All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
8
votes
0
answers
357
views
Does Stepanov's method extend to complete intersections?
Stepanov (circa 1970) created the polynomial method to limit the rational points of an algebraic curve over $\mathbb{F}_q$, leading to one of several alternative proofs of Weil's Riemann hypothesis ...
- 13.8k
21
votes
2
answers
1k
views
CM $j$-invariants in $p$-adic fields
I'm trying to understand the $p$-adic distribution of $j$-invariants for elliptic curves with complex multiplication.
Specifically, suppose $\sigma$ is some embedding $\sigma:\overline{\mathbb Q}\to \...
- 1,021
3
votes
0
answers
95
views
Sign of bivariate polynomial evaluated over two algebraic numbers
I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
- 151
12
votes
0
answers
729
views
Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)
Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$.
I have seen another post on ...
- 3,539
2
votes
0
answers
119
views
A reference about a problem of the number of the rational points on a projective scheme
Let $X\hookrightarrow\mathbb{P}^n_{\mathbb{F}_q}$ be a pure dimensional projective scheme of dimension $d$. So we know a trivial estimate of the number of $X(\mathbb F_q)$ is that $\#X(\mathbb F_q)\...
- 403
0
votes
1
answer
456
views
How do Modular Forms and the Geodesic Flow interact? [closed]
Textbooks talk at length of the modular properties of $\theta(z)$ or $\tau(z)$ and the prominent role of $SL(2,\mathbb{Z})$ or one of the congruence groups.
In that case, aren't the basic objects ...
- 22.8k
21
votes
3
answers
1k
views
Does X(13) have potentially good reduction at 13?
The complete level modular curve $X(p)$ does not have potentially good reduction at $p$ for any $p \neq 2,3,5,7,13$ because then there are cusp forms on $X_0(p)$ showing up in the cohomology of $X(p)$,...
- 148k
1
vote
1
answer
583
views
Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?
It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type:
https://en.wikipedia.org/wiki/...
- 179
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
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
0
answers
386
views
Universal vector extension of a p-divisible group and the exponential.
Given a coalgebra $U$ with a marked element $1\in U$, define its lie algebra to be the set of primitive elements ($u \in U|\Delta(u) = u \otimes 1 + 1\otimes u$, where $\Delta: U \rightarrow U\otimes ...
- 41
12
votes
1
answer
565
views
Parametrizing all cyclic extensions of the rational numbers of degree 5
Is there a polynomial $f(T,X) \in \mathbb{Q}(T)[X]$ in the indeterminate $X$ over the field $\mathbb{Q}(T)$ with $\mathrm{Gal}(f/\mathbb{Q}(T)) \cong \mathbb{Z}/5\mathbb{Z}$ such that for every Galois ...
- 11.3k
11
votes
1
answer
624
views
Weak Mordell-Weil over number fields
I have a question regarding the Mordell Weil theorem a number field $K$.
I read the proof of the Mordell Weil theorem in "rational points on elliptic curves" by Tate and Silverman.
They presented a ...
user78817
70
votes
7
answers
28k
views
Have there been any updates on Mochizuki's proposed proof of the abc conjecture?
In August 2012, a proof of the abc conjecture was proposed by Shinichi Mochizuki. However, the proof was based on a "Inter-universal Teichmüller theory" which Mochizuki himself pioneered. It was known ...
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
1
vote
0
answers
201
views
Number of minimal primes for UFD
Let $R$ be a UFD which is NOT noetherian. It is well-known that $R$
is a Krull ring. Let $I$ be an ideal of $R$ such that the height of $I$
is $d$ which is finite.
Question. Is the number of minimal ...
- 781
10
votes
1
answer
635
views
Convergence of zeta functions for schemes of finite type over the integers
In his lecture "Zeta functions and $L$-functions", Serre presents a very elegant proof of the convergence of the zeta function
$ \zeta (X,s) = \prod_{x \in |X|} (1- N(x)^{-s})^{-1}$ in the half plane ...
user76025
6
votes
1
answer
417
views
Levelt-Turrittin Theorem over p-adics (or the monodromy theorem)
Let $V$ be a finite dimensional vector space over $\mathbb{C}((t))$. Let $D:V\rightarrow V$ be a differential operator; i.e., an additive $\mathbb{C}$-linear map satisfying
$$
D(a.v)=(t\frac{d}{dt}a)....
- 2,751
3
votes
0
answers
204
views
Exponential analogue of formal connections
Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation:
$$
g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).
$$
Here, $\...
- 2,751
10
votes
0
answers
347
views
The non-abelian Gauss-Manin connection; non-abelian M_dR; a Grothendieck lemma for cyrstals
I'm interested in understanding the non-abelian Gauss-Manin connection on Carlos Simpson's relative de Rham Moduli space $M_{dR}(X/S,n)$ for a smooth projective morphism of schemes $X/S$. The scheme $...
- 359
2
votes
0
answers
95
views
Pairing for non-uniformizable Anderson T-motives
Let $M$ be an Anderson T-motive (the simplest case, i.e. abelian in the meaning of [G] Goss, Basic structures of function field arithmetic, Def. 5.4.12, over $A^1$, having $N=0$), and let $H_1(E)$ ([G]...
- 305
5
votes
0
answers
486
views
Computing intersection number of two arithmetic line bundles
I have some questions in Arithmetic Arakelov geometry
Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and $\...
user21574
5
votes
0
answers
224
views
Comparison of sheaves of modular forms
Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$:
$e^*\Omega^1_{E/X}$ and $\...
- 845
6
votes
0
answers
666
views
Interuniversal Teichmuller theory's applications
Apart from a proof of the ABC conjecture -and its accepted consequences- are there applications of Mochizuki's IUT? In particular are there already widely accepted applications? Does it shed ...
- 982
2
votes
1
answer
249
views
Question related to Fermat curve: Does the equation $A x^n + By^n = C z^n$ have any solution in $\mathbb{N}$?
Let $A, B, C \in \mathbb{N}$ be such that $\gcd(A,B,C)=1$. Is it known if the equation $A x^n + By^n = C z^n$ has any non-trivial solutions $x,y,z \in \mathbb{N}$? I know there are no such solutions ...
- 3,625
12
votes
1
answer
329
views
Analogue of Tate curve for $g>1$
Is there any analogue of the Tate curve for (principally polarized) abelian varieties of dimension $g$ ?
user85435
6
votes
0
answers
354
views
Sporadic and Exceptional
I have been reading this recent paper of J.McKay and YH. He (they've written a number of papers recently, including a fun and joking one on 42 which overflow commented on) called "Sporadic and ...
- 151
1
vote
1
answer
197
views
A certain invariant of non-singular algebraic surfaces
Let $X \subset \mathbb{P}^3$ be a non-singular surface defined over $\mathbb{Q}$ of degree $d \geq 3$. It is a theorem of Colliot-Thelene (see the appendix to this paper: http://www.jstor.org/stable/...
- 26.9k
22
votes
2
answers
1k
views
Describing the crystalline extension of $\mathbb{Q}_p$ by $\mathbb{Q}_p$
Let $K$ be a finite extension of $\mathbb{Q_p}$. The group $\ker H^1(G_K, \mathbb{Q}_p) \rightarrow H^1(G_K, B_{crys})$ is one-dimensional, which tells us that among all extensions of Galois modules
...
- 2,809
10
votes
2
answers
1k
views
Algebraic independence of exponentials
First of all, a happy new year. Be it better than 2015,
healthy, wealthy, fruitful and cross-fertilizing
for you, familly and friends.
In order to cope with families of solutions of evolution ...
- 3,738
3
votes
1
answer
664
views
Colmez conjecture and endomorphism rings
It is given by the Colmez conjecture that if $A$ has a CM-type $(k, \phi)$ and $\text{End}(A) = \mathcal{O_k}$ $$\displaystyle h_{\text{fal}}(A) = \sum_{\text{irr} \hspace{1 mm}\rho: G(k^{\text{nor}}/\...
- 201
7
votes
0
answers
279
views
Quadratic twists of 1-motives
Quadratic twists of elliptic curves (or, more generally, abelian varieties) are familiar objects in arithmetic geometry. I would like to extend that definition to the category of 1-motives over global ...
4
votes
1
answer
311
views
Generalization of a theorem of Mahler to higher dimensions
A seminal theorem of Kurt Mahler, in his papers Zur Approximation algebraischer Zahlen. I-III., is the following:
Let $F(x,y) \in \mathbb{Z}[x,y]$ be a binary form of degree $d \geq 3$ and non-zero ...
- 26.9k
1
vote
0
answers
236
views
Canonicity of Čech cohomology
For a topological space $X$, consider the Leray covering $U_\lambda$ (i.e. $\cap U_\lambda$ is sufficiently fine, e.g. affine for Zariski topology) of $X$.
For a sheaf $F$ on $X,$ the cohomology $H^...
- 781
2
votes
2
answers
440
views
Rational points on towers of curves
Let $\ldots \to X_n \to X_{n-1} \to \ldots \to X_0$ be etale maps between smooth projective curves of genera $g(X_n)>1$, all defined over a fixed number field $K$. By Faltings' Theorem, we know ...
- 16.1k
3
votes
1
answer
258
views
Torsion point on jacobian of ramified cover
Suppose $C$ is a hyperelliptic curve. Then the set of two-torsion points on the jacobians is generated by the set of difference of Weierstrass points.
Suppose $C'$ is another hyperelliptic curve. Is ...
- 31
20
votes
5
answers
2k
views
Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$
As a generalisation to the equation of Fermat, one can ask for rational solutions of $X^n+Y^n+Z^n=1$ (or almost equivalently integer solutions of $X^n+Y^n+Z^n=T^n$).
Contrary to the case of Fermat, ...
- 7,722
21
votes
1
answer
757
views
What should motives for $L(E,n)$ look like?
Goncharov and Manin showed in this paper that the zeta values $\zeta(n)$ can be realized as periods of framed mixed Tate motives constructed from moduli spaces $\overline{\mathcal{M}}_{0,n+3}$ of ...
- 17.4k
2
votes
0
answers
197
views
Gromov-Witten invariants for arithmetic surfaces counting sections passing through points
Suppose we are given an arithmetic surface, $X\to \text{Spec}\mathbb{Z}[1/N]$ smooth and quasi-projective, and a finite set of closed points all in different vertical fibers.
Can we count the number ...
- 845
8
votes
3
answers
503
views
polynomials and symmetric functions
Suppose I have a polynomial function $f\in \mathbb{Z}[x_1, \dotsc, x_k],$ such that whenever $r_1, \dotsc, r_k$ are roots of a monic polynomial of degree $k$ with integer coefficients, we have $f(r_1, ...
- 96.4k
11
votes
4
answers
1k
views
Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$
David Speyer commented the following here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $a(t)...
user61522
1
vote
0
answers
520
views
Euler Characteristic of simple sheaves
Let $X$ be a projective curve over a field $K$ (any characteristic). Let $\mathcal{F}$ be a coherent simple sheaf
(In the sense, that $\mathcal{F}$ doesn't have non-trivial subsheaves). What is the ...
7
votes
1
answer
1k
views
Analytic continuation for $L$-functions of elliptic curves
Let $E$ be an elliptic curve over a number field.
When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...
- 509
7
votes
2
answers
776
views
No nonconstant coprime polynomials $a(t)$, $b(t)$, $c(t) \in \mathbb{C}[t]$ where $a(t)^3 + b(t)^3 = c(t)^3$
See David Speyer's answer here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)$...
user80013
0
votes
0
answers
151
views
Are there unconditional results for boundedness of finitely many rational points on $f(x,y)=n$ for all $n$?
Major rewrite due to comments.
Let $f(x,y) \in \mathbb{Q}[x,y]$ and $f$ depends on both $x,y$.
Q1 Is it possible the number of rational solutions to $f(x,y)=n$
to be uniformly bounded for all ...
- 25.4k
7
votes
1
answer
508
views
What is the exact statement about uniform boundedness of rational points on curves of genus greater than one? Singular points can be unbounded
According to several sources, it is conjectured (or at least believed)
that the rational points of curves over the rationals of genus $g > 1$
are uniformly bounded by $g$. E.g. here p. 1.
Assuming ...
- 25.4k
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-...
4
votes
1
answer
358
views
Examples of perfect pseudo algebraically closed fields in positive characteristic
Is there any known example of a perfect pseudo algebraically closed field of positive characteristic containing $\overline{\mathbb{F}_p}$ but is not algebraically closed?
- 2,323
1
vote
1
answer
234
views
Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvable extension?
Are there counterexamples to the following:
Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over $\...
- 1,805
10
votes
1
answer
582
views
Intuitive reasons for the existence of modular parametrizations
Whenever I encounter anything about modular parametrizations, I have a feeling it is something very unnatural: you have some kind of moduli space and all of a sudden it parametrizes an object ...
- 18.4k