# Tagged Questions

**6**

votes

**0**answers

119 views

### Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$

$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying
$b$ is bilinear,
...

**1**

vote

**0**answers

169 views

### Average rank of elliptic curves over function fields

De Jong showed in 2002 if the finite field $\mathbb{F}_q$ has characteristic not equal to 3, then the limsup of the average of 3-Selmer rank is bounded above, where the average is taken over the ...

**2**

votes

**0**answers

81 views

### The number of different lattice triangles

Two convex lattice polygons are equivalent if there is a lattice-preserving affine transformation mapping one of them to the other. Equivalent polygons have the same area. Let $H(A)$ denote the number ...

**3**

votes

**0**answers

88 views

### On one class of Euclidean lattices

Let $\Lambda\subset \mathbb Z^3$ be 3D lattice with a basis
$$a_1=\left(\begin{smallmatrix} a_{11} \\ a_{21}\\
a_{31}
\end{smallmatrix}\right),a_2=\left(\begin{smallmatrix} a_{12} \\ a_{22}\\
a_{32}
...

**5**

votes

**1**answer

224 views

### Regular lattice polygons

Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within ...

**18**

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 ...

**15**

votes

**1**answer

385 views

### Geometry of numbers for three by three matrices?

While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem:
What is the volume of the largest symmetric convex subset ...

**6**

votes

**3**answers

555 views

### Sums of inverse determinants over matrices

Let $A \in M_n(\mathbb Z)$ and $\|A\| = \max |a_{ij}|$.
Denote $$ S(r) = \sum_{\substack{\|A\| \leq r \\\ \det{A} \neq 0}} \dfrac{1}{|\det{A}|} $$
- the sum over all matrices $A \in M_n(\mathbb Z)$ ...

**7**

votes

**0**answers

98 views

### Minimal number of colours for colouring Voronoi-cells of a $d-$dimensional lattice

There are arbitrarily large finite sets of points in $\mathbb R^3$ whose Voronoi-domains
intersect all pairwise in $2-$dimensional polytopes. This shows that one needs infinitely many
colours in order ...

**5**

votes

**2**answers

394 views

### On Weil's characters of type (A)

In Weil's paper
"On a certain type of characters of the idele-class group of an algebraic number field",
Weil introduces a class of characters on the Idele class group (of not necessarily finite ...

**5**

votes

**2**answers

405 views

### Still more generalized Dirichlet Theorem

Dirichlet proved a classical theorem about approximating irrational real numbers with rational numbers, saying that for any irrational real number $\alpha$, you can find infinitely many rational ...

**29**

votes

**0**answers

851 views

### (Approximately) bijective proof of $\zeta(2)=\pi^2/6$ ?

Given $A,B\in {\Bbb Z}^2$, write $A \leftrightarrows B$ if the
interior of the line segment AB misses
${\Bbb Z}^2$.
For $r>0$, define
$S_r:=\{ \{A, B\} | A,B\in {\Bbb Z}^2,||A||<r,||B||<r, ...

**7**

votes

**0**answers

428 views

### Higher-dimensional analogs of the Farey sequence/Riemann hypothesis connection?

See here for Franel and Landau's equivalent forms of the Riemann
hypothesis in terms of the uniformity of distribution of Farey sequences.
...

**7**

votes

**1**answer

486 views

### Verifying my other example in the Geometry of Numbers and Quadratic Forms

In answer to Pete L. Clark's question Must a ring which admits a Euclidean quadratic form be Euclidean? on Euclidean quadratic forms, I also gave an example in six or fewer variables, repeated below. ...