All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
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
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
3
votes
0
answers
443
views
What is the definition of smoothness in scheme theory?
I want to ask if there is a somewhat desirable definition of "smoothness".
Definition. Let $k$ be a field and $X$ be a separated finite type scheme over $k$. Then $X$ is smooth if the ...
user196717
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
12
votes
2
answers
758
views
An isogeny between Jacobians of hyperelliptic curves
Let $\mathbf{F}_q$ be a finite field of odd characteristic. Let $X_t$ be the hyperelliptic curve over $\mathbf{F}_{q^2}(t)$ with affine equation
$$y^2 = \left((x^{(q+1)/2}-(x-1)^{(q+1)/2})^2 - t\...
- 4,487
3
votes
0
answers
312
views
Eichler orders in a certain quaternion algebra
Let us consider a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We shall consider the quaternion algebra $D$ over $K$ such that $D$ splits everywhere at finite places ...
- 781
3
votes
1
answer
244
views
A local-to global principle for splitting of Azumaya algebras
Let $S$ be a finitely generated domain with the field of fractions $F.$ Let X be a smooth,
geometrically connected affine variety over $S.$ Let $A$ be an Azumaya algebra over $X.$
Assume that for all ...
- 33
4
votes
0
answers
204
views
$\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the poles of $\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts ...
- 563
3
votes
0
answers
150
views
When is the Fermat Catalan surface a rational surface?
Related to Fermat Catalan conjecture and scholar.google.com didn't return any results.
Define the Fermat Catalan surface
$$ S_{m,n,k}: x^m+y^n=z^k$$
Where $\frac1m+\frac1n+\frac1k < 1$.
Q1 When is ...
- 25.4k
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
11
votes
3
answers
552
views
When $k = \mathbb{F}_q$ finite field, $X$ always has $k$-rational point, and so $A \simeq X$?
Let $k$ be an arbitrary field. Let $(A, e)$ be an abelian variety over $k$, and let $X$ be a torsor for $A$, i.e. $X$ is a proper smooth $k$-variety, and there is an $A$-action acting $:A \times X \to ...
user97565
14
votes
3
answers
3k
views
Does isomorphic generic fibre imply isomorphic special fibre for smooth morphisms?
Let $X$ and $Y$ be regular integral Noetherian schemes. Assume that $X$ and $Y$ are smooth and proper over a base scheme $S=Spec R$, where $R$ is a discrete valuation ring.
If $X$ and $Y$ have ...
- 21.3k
14
votes
3
answers
666
views
Patterns in solutions to $a^2 + b^2 + c^2 = n$
I have plotted solutions $(a,b,c)$ to $a^2 + b^2 + c^2 = n$ for $12000 \leq n < 12100$, rescaled to $S^2$ and projected onto the first two coordinates. (these are read from the lower left, across ...
- 22.8k
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
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
13
votes
2
answers
1k
views
Galois cohomologies of an elliptic curve
I asked this question on Mathematics Stack Exchange but did not get any answer and I was suggested to post the question here.
I am studying basic theory of elliptic curves. I encountered Galois ...
user46846
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
3
votes
1
answer
153
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
1
answer
227
views
Global section of vertical differential 1 forms on universal elliptic curve
Let $B$ be a modular curve (of some level) over a number field $K$ (here, we implicitly assume that $K$ is large enough to make sense the phrase "$B$ is a $K$-variety"). Let $E\to B$ the ...
- 1,428
19
votes
2
answers
2k
views
What is the relationship between these two notions of "period"?
The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where ...
- 9,061
15
votes
0
answers
673
views
Exposition of Drinfeld's proof of function field Langlands for GL(2)
I know, or think I know, the vague outline of the proof: the Galois-to-automorphic direction is "classical," i.e. follows from converse theorems due to Grothendieck et al., and for the ...
- 311
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
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
214
views
Distribution of the rank of $y^2=x^4+x+b^2$
For positive integer $b$ define the curve $C_b : y^2=x^4+x+b^2$.
$C_b$ is genus one and has the rational points: $(0,\pm b),(-1,\pm b)$
and one more point from the reciprocal of the polynomial y=0
...
- 25.4k
1
vote
0
answers
255
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
9
votes
0
answers
607
views
Geometric meaning of twist
It is sometimes the case that a Galois representation or a motive acquires a desirable property only after a twist by a character, usually a Tate twist. The latest example of this I have come across ...
- 1,267
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
10
votes
1
answer
425
views
When is a twisted form coming from a torsor trivial?
Consider a sheaf of groups $G$, equipped with a left torsor $P$ and another left action $G$ on some $X$. Form the contracted product $P \times^G X := (P \times X)/\sim$ where $\sim$ is the ...
- 1,094
37
votes
3
answers
2k
views
Unexpected applications of transcendental number theory?
In the last pages of "Equations Différentielles à points singuliers réguliers", Deligne provides a proof, attributed to Brieskorn, of the so-called local monodromy theorem (on the quasi-unipotence of ...
user85435
37
votes
7
answers
8k
views
Model theoretic applications to algebra and number theory(Iwasawa Theory)
One of my favorite results in algebraic geometry is a classical result of AX (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/) I'll recall ...
- 2,396
4
votes
0
answers
277
views
Explicit computations of the fundamental groups of perfectoid spaces
If $X$ is a perfectoid space then it has the same étale site as its tilt $X^\flat$. This means that the fundamental groups (suitably defined) of $X$ and $X^\flat$ are isomorphic.
Can you give ...
- 41
12
votes
2
answers
1k
views
How to visualize finiteness of class number?
As the question title asks for, how do others "visualize" the finiteness of class number with algebro-geometric insight? I just think of it as a result in algebraic number theory and not one in ...
2
votes
1
answer
303
views
Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases?
Help me please.
Let $k$ be an algebraically closed field (I am mainly interested in $k = \overline{\mathbb{Q}}, \overline{\mathbb{F}_q}$). Consider a plane curve $C \subset \mathbb{A}^2$ of degree $d$ ...
- 1,833
5
votes
1
answer
213
views
Can the strongest Hensel lemma over integer rings imply smoothness over $\mathbb Z_p$?
Let $X \rightarrow \mathbb Z_p$ be a flat finite type morphism, with reduced special fiber and smooth generic fiber.
Assume $X(O_K) \rightarrow X(O_K/m_K)$ is surjective for all fintie extension $K$ ...
- 461
6
votes
2
answers
1k
views
About different cohomology theories used to study Shimura varieties
The classical theory of Eichler-Shimura realizes the space of cusp forms of certain weights and levels in the parabolic cohomology of modular curves, which is the image of the cohomology with compact ...
- 159
3
votes
1
answer
356
views
Understanding the implementation of the $p$-adic(?) sigma function in SageMath
I'm trying to understand the (pretty undocumented) method .sigma() method for formal groups of elliptic curves, as listed here. The source code looks like this:
<...
- 2,044
5
votes
0
answers
328
views
Ramification behavior of field given by adjoining $p$-torsion point on formal group of abelian variety
Setup. Let $p > 2$ be a prime, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and fix an algebraic closure $\overline{K}$ of $K$. Let $A/K$ be an abelian variety ...
- 998
29
votes
3
answers
3k
views
how to find the varieties whose cohomology realizes certain representations?
The cohomology of Shimura varieties and Drinfeld shtukas is conjectured to realize the representations sought for in the Langlands programme/conjectures, the cohomology of Deligne-Lusztig varieties ...
user19475
12
votes
3
answers
1k
views
Chow Groups of varieties over number fields
I believe that there is a conjecture that for any smooth projective variety $X$ over a number field $K$, its Chow groups $CH^i(X)$ (or at least $CH^i(X)\otimes_{\mathbf Z} \mathbf Q$) are finitely ...
- 2,923
6
votes
1
answer
465
views
Etale fundamental group of an order in a number field
Let $\mathcal{O}$ be an order in a number field $K$, that is a subring of $K$ with rank as abelian group equal to $[K:\mathbb{Q}]$. What is known about the SGA3-étale fundamental group of $X=\mathrm{...
- 617
32
votes
2
answers
2k
views
Etale cohomology can not be computed by Cech
It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H_{et}^{\bullet}}(X,\...
- 7,377
4
votes
0
answers
289
views
Formal integration (?) in Chabauty’s method
In Mccallum, Poonen’s paper “The method of Chabauty and Coleman”,
the authors define, for the Jacobian $J$ of a geometrically connected smooth proper curve over the rational field and for $\omega \in ...
- 1,364
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
2
votes
1
answer
187
views
Is every sufficiently general monic quartic rational square infinitely often?
Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$.
Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$.
$C$ is birationally equivalent to $f(x)=y^2$.
The constant coefficient of $g(x)$ is $1$ since $f$ is monic
and $(...
- 25.4k
48
votes
6
answers
5k
views
Smooth linear algebraic groups over the dual numbers
It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, ...
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
20
votes
2
answers
2k
views
Tate's definition of residues
In http://www.numdam.org/article/ASENS_1968_4_1_1_149_0.pdf, Tate defines residues on a curve over an arbitrary field as a trace of some commutator. What is the intuition for the definition? If I knew ...
- 431
6
votes
1
answer
652
views
$l$-adic periods?
For an algebraic variety $X$ over $\mathbb{Q}$ the comparison isomorphism between Betti and de Rham cohomologies provides the theory of periods with a motivic context whose reformulation as motivic ...
- 61
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
26
votes
1
answer
4k
views
Underlying structure behind the infamous IMO 1988 Problem 6
This is the infamous Problem 6 from the 1988 IMO which has recently been popularised by the YouTube channel Numberphile:
Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^{2} + b^{...
user117786