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

**61**

votes

**9**answers

7k views

### Why should I believe the Mordell Conjecture?

It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points.
I am interested to know why Mordell and ...

**22**

votes

**2**answers

1k views

### Are most curves over Q pointless?

Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points. Don Zagier suggests the paraphrase : Most ...

**11**

votes

**5**answers

2k views

### Rational points on a sphere in $\mathbb{R}^d$

Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers.
Q1.
Is the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$ dense in rational points, i.e. does $S$ include a ...

**42**

votes

**8**answers

5k views

### Questions about analogy between Spec Z and 3-manifolds

I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...

**22**

votes

**2**answers

4k views

### Intuition behind the Eichler-Shimura relation?

The modular curve $X_0(N)$ has good reduction at all primes $p$ not dividing $N$. At such a prime, the Eichler-Shimura relation expresses the Hecke operator $T_p$ (as an element of the ring of ...

**37**

votes

**1**answer

13k views

### What is inter-universal geometry?

I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...

**21**

votes

**2**answers

1k views

### unboundedness of number of integral points on elliptic curves?

If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and ...

**13**

votes

**6**answers

1k views

### SAT and Arithmetic Geometry

This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...

**10**

votes

**3**answers

2k views

### Existence of fine moduli space for curves and elliptic curves

For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's ...

**112**

votes

**1**answer

10k views

### What is a Frobenioid?

Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion ...

**39**

votes

**6**answers

4k views

### Intuition for the last step in Serre's proof of the three-squares theorem

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m ...

**61**

votes

**1**answer

7k views

### What is an étale theta function?

Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, ...

**38**

votes

**4**answers

3k views

### Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here ...

**9**

votes

**5**answers

1k views

### The significance of modularity for all Galois representations

On pg. 1 of the slides of a talk, Henri Darmon wrote:
Question: What is an interesting Diophantine equation?
A “working definition”. A Diophantine equation is interesting
if it reveals or ...

**18**

votes

**1**answer

1k views

### De Rham cohomology of formal groups

Let $G$ be some (dimension $1$, to simplify) formal group over a characteristic $0$ field $K$. The law of $G$ is denoted by $\oplus$. If $w(X) \in K[[X]] dX$ is a differential form, let $F_w(X)$ be ...

**11**

votes

**3**answers

710 views

### What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the ...

**19**

votes

**4**answers

863 views

### Everywhere locally isomorphic abelian varieties

Is there a standard example of two abelian varieties $A$, $B$ over some number field $k$ which are $k_v$-isomorphic for every place $v$ of $k$ but not $k$-isomorphic ?

**9**

votes

**2**answers

2k views

### “Fermat's last theorem” and anabelian geometry??

Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If ...

**21**

votes

**3**answers

2k views

### Over which fields does the Mordell-Weil theorem hold?

According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...

**11**

votes

**4**answers

5k views

### Is the ABC conjecture known to imply the Riemann hypothesis?

I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...

**10**

votes

**4**answers

1k views

### Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$.
An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...

**9**

votes

**0**answers

199 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 ...

**3**

votes

**4**answers

1k views

### reduction of CM elliptic curves

Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$:
(i) $p$ is inert in End($E$)
(ii) $E_p$ is supersingular
(iii) The trace of the Frobenius at $p$ is $0$ ...

**11**

votes

**1**answer

602 views

### Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...

**10**

votes

**3**answers

575 views

### what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)
We are ...

**6**

votes

**0**answers

687 views

### Can you get Siegel's theorem “for free” from modularity and Mazur's Eisenstein Ideal paper?

There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...

**12**

votes

**0**answers

621 views

### Points of bounded height in a number field

Let $K$ be a number field of absolute degree $d$, let $B$ be a positive real number, and write $S(K, B) = \{x \in K : H(x) \leq B\}$. Here $H$ is the absolute multiplicative height of an algebraic ...

**11**

votes

**2**answers

517 views

### Belyi's theorem for function fields

Belyi's theorem states that every smooth projective algebraic curve $C$ defined over $\bar{\mathbb{Q}}$
admits a map $C\to\mathbb{P}^1$ ramified only over $0,1,\infty$.
Is there an analogue of this ...

**10**

votes

**3**answers

391 views

### Circles avoiding rational points of height $\le h$

Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$)
of radius $r < 1$ avoid all rational points
of height $\le h$?
A rational point is a point all of whose coordinates ...

**7**

votes

**0**answers

177 views

### Lattice radial-step (ratchet) spirals

(30Oct13: Now solved; see Addendum.)
Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.
$S(r_0,\epsilon)$ begins with the arc ...

**6**

votes

**2**answers

862 views

### “Bad” reduction of Shimura curves via dual graphs

I have the following naive (and inexpert) question about the
reduction of Shimura curves at primes dividing the discriminant
of the underlying quaternion algebra. It requires some background
to ...

**12**

votes

**0**answers

923 views

### Effective proofs of Siegel's theorem using arithmetic geometry

This is a speculation and perhaps naive. The theorem of Siegel that
There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is ...

**9**

votes

**2**answers

335 views

### Rational points on circular spirals

Is it the case that every unit-radius circular spiral,
$$x = \cos(t)$$
$$y = \sin(t)$$
$$z = c \cdot t$$
for $c \in \mathbb{R}^+$
is dense in rational-coordinate points
(i.e., all three coordinates ...

**4**

votes

**0**answers

591 views

### Soft proof of multiplicity one for character groups of Shimura curves?

Is it not possible to prove mutiplicity one type statements for character groups of quaternionic Shimura curves by simply using Raynaud's description for character groups at primes dividing the ...

**2**

votes

**2**answers

345 views

### transcendence of canonical heights

Are there known examples of rational points on elliptic curves/abelian varieties
over number fields with transcendental canonical height? Thanks.

**11**

votes

**0**answers

368 views

### Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal ...

**9**

votes

**1**answer

540 views

### Can we always find a curve which doesn't have semi-stable reduction

Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction ...

**8**

votes

**1**answer

500 views

### Potentially good, semi-stable reduction => good reduction ?

Does a smooth proper variety having semi-stable reduction as well as potentially good reduction have good reduction ?
Note that over a $p$-adic field, this is true for the Galois representations in ...

**7**

votes

**0**answers

235 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 ...

**11**

votes

**1**answer

703 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} ...

**5**

votes

**0**answers

174 views

### Semiabelian actions appearing in the toroidal campactification of a degenearting abelian varieties

Given a totally degenerated abelian variety $A_K$ (to make it easier) over a complete discrete valuation field $K$ with $R$, $\pi$ and $k$ the corresponding discrete valuation ring, uniformiser and ...

**5**

votes

**5**answers

1k views

### The use of embedding a curve into its Jacobian

I'm looking for as many examples/applications as possible of the use of embedding a smooth projective geometrically connected curve $X$ over a number field $k$ with $X(k)\neq \emptyset$ into its ...

**3**

votes

**2**answers

142 views

### Lattice-point-free buffers around circles

Let $C(r)$ be the origin-centered circle of radius $r$,
and let $\beta(r)$ be the exterior buffer around $C(r)$:
the distance from $C(r)$ to the closest lattice point exterior to $C(r)$:
...

**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 $7$ ?

**1**

vote

**2**answers

172 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 ...