Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, ...

learn more… | top users | synonyms

11
votes
1answer
723 views

What is the first cohomology $H_{fppf}^{1}(X, \alpha_{p})$?

Let $X$ be a smooth projective curve of genus $g>1$ over an algebraically closed field $k$ of characteristic $p>0$. Let $\alpha_{p}$ be the group scheme of the kernel of $F: \mathbb{G}_{a} \...
2
votes
2answers
374 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 ...
4
votes
1answer
117 views

Exceptional specializations of Galois groups in the Hilbert Irreducibility Theorem

Suppose $f(x,t)\in\mathbb{Q}(t)[x]$ is an irreducible polynomial with Galois group G. For any rational number $a$ we may consider the polynomial $f(x,a)\in\mathbb Q[x]$ and its corresponding Galois ...
10
votes
1answer
429 views

Galois representations for the curve $y^{2} = x^{3} - x$

Let $E / \mathbb{Q}$ be the elliptic curve given by $y^{2} = x^{3} - x$. I would like to know explicitly what the field of all $2$-power torsion looks like, as well as the image in $\mathrm{GL}(T_{2}(...
4
votes
0answers
122 views

Ranks of elliptic curves over Q(t)

I have an elliptic curve $E/\mathbb{Q}(t)$, and I want to compute its rank. Does knowing the rank over $\mathbb{F}_p(t)$ for some prime of good reduction give a bound on the rank over $\mathbb{Q}(t)$?...
19
votes
2answers
726 views

Rational points on the “quintic circle” $x^5 + y^5 = 7$

I suspect that the curve $x^5 + y^5=7$ has no $\mathbb Q$ points, and a brief computer search verifies this hypothesis for denominators up to $10^4$. What techniques can be used to show that there are ...
9
votes
2answers
579 views

BSD and congruent numbers

Let $n$ be a positive integer, and let $E_n$ denote the elliptic curve $y^2=x^3-n^2x$. By work of Tunnell, it's known that if $E_n$ satisfies the BSD conjecture, then there is an algorithm to decide ...
4
votes
1answer
211 views

Checking whether two rational points of infinite order are generating the torsion free part of an elliptic curve

Let an elliptic curve be given. As the title says I want to know if we can show that two independent points $P$ and $Q$ are generators of the torsion free part of $E$. For instance let $E:y^2=x^3-...
4
votes
1answer
214 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?
5
votes
1answer
155 views

curve over higher dimensional basis with 0-dimensional locus of bad reduction

Is there an example of a flat proper relative curve $X/S$ with geometrically connected fibres and with $\mathrm{dim} S > 1$ and $S$ regular and connected with $0$-dimensional locus of bad reduction ...
12
votes
1answer
539 views

Arithmetical results to help study arithmetic geometry?

I'm very keen to deepen my understanding of arithmetic and diophantine problems. In the past I studied some algebraic, analytic and sieve based number theory. Recently I've been reading Weil - Basic ...
1
vote
0answers
67 views

Shortest paths stepping on rational points of height $h$

Q. Do shortest paths walking between rational points of height $h$ ever properly cross themselves? Explaining this question takes a bit of definitional exposition. First, I copy definitions from ...
10
votes
1answer
311 views

Integral points on elliptic curves of the form $y^2=x^3+px$

As the title says. Can we determine all the integral points on elliptic curves of the form $$y^2=x^3+px$$ for a prime $p$? If yes, can someone explain me how? A good reference would also be ...
5
votes
1answer
169 views

Compact hyperbolic 3-manifolds with prescribed quaternion algebra, quaternion parameters as ramification condition

What is an interesting class of examples of hyperbolic 3-manifolds, each of which satisfies the following conditions? 1. It is compact 2. Its trace field contains a unique imaginary quadratic ...
5
votes
1answer
172 views

Congruences between modular forms and the eigencurve construction

This question might be too conceptual. Congruences between modular forms (due to Shimura, Hida, etc) are really amazing. I know that the eigencurve construction are closely related to these ...
5
votes
3answers
281 views

Smooth complete intersections and sharpness of the Chevalley-Warning theorem

Let $X$ be a complete intersection in $\mathbb{P}^n$ of multidegree $(d_1,\ldots,d_r)$. If we're working over a finite field $\mathbb{F}_q$, the Ax-Chevalley-Warning theorem says that if $X$ is in the ...
14
votes
1answer
299 views

Number of height-limited rational points on a circle

Consider origin-centered circles $C(r)$ of radius $r \le 1$. I am seeking to learn how many rational points might lie on $C(r)$, where each rational point coordinate has height $\le h$. For example, ...
7
votes
2answers
323 views

Order of reduction of infinite order rational point on an Elliptic Curve

Let $E/$ℚ be an elliptic curve and $P$ ∈ $E($ℚ$)$ a rational point of infinite order. Does the reduction of $P$ mod $p$ generate a maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ for ...
9
votes
0answers
204 views

Extension of Messing-Mazur-Oda to general groups

The following may be well-known (or obviously false), but I can't find a counterexample or a reference. Suppose that $k$ is some perfect field (one can assume algebraically closed, if that makes you ...
4
votes
1answer
203 views

How do non-trivial global differentials give non-trivial cohomology classes in positive characteristic

Let $k$ be an algebraically closed field and let $X$ be an $n$-dimensional smooth projective variety over $k$. If $k= \mathbb C$, there is a natural injective morphism of vector spaces $$H^0(X,\...
6
votes
0answers
251 views

A problem on universally locally acyclic

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ and $S$ be two smooth varieties over $k$ and $\mathcal F$ a constructible \'etale sheaf of $\mathbb F_\ell$-modules on $X$ (...
5
votes
0answers
113 views

Counting square zero forms over finite fields

Let $p$ be an odd prime and let $R=\Lambda_{\mathbb{F}_p}[x_1,\dots,x_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements. This is a graded-commutative ring. Is ...
2
votes
1answer
251 views

On quasi-algebraically closed fields

By Lang's theorem, a complete valued field which is the fraction field of a discrete valuation ring with an algebraically closed residue field is quasi-algebraically closed (or $C_1$). How much is ...
12
votes
1answer
262 views

To what extent are modular parametrizations expected to generalize?

By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) ...
1
vote
0answers
119 views

On the Weil Chatelet Group

Let $A$ be a abelian curve over a number field $K$. The Weil Chatelet group parametrizes the twists of $A$, modulo the twists with a $K$ rational point. We can assume that $A$ is a plane curve. My ...
10
votes
0answers
240 views

comparison of completion and Henselization in class field theory

Given a ring $R$ with maximal ideal $\mathfrak{m}$, we can form the localization $R_\mathfrak{m}$, the completion $\hat{R}_\mathfrak{m}$ or the Henselization $\hat{R}^h_\mathfrak{m}$ of $R$ with ...
9
votes
0answers
364 views

Does bounded-degree base extension yield Zariski-dense Mordell-Weil group?

If $d$ and $n$ are positive integers, does there exist a constant $B=B(d,n)$ with the following property? For any $n$-dimensional abelian variety $A$ over a degree-$d$ number field $K$, there is an ...
16
votes
1answer
3k views

A road to inter-universal Teichmuller theory

What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...
1
vote
2answers
173 views

Counting number of $2\times 2$ unimodular matrices of particular type

From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form? \begin{bmatrix} a^2 &...
1
vote
0answers
75 views

Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...
5
votes
0answers
162 views

Expressing every algebraic number using roots of trinomials?

This question is a continuation of Is every polynomial a factor of a trinomial? We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m ...
12
votes
0answers
197 views

$p$-Adic or arithmetic variants of Khovanskii's “low complexity $\Rightarrow$ tame topology” theory

This question is prompted by a remark I made in a comment to Is every polynomial a factor of a trinomial?, which was that Descartes's observation (cf. his rule of signs, etc.), that the number of real ...
3
votes
1answer
168 views

Can we define a height function for a variety over a finite field?

That is, is there a way to measure the complexity of a point over a finite field the same way we do it over number fields?
1
vote
0answers
100 views

In how many ways can one extend the zero section of the affine line with a double origin

Let $X$ be the affine line with a double origin over Spec $\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over Spec $\mathbb Q$. Let $0$ be one of the origins of $...
8
votes
3answers
674 views

Ranks of elliptic curves depend only on the field?

Let $K/\mathbb{Q}$ be an algebraic extension, and let $E_1,E_2/\mathbb{Q}$ be elliptic curves. Is it possible that the Mordell-Weil rank of $E_1(K)$ is finite while that of $E_2(K)$ is infinite?
2
votes
0answers
156 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 $7$ ?
43
votes
1answer
3k views

What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...
7
votes
0answers
237 views

Is the compositum of all quadratic extensions of the rationals an ample field?

Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$ Is there a (geometrically irreducible) smooth variety $V/\mathbb{...
6
votes
1answer
302 views

Isotrivial families with non-zero Kodaira spencer map

Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this ...
6
votes
1answer
463 views

Pure motives and compatible systems of $\ell$-adic representations

I am trying to understand the statement of the conjectures of Deligne on special values of certain $L$-functions, from his article titled, "Valuers de Fonctions L et periodes d'integrales" which ...
3
votes
4answers
511 views

Integral points on a particular family of curves

This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that $$ \prod_{i=1}^...
4
votes
1answer
245 views

Lifting torsors in characteristic $p$ to characteristic zero

Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with ...
6
votes
1answer
173 views

Question on paper of Stewart and Top about ranks of elliptic curves over Q(t)

I'm reading "On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms" by Stewart and Top, and struggling to understand the argument on pg 962 which shows that the rank of a ...
0
votes
1answer
175 views

Sections of proper, flat morphism

Let $f:X \to Y$ be a proper, flat morphism of projective scheme and $Y$ is an irreducible, non-singular surface. Assume further that there exists a Zariski open subset $U$ of $Y$ whose complement is ...
5
votes
2answers
429 views

Order of vanishing of an integer polynomial at a point

Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha_1,\alpha_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless $...
3
votes
0answers
119 views

Is there a difference between the inertia stack and the universal automorphism group

Let $\mathcal M$ be a stack representing some moduli problem. Let $\mathcal X\to \mathcal M$ be the corresponding universal family. What is the difference between the inertia stack $I\to \mathcal M$ ...
5
votes
1answer
293 views

Algebraic spaces which are automatically schemes

Let $S$ be a scheme, and let $f:X\to S$ be a morphism of algebraic spaces. If $f$ is smooth proper curve of genus at least two, then $X$ is a scheme. (Here I mean that $f$ is a smooth proper morphism ...
8
votes
1answer
235 views

Distribution of Mordell–Weil ranks of higher genus curves

By "nice curve", I mean a smooth, projective, geometrically integral curve over $\newcommand{\Q}{\mathbb{Q}}\newcommand{\Jac}{\operatorname{Jac}}\Q$ with at least one $\Q$-rational point. The Mordell–...
7
votes
0answers
173 views

Almost rational point

Let $X$ be a variety over a number field $K$. Let $S$ be a finite set of places of $K$. Is there a notion of a point $p \in X(\overline{K})$ to be "almost rational" in the following sense?: $p$ and $^...
3
votes
0answers
119 views

Classification of compact Shimura curves

Is there a classification that determines all isomorphism classes of compact Shimura curves at least Shimura curves in $A_g$? I did not find this in the literature and appreciate any helpful ...