All Questions
Tagged with nt.number-theory ag.algebraic-geometry
196 questions
6
votes
1
answer
486
views
Strengthened supercongruences for Ramanujan-type formulas for $1/\pi^k$
The question below is again a follow-up of an old question.
Motivation: Zhi-Wei Sun listed a number of supercongruences attached to Ramanujan-type $1/\pi$ formulas in the arXiv paper which can be ...
- 3,337
6
votes
0
answers
437
views
Are there infinitely many integer solutions to $a^4+b^4-c^4=N$?
Is there non-zero integer $N$ such that
$$ a^4+b^4-c^4=N \qquad (1)$$
has infinitely many integer solutions $(a,b,c)$ with $a,b \ne \pm c$?
(1) is a surface, so possible approach is to find genus 0 ...
- 25.4k
6
votes
1
answer
566
views
Public key cryptography based on non-invertible matrices?
Added Wed 13 Apr 2022
I have written a short note with experimental data,
which shows not all pseudo keys are good keys.
Public key cryptography based on non-invertible matrices
We got public key ...
- 25.4k
6
votes
0
answers
434
views
Local Fontaine--Mazur?
Given a finite extension $K$ of $\mathbb{Q}_p$, is there some conjectural statement characterizing which finite-dimensional $p$-adic representations of the absolute Galois group of $K$ are (Tate ...
user138661
6
votes
1
answer
821
views
Does there exist a discrete valuation subring $R$ of $K((t))$ ($K$ a number field) of residue characteristic $p$ with $\mathrm{Frac}(R) = K((t))$?
Let $K$ be a number field, and let $K((t))$ be the field of formal Laurent series. Let $p > 0$ be a prime.
I have two questions:
Does there exist a discrete valuation subring $R$ of $K((t))$ of ...
- 10.7k
6
votes
1
answer
367
views
Lifting of Frobenius on torsors over abelian varieties
This is related to my previous question Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and $\mathcal{L}...
- 7,377
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
5
votes
3
answers
448
views
Variation of centraliser in $\operatorname{GL}(n,\mathbb{Z})$
$\DeclareMathOperator\GL{GL}$Let $n$ be a positive integer $\geq 2$. The setting is that $K \in \GL(n,\mathbb{Z})$, and people are interested in understanding the centralizer:
$$
C(K)=\{ B \in \GL(n,\...
- 145
5
votes
2
answers
250
views
What integer value can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$?
Fix a positive integer $g$. What positive integer $N$ can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$ ?
For example, as there is no abelian varieties over $\mathbb Z$, $N$ ...
- 6,622
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
5
votes
1
answer
1k
views
Misunderstanding of Hodge conjecture
I ask a question on math stack exchange about Hodge conjecture and have not got any reply or comment.
https://math.stackexchange.com/questions/2704988/clarify-hodge-conjecture
so I decide to post ...
- 2,971
4
votes
1
answer
354
views
Can a general quintic be solved using inverse beta regularized function?
Tyma Gaidash has recently posted solutions to some quintics in terms of Inverse Beta Regularized function. He also found the closed form for the equation $\cos x=x$ using the same Inverse Beta ...
- 10.1k
4
votes
2
answers
1k
views
Fricke groups and Fricke curves
The congruence subgroup $\Gamma_{0}(n) \subset PSL_{2}(\mathbb{Z})$ is normalized by the Fricke involution $F_n: z \mapsto -1/nz$ and so we may form the Fricke modular group $\Gamma_{0}^{+}(n)\langle \...
- 1,859
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
4
votes
1
answer
599
views
Reference request, zeta function is rational function via Riemann-Roch?
I am looking for a reference to a proof that the zeta function of a function field in one variable over a finite field $\mathbb{F}_q$ is a rational function in $q^{-s}$ by using the Riemann-Roch ...
- 75
4
votes
1
answer
579
views
Find all rational solutions of $x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1)$
Find all rational solutions of
$$x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1).$$
Clearly the following six solutions hold:
$$(x,y)=(1,1),(-1,-1),(-1,1),(1,-1),(0,1),(0,-1)$$
But how to find all rational ...
- 4,280
4
votes
1
answer
366
views
Splitting the Witt vectors of $\overline{\mathbb{F}_p}$
Let $\overline{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W({-})$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \overline{\...
- 2,052
4
votes
1
answer
567
views
Lifting of Frobenius on semi-abelian varieties
Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume ...
- 7,377
4
votes
1
answer
600
views
Is it expected that every natural number is the rank of some elliptic curve over the rationals?
It is a well-known problem on the theory of elliptic curves that the rank of an elliptic curve (the number of generators of the free part of the Mordell group of the elliptic curve) cannot be ...
- 26.9k
3
votes
0
answers
87
views
Norm $-1$ elements of quaternion algebras and Shimura curves [duplicate]
Let $Qa$ be an indefinite quaternion algebra over $\mathbb{Q}$.
Let $O$ be an order of $Qa$. The moduli space of abelian surfaces with quaternionic multiplication by $O$ is usually designed as the ...
- 91
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
1
answer
231
views
Relative density of images of diophantine polynomials
My current research project involves sampling a sequence along non-constant polynomials with non-negative integer coefficients defined over the natural numbers, and I'd like to be able to say when two ...
- 155
3
votes
0
answers
154
views
Examples of subspaces singled out by modular forms
I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature.
To be more clear let me explain the example I have in mind.
Let $N\...
- 231
3
votes
2
answers
530
views
isogeny and congruence subgroup
Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$.
By ...
- 699
3
votes
1
answer
665
views
Reducibility of resultants
This is closely related to this question. Suppose I have the resultant $\mathcal{R}$ of two (or more polynomials) over $\mathbb{Q},$ and suppose $\mathcal{R}$ is not irreducible. What is the ...
- 96.4k
3
votes
1
answer
182
views
Does a modular function primitive for $\Gamma$ generate the function field of $\mathcal{H}/\Gamma$?
Let $f$ be a modular function (that is, a meromorphic modular form of weight 0) holomorphic on $\mathcal{H}$ which is invariant under $\Gamma\le SL_2(\mathbb{Z})$ (not necessarily congruence!), and ...
- 10.7k
3
votes
1
answer
420
views
Automorphisms of complete discrete valuation ring
Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb F}...
- 87
3
votes
1
answer
230
views
Cubic polynomial over $\mathbb{Z}_p$
Let
$$
f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b,
$$
where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all pairs $(a,b)\in\mathbb{Z}_p^2$ such that $f_{a,b}(x)$ factor linearly. Then what ...
user126352
3
votes
2
answers
751
views
hyperelliptic curves over finite fields
What is information about the existence of rational points on hyperelliptic curves over finite fields available?
- 2,576
3
votes
2
answers
907
views
Finding coefficient of multivariate polynomial
$f(x_1,x_2,\ldots x_n)$ is polynomial with integer coefficients.
$f$ is rather large to be computed explicitly, but an algorithm can
compute it efficiently at integers and complex number and "...
- 25.4k
3
votes
1
answer
213
views
5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians
I would like to know if there is something I can read to compute the following:
Let $H$ be a hyperelliptic curve of genus $2$ given by $y^2=x^5 + 10$ and let $J$ be its Jacobian.
How can I prove ...
2
votes
0
answers
182
views
An elliptic curve trivial over any extension unramified outside 7 and infinity?
Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?
- 11.3k
2
votes
3
answers
1k
views
Finiteness of étale Cohomology Groups
Mr. Milne, in "Étale Cohomology", gives the following proposition (p.224, Corollary VI.2.8):
Proposition: Let $F$ a constructible sheaf on $X_{et}$, the small étale site of $X$, $X$ proper over a ...
- 23
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
2
votes
0
answers
109
views
Shimura curves and quaternion orders without elements of norm -1
Let $O$ be an order of a quaternion algebra over $\mathbb{Q}$ such that $O$ does not contain elements of norm $-1$. Such orders exist, but seems less used, in particular these orders are not Eichler.
...
- 91
1
vote
0
answers
146
views
Can we find curves with many rational points using linear algebra?
Probably this is impossible, but let us try.
Working over $\mathbb{Q}[x_1,...,x_n]$.
Let $T_i$ be $n$ sets of rationals with cardinality $B$.
Assume we are given $n-2$ linear equations $f_i$ which are ...
- 25.4k
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
1
vote
0
answers
175
views
Algebraic numbers with a polynomial property
In my research I faced with an intricate construction of an algebraic number with some properties.
Problem. For which classes of polynomials $P(X,Y)\in \mathbb{Z}[X,Y]$, we have the following property....
- 515
1
vote
0
answers
152
views
How difficult is to find rational points on these genus 3 curves:
How difficult is to find all rational points on these genus 3 curves:
$$
(a) \quad \quad x^3 + y^3 x +y^2 - y = 0
$$
$$
(b) \quad \quad x^4 - y^3 + x y + x = 0
$$
Short motivation. Consider the ...
- 7,182
1
vote
0
answers
102
views
Maximum number of bounded primitive integer points in a zero-dimensional system
Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...
- 1,826
1
vote
1
answer
536
views
Functional equations of zeta functions over global fields
The functional equations for Dedekind zeta functions (zeta functions attached to rings of integers in algebraic number fields) come from functional equations of theta functions like $\sum_{n \in \...
- 7,062
1
vote
2
answers
349
views
Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
- 7,182
0
votes
1
answer
3k
views
Can one expect the existence of a relevant approach for a proof of the Riemann hypothesis using Mochizuki's theory? [closed]
Next month at Oxford university, there will have the first workshop outside Asia on the Inter-Universal Teichmuller theory of Shinichi Mochizuki: http://www.claymath.org/events/iut-theory-shinichi-...
0
votes
0
answers
107
views
Cubic monic polynomial over z_p
Let
$$
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
0
votes
0
answers
96
views
Elementary constraints for the solutions of $z^{n-2}y(y+z)=x^n$?
Related to FLT and this question.
For natural $n > 4 $ define the curve $C_n : z^{n-2}y(y+z)=x^n$.
$C_n$ has the trivial points with $x=0$ for all $n$.
The answer in the linked question shows ...
- 25.4k
-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