All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
0
votes
1
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323
views
An identity for Weierstrass elliptic functions evaluation
Let $\wp(z), \zeta(z)$ and $\sigma(z)$ be the Weierstrass $\wp$, zeta and sigma functions associated to the ODE:
$$\wp'(z)^2 = 4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3)$$
and we assume $e_1=\frac{2-c}3>...
- 43.2k
75
votes
4
answers
5k
views
What are reasons to believe that e is not a period?
In their 2001 paper defining periods, Kontsevich and Zagier (pdf) without further comment state that $e$ is conjecturally not a period while many other numbers showing up naturally (conjecturally) are....
- 2,493
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
0
votes
0
answers
107
views
Cubic monic polynomial over z_p
Let
$$
f_{a}(x)=x^3+(u-2-a)x^2+ax+1,
$$
where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all $a\in\mathbb{Z}_p$ such that $f_{a}(x)$ factor linearly. Then what is the cardinality ...
user126352
5
votes
0
answers
212
views
Relations between Whittaker functions/W algebras and Stokes data/resurgence
Skippable background: A Whittaker function is more or less a function on a flag manifold which is twisted-invariant for the action of a unipotent subgroup. E.g. consider functions $f$ on $\mathbf{P}^1$...
- 5,701
1
vote
0
answers
210
views
What is a definition of $A(P_v)$ in the definition of Brauer-Manin obstruction?
This is a question related to the definition of Brauer-Manin obstruction.
Let $K$ be a number field. $X/K$ be an algebraic variety over $K$.
Let $O_{X,P}$ be a local ring of $X$ at $P$. Let $Br(X)=\...
- 1,541
4
votes
1
answer
442
views
Trigonometric Diophantine equation
Is there a general method to solve the equation $P(x_1,x_2,...,x_n)=0$ with $P$ is a polynomial in $n$ variables with integer coefficients and $x_k=\cos(q_k\pi)$ with $q_k$ is a rational number?
This ...
- 1,462
1
vote
0
answers
182
views
Crystalline exact sequence in Galois cohomology
Let $G$ be the absolute Galois group of $\mathbb{Q}_p$, and let $1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$ be a short exact sequence of (non-abelian) algebraic group ...
- 2,907
33
votes
4
answers
6k
views
Is there any theory why (for Bitcoin) the discrete logarithm problem is so hard to solve?
Note I am an active member and contributor at the sister site https://bitcoin.stackexchange.com while studying Bitcoin and as a person who studied mathematics 10 years ago there is one thing I kept ...
- 439
8
votes
1
answer
2k
views
Some questions from the paper by Scholze-Weinstein
The following is from the paper by Scholze-Weinstein on moduli of $p$ divisible groups.
My question is from a part of Lemma 4.1.7: If $R$ is a semiperfect ring, then the canonical map $W(R^{\flat}) \...
- 81
6
votes
0
answers
629
views
On the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties"
I am trying to understand section (3) of the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties" in detail. In particular, there is the following sentence on page ...
- 507
3
votes
0
answers
131
views
Rational points on a cubic surface with small coefficients
Do there exists integers $(x,y,z,t)\neq (0,0,0,0)$ such that
$$
2x^3+2y^3+z^3+t^3+2x^2y-2z^2x-y^2z-z^2t = 0 ?
$$
A short motivation: there are many known counterexamples to the Hasse principle for ...
- 7,182
1
vote
1
answer
315
views
About simple motives
I'm reading through Jannsen's paper Motives, numerical equivalence, and semi-simplicity and I'd like to pose two questions.
Suppose all motives are $F$-linear, for some characteristic zero field $F$, ...
user497064
2
votes
0
answers
149
views
Order $4$ element of Tate-Shafarevich group
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Tate-Shafarevich group $\mathit{Sha}(E/\Bbb{Q})$ is defined as follows.
$$\mathit{Sha}(E/\Bbb{Q})\stackrel{\text{def}}{=} \operatorname{Ker}...
- 1,541
3
votes
0
answers
280
views
The closed unit adic disk
I am reading the Scholze-Weinstein Berkeley lecture notes on "Perfectoid Spaces", and in particular I am stuck trying to understand the closed adic unit disk, which is the second example of ...
- 2,907
2
votes
1
answer
164
views
What is the sum operation on torsors induced by Weil uniformization?
Let $k$ be an algebraically closed field, $G$ a reductive group, and $C$ a curve. The algebraic version of the Weil uniformization theorem (see e.g. arXiv:1511.06271v2) says that groupoid of $G$-...
- 1,969
5
votes
1
answer
430
views
Where can I find that Weil suggested a cohomology theory for characteristic $p>0$?
I have seen that in Grothendieck's paper "THE COHOMOLOGY THEORY OF ALGEBRAIC VARIETIES", he says "The need of a theory of cohomology for 'abstract' algebraic varieties was first ...
- 121
7
votes
1
answer
421
views
Integral refinements of rigid cohomology
Disclaimer: I know absolutely nothing about p-adic cohomology, so it is possible that even the premises of this question are incorrect. But it turns out that I need to apply the theory of rigid ...
- 5,958
4
votes
0
answers
135
views
Analog of a theorem on equidistribution in adeles
Is there a reference anywhere for the analog of Theorem 6 in chapter XV of Langs Algebraic Number Theory for global function fields?
In my research I have been using this theorem to prove density ...
- 337
3
votes
0
answers
176
views
Rational points on curves x^p + y^p = 1 in the open unit square, for p > 0
For each real p > 0, let Cp = {(x,y) ∊ (0, 1)2 | xp + yp = 1}.
It is easy to see that each Cp is topologically an open interval that is the interior of a smoothly embedded closed interval with ...
- 2,858
12
votes
2
answers
2k
views
how do we prove that a sum of two periods is still a period?
Kontsevich and Zagier define periods as the values of absolutely convergent integrals $\int_\sigma f$ where $f$ is a rational function with rational coefficients and $\sigma$ is a semi-algebraic ...
- 129
28
votes
1
answer
2k
views
SOS polynomials with integer coefficients
A well known theorem of Polya and Szego says that every non-negative univariate polynomial $p(x)$ can be expressed as the sum of exactly two squares: $p(x) = (f(x))^2 + (g(x))^2$ for some $f, g$. ...
- 1,703
2
votes
0
answers
96
views
Cryptography signature scheme based on hardness of finding points on varieties?
Related to this question Complexity of finding solutions of trapdoored polynomial.
I am trying to build signature scheme based on hardness
of finding points on varieties.
Let $K$ be field and $M=K[x_1,...
- 25.4k
3
votes
1
answer
249
views
Completely reducible subgroups over local field in terms of closed orbits
$\DeclareMathOperator\GL{GL}$Let $ \overline{\mathbb{Q}}_{p} $ be an algebraic closure of $ p $-adic numbers $ \mathbb{Q}_{p} $. A closed subgroup $ H $ of a general linear group $ \GL_{n}(\overline{\...
- 667
2
votes
1
answer
207
views
The reduction map on the $\ell$-primary torsion of abelian variety
Let $A/\mathbb{Q}$ be an abelian variety with good reduction at a prime $p$. Assume $\mathcal{A}/\mathbb{Z}_{(p)}$ is an integral model at $p$(hence proper smooth).
For any number field $K$ and any ...
- 547
1
vote
0
answers
98
views
Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
- 25.4k
6
votes
1
answer
438
views
$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves
Do there exists rational numbers $x$ and $y$ such that
$$
y^3 = x^4 + x + 2 ?
$$
Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
- 7,182
23
votes
2
answers
1k
views
Theta functions on an elliptic curve and Serre duality
Given an elliptic curve $E$ (over $\mathbb{C}$) and line bundle $L$, one can identify $H^0(E,L)$ with a particular space of theta functions.
Serre duality gives a perfect pairing between $H^0(E,L)$ ...
- 411
42
votes
3
answers
5k
views
The Origin(s) of Modular and Moduli
In mathematics and in physics, people use the terms "modular..." and "moduli space" very often. I was puzzled by the etymology, the origins and the similarity/equivalence/differences for these usages/...
- 10.5k
2
votes
0
answers
87
views
Complexity of finding solutions of trapdoored polynomial?
Related to this question Cryptography signature scheme based on hardness of finding points on varieties.
Working over $K=\mathbb{Q}[x_1,...,x_n,y_1,...y_m]$.
By abuse of notation, for polynomial $f$, ...
- 25.4k
1
vote
2
answers
349
views
Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
- 7,182
7
votes
0
answers
259
views
K3 surfaces with no −2 curves
I seem to remember that a K3 surface with big Picard rank always
has smooth rational curves.
This question is equivalent to the following question about integral quadratic lattices. Let us call a ...
- 9,237
1
vote
0
answers
88
views
Is there a definition of lcm of ideal sheaves?
Let $X$ be a scheme, let $\text{Div} (X)$ be its Weil divisors. For two Weil divisors $D,E$ we can easily define their gcd and lcm:
$$\gcd(D,E)=\sum\min\{\text{ord}_F D,\text{ord}_F E\}F,\\
\text{lcm}(...
- 143
55
votes
3
answers
11k
views
What is precisely still missing in Connes' approach to RH?
I have read Connes' survey article http://www.alainconnes.org/docs/rhfinal.pdf
and I am somewhat familiar with his classic paper on the trace formula: http://www.alainconnes.org/docs/selecta.ps
Very ...
- 1,338
0
votes
0
answers
179
views
Points at which a polynomial becomes reducible
Let $n \geq 10$ and set $\mathbf{y} = (y_1,\ldots,y_n)$. Let $Q_1(\mathbf{y}),\ldots,Q_5(\mathbf{y})$ be non-zero quadratic forms with integer coefficients such that the cubic form $x_1Q_1(\mathbf{y})+...
- 105
0
votes
0
answers
138
views
Is the set of common-zeros of the systems of polynomials Zariski dense?
Let $K$ be an algebraically closed field of characteristic $0$.
Let $f: K^m \to K^m$ be a map defined by $f(x_1,x_2,\cdots,x_m)=(f_1,f_2, \cdots,f_m)$ where $f_i \in K[x_1,x_2, \cdots, x_m]$. Let $Z$ ...
- 930
6
votes
1
answer
543
views
Chebotarev density theorem and pure weight local systems
How do we deduce the following statement from the Chebotarev density theorem? The statement is from Ngo's Fundamental Lemma paper.
Let $U$ be a scheme of finite type over $\mathbb{F}_q$. Let $\...
- 677
7
votes
0
answers
296
views
Connections between Borger's absolute geometry and Connes' and Consani's $\Gamma$-spaces
As the idea of an absolute geometry over the field with one element $\mathbb{F}_1$ becomes more clear, two approaches seem to have crystallized, being based on different assumptions and going into ...
-1
votes
1
answer
342
views
Finite or polynomial number integral points clarification on Coppersmith's theorems (possibility of ellipse counter example?)
Coppersmith states if $f(x,y)$ is an irreducible bivariate with total degree $\delta$ then if he can list all roots $(X,Y)$ of the polynomial in $\mathsf{poly}(\log D,\delta)$ time if the roots ...
- 13.9k
3
votes
1
answer
470
views
Why does the Manin-Mumford conjecture over number fields imply the conjecture over arbitrary fields of characteristic 0?
The Manin-Mumford conjecture states that for an abelian variety A over a field F of characteristic 0 the torsion points are dense in an integral closed subvariety Z if and only if it is an abelian ...
- 193
1
vote
0
answers
162
views
Difficulty understanding a step in the proof of multiset version of Cauchy-Davenport Theorem
In a paper "G. Kós, L. Rónyai, Alon’s Nullstellensatz for multisets, Combinatorica, 32(5) (2012) 589-605", the authors prove a multiset version of the Cauchy-Davenport Theorem (please see ...
- 167
4
votes
0
answers
225
views
The coarse moduli schemes of the "Shimura stacks" are the canonical models of the corresponding Shimura varieties
Let $F$ be a number field, $B$ a central simple algebra over $F$, $*$ a positive involution on $B$ which fixes $F$, and
$O_B$ a maximal $O_F$-order of $B$ which is stable under $*$.
Assume that $(B, *)...
- 1,364
14
votes
2
answers
567
views
Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?
Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
- 1,353
75
votes
5
answers
3k
views
When the automorphism group of an object determines the object
Let me start with three examples to illustrate my question (probably vague; I apologize in advance).
$\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M,...
7
votes
1
answer
916
views
Why are Shimura varieties the "right" objects?
So this is probably blasphemist to ask and I've resisted asking this for a while. Essentially my question is why are locally symmetric spaces/Shimura varieties the "right" object to study ...
- 4,418
6
votes
1
answer
512
views
Mordell conjecture over function fields
So I've read (for instance in the introduction to R.S de Jong's thesis ) that the naive adaptation of the proof of the Mordell conjecture over function fields fails, even using Arakelov intersection ...
- 4,418
7
votes
2
answers
605
views
ℤ/18ℤ elliptic curves over cubic fields
I am working on $\mathbb{Z}/18\mathbb{Z}$ elliptic curves over cubic fields. The curves are created using the formulas on p. 584 of
D. Jeon, C. H. Kim, Y. Lee, Families of elliptic curves over cubic ...
- 913
2
votes
1
answer
571
views
What goes wrong with this proof of Dirichlet's Theorem?
I am curious what goes wrong with this invariant theoretic argument of Dirichlet's Theorem of primes in arithmetic progression. I know that something goes wrong, but I am really curious about what ...
- 912
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
0
votes
0
answers
182
views
Does a $p$-adic power series $F(x,y)=\sum_{i,j \geq 0}b_{ij}x^iy^j \in \mathbb Z_p[[x,y]]$ have finitely many zeros in $\mathfrak{m}_{\mathbb C_p}$?
Let us consider the $p$-adic field $\mathbb Q_p$ with ring of integers $\mathbb Z_p$ and maximal ideal $\mathfrak{m}$.
Then any $p$-adic power series $f(x)=\sum_{n>0}a_nx^n \in \mathbb Z_p[[x]]$ ...
- 930