $\Bbb Z$.
Can there be analogies between arithmetic in $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\...
user76479
11
votes
1
answer
2k
views
Quasi-split tori and algebraic groups
Let $k$ be a perfect field.
Recall that an algebraic torus $T$ over $k$ is called quasi-split if there exists some finite étale $k$-algebra $A$ such that
$$T \cong \mathrm{R}_{A/k} \mathbb{G}_m.$$
A ...
- 21.3k
3
votes
1
answer
434
views
Tate modules of elliptic curves with complex multiplications
Let $E/K$ be an elliptic curve with complex multiplication
over an imaginary quadratic field $K$. Then, I heard that
it is well-known that the Tate module $V_{p}(E)$ over
$\mathbb{Q}_{p}$ ...
- 31
0
votes
2
answers
178
views
Products of varieties of index 1
Let $k$ be a field of characteristic 0 and let $X$ and $Y$ be smooth, projective and geometrically integral $k$-schemes of finite type. Assume that both $X$ and $Y$ have 0-cycles of degree 1. Does $X\...
0
votes
0
answers
283
views
Normalizer of non-split tori
Let $\mathbb{G}$ be a connected reductive group over $\mathbb{C}$. Let $G:=\mathbb{G}(\mathbb{C}(\!(t)\!))$. Let $T$ be a maximal torus in $G$.
Question: What do we know about the normalizer $N_G(T)$...
- 2,751
8
votes
0
answers
980
views
What is an example of a non-mixed $\ell$-adic sheaf?
$\def\FF{\mathbb{F}}\def\cG{\mathcal{G}}\def\QQ{\mathbb{Q}}\def\CC{\mathbb{C}}$I've been attending a reading seminar at Michigan on Kiehl and Weissauer's book Weil conjectures, perverse sheaves and l’...
- 156k
12
votes
5
answers
2k
views
Clarification on the weak BSD conjecture
It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to ...
user85435
8
votes
1
answer
308
views
Algebraic points of uniformly bounded degree on an algebraic variety
Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$.
Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$.
Does there exist a natural number $d=d(\...
- 14.2k
18
votes
1
answer
881
views
Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$
I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)=\mathrm{Aut}\underline\pi^{temp}_{{\mathbb{C}}_p}$), ...
- 17.6k
9
votes
1
answer
617
views
Characters of simply connected semsimple algebraic groups over local fields
Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$.
However, it is quite possible ...
- 21.3k
2
votes
0
answers
294
views
"Algebrazing" canonical subgroups of elliptic curves
I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but ...
- 845
4
votes
2
answers
792
views
Adjoint semi-simple algebraic groups over non-algebraically closed fields
Let $k$ be a field of characteristic zero and let $G$ be an adjoint semi-simple algebraic group over $k$.
On p34 of the paper "Sansuc - Groupe de Brauer et arithmétique des groupes
algébriques ...
- 21.3k
9
votes
2
answers
532
views
Neron models and ramification
I encountered this result while reading a few things, and it was stated without reference. I am having a hard time finding a reference for it (or a simple proof), so maybe you can help me:
let $E$ be ...
- 203
3
votes
1
answer
326
views
Connected-étale sequence for ordinary CM elliptic curves
Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$.
Suppose that $p$ is ...
- 845
6
votes
3
answers
912
views
Hypersurface missing just one point
Let $\mathbb F_q$ be a finite field and $n$ an integer.
What is the minimal degree $d = d(q,n)$ of a polynomial $f \in \mathbb F_q[X_1,\dots,X_n]$ such that the set $Z(f)$ of zeros of $f$ in the ...
- 26k
5
votes
2
answers
1k
views
BSD and generalisation of Gross-Zagier formula
The classical Gross-Zagier formula and the modularity theorem leads to a proof of half-BSD (i.e. an inequality and not equality) for elliptic curves of analytic rank 0.
The Gross-Zagier formula gives ...
- 329
11
votes
0
answers
2k
views
Is it worth the efforts to read books/papers written in Weil's algebraic geometry language
There is much important work written in Weil's language of algebraic geometry rather than schemes (besides Weil himself, I can think of Shimura, Neron immediately).
My question is: is it worth the ...
- 111
9
votes
2
answers
2k
views
Langlands program vs Shimura-Taniyama-Weil conjecture
Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves"
I hope I'm not distorting his phrase, can someone ...
- 1,613
1
vote
0
answers
485
views
On the coherence of formal power series ring
Let $A = {\Bbb F}_p[[X_1,X_2,...]]$
be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$
$A$ consists of such formal sum elements as $\sum c_{e_1,.....
- 781
3
votes
1
answer
274
views
Number of rational points in a non-smooth variety
Let $X$ be an algebraic variety over $\mathbb{F}_q$ with dimensional $n$. We know that if $X$ is smooth than $X$ has about $q^{nk}$ rational points over $\mathbb{F}_{q^k}$ (Weil hypothesis). Is there ...
- 2,576
3
votes
0
answers
408
views
Finding the number of rational points effectively
Consider $\# P$ and $\oplus P$.
There is a $\# P$-hard problem: to find number of rational solutions of a system of polynomial equations over $\mathbb{F}_2$. The corresponding $\oplus P$-hard ...
- 2,576
1
vote
0
answers
86
views
Are there only finitely many fixed degree nontrivial polynomial parametrizations of the surface $x^4+y^4=z^4+t^4$?
Consider the surface over the rationals
$$ x^4+y^4=z^4+t^4 \qquad (1)$$
Consider parametrizations of form:
$$ f_1(u)^4+f_2(u)^4=f_3(u)^4+f_4(u)^4 \qquad(2) $$
where $f_i$ are polynomials with integer ...
- 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
4
votes
0
answers
275
views
Equations for Elliptic Curves
An elliptic curve $C$ over a field $k$ is a smooth, genus 1 curve defined over $k$ with an associated $k$-rational point. If char$(k) \ne 2$, we can show that $C$ has a model of the form $y^2 = f(x)$ ...
- 109
3
votes
0
answers
312
views
Comparison of algebraic and analytic q-expansion
I would like to check that algebraic and analytic q-expansion of a modular form coincide.
I'm thinking about modular forms as global sections of some sheaf on modular curves. If $X$ is a modular ...
- 845
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
5
votes
1
answer
473
views
A question regarding lines on a cubic surface
Let $X$ be a smooth cubic surface in $\mathbb{P}^3$. It is a classical theorem of Cayley and Salmon that $X$ contains exactly 27 lines over an algebraically closed field.
In 2002, Heath-Brown proved ...
- 26.9k
1
vote
0
answers
82
views
Reference request: Structure of $H^1({{\mathbf{Q}}_{q}},{{{{E}}_{{p}^{\infty}}}})$
I need reference on the structure of ${H}^{1}({{\mathbf{Q}}_{q}},{{E}_{{{p}^{\infty}}}})$, in particular when:
(1.) $q=p$
and/or
(2.) $E$ has multiplicative reduction at $q$.
Here, $E$ is an ...
- 1,805
2
votes
1
answer
165
views
Local nontriviality of genus-one curves over extensions of degree dividing $6^n$
Suppose $p\geq 5$ is a prime, and $C$ a genus-one curve, defined over $\mathbf{Q}$. Is there always an extension $K/\mathbf{Q}_{p}$ whose degree divides a power of $6$, so that
$C(K)$ is not empty?
(I ...
- 1,805