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

**3**

votes

**0**answers

166 views

### splitting property of etale covering

Theorem (Global Splitting): Let $X$ be an integral separated normal scheme flat and of finite type over $\mathbb Z$. Let $\phi: Y\rightarrow X$ be a connected etale covering which splits completely ...

**3**

votes

**0**answers

168 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 $\...

**18**

votes

**1**answer

558 views

### Is hyperelliptic cryptography “practical”?

Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over ...

**9**

votes

**0**answers

134 views

### Does the Tate pairing agree with the Brauer-Manin pairing

Let $X$ be a proper, smooth, geometrically integral variety over a field $k$. Let $A$ be (the identity component of) its Picard variety and let $B$ be (the identity component) of its Albanese variety. ...

**1**

vote

**1**answer

210 views

### Finding cyclic subgroups of points on elliptic curves for isogeny based cryptography

Isogeny based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is a theorem:
Elliptic curves ...

**5**

votes

**2**answers

313 views

### Lifting the Hasse invariant mod $2$

Katz defines in Section 2.0 $p$-adic properties of modular schemes and modular forms the Hasse invariant as a mod $p$ modular form $A$ of weight $p-1$. In other words, it is a section of $\omega^{\...

**10**

votes

**0**answers

112 views

### Finiteness of torsion in $\mathcal{K}_2$-cohomology

Let $F$ be a number field, $C$ be a smooth projective curve over $F$ and $\mathfrak{C}$ be a proper regular model. I am interested in $\mathcal{K}_2$-cohomology, i.e., Zariski cohomology of the sheaf ...

**1**

vote

**1**answer

230 views

### Solving Non-Linear Equations over a Finite Field of a Large Prime Order

I want to know is there is an efficient way to figure out whether or not a ( underdetermined) system of non-linear equations have a solution over a finite field of large prime order. The equations ...

**11**

votes

**1**answer

729 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

**2**answers

375 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

**1**answer

118 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

**1**answer

432 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

**0**answers

123 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

**2**answers

732 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

**2**answers

590 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

**1**answer

214 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

**1**answer

217 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

**1**answer

156 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

**1**answer

548 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

**0**answers

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

**1**answer

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

**1**answer

172 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

**1**answer

176 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

**3**answers

284 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

**1**answer

303 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

**2**answers

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

**0**answers

205 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

**1**answer

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

**0**answers

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

**0**answers

114 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

**1**answer

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

**1**answer

263 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

**0**answers

120 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

**0**answers

241 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

**0**answers

365 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 ...

**18**

votes

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

163 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

**0**answers

199 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

**1**answer

170 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

**0**answers

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

**3**answers

682 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

**0**answers

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

**1**answer

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

**0**answers

241 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

**1**answer

304 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

**1**answer

467 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

**4**answers

514 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

**1**answer

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 ...