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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic
This is really more of an extended comment, but it would be unpleasant to write it in that format. I am commenting on the suggestion of Peter Mueller to argue that the boundary of the triangles he con …
6
votes
Accepted
binomial/factorial identity mod p
Sorry, don't know a reference, but here is a quick argument.
If $M=p^ab+c$ with $0\leq c\leq p^a-1$, then
$$(1+x)^M=(1+x)^{p^ab}(1+x)^c
=(1+x^{p^a})^b(1+x)^c
\mod p.
$$
In turn, this equals
$$
(1+bx …
4
votes
0
answers
147
views
Zeta functions with Brauer class
In algebraic geometry there are examples when a variety $X$ is somehow related (I call it double-mirror) to another variety $Y$, together with
a 2-torsion Brauer class. By "related" I mean statements …
3
votes
0
answers
93
views
Finding a divisor on a curve in a given linear equivalence class made with points over a "sm...
Suppose I have a curve $C$ defined over $\mathbb Q$ and a line bundle $\mathcal L$ on it,
also defined over $\mathbb Q$. I am interested in trying to find a (non-effective) divisor $D=\sum_i a_ip_i$ o …
10
votes
1
answer
620
views
K3 surfaces that correspond to rational points of elliptic curves
In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the quo …
10
votes
1
answer
558
views
Newton polygons of modular polynomials
This is pretty much straightforward curiosity. Is there anything known about Newton polygons of classical modular polynomials (polynomial relations between $j(\tau)$ and $j(n\tau)$)? I understand that …
1
vote
0
answers
179
views
Interpretation of the Gross-Zagier formula for Green function
I am reading the paper of Gross and Zagier on heights of Heegner points and would like to check with the experts whether the following (meta?)mathematical statement makes sense.
In the calculation of …
4
votes
0
answers
183
views
Arithmetic analogs of vertex algebras?
Has anyone successfully defined and studied analogs of vertex algebras where the grading of the fields is by $(\log \mathbb Q)$ rather than $\mathbb Z$? What I mean is that the usual fields
$$
a(z) = …
1
vote
Accepted
Decimal digits multiplied by powers of 2: leads to mod 8?
Just a simple proof of the formula for $2$ and $10$. The remainder modulo $8$ is unchanged under the operation $f$, since $2^k=10^k$ modulo $8$. On the other hand, you will never reach $0$ or $1$ if y …
1
vote
Models of the modular curve $Y_1(N)$
There are in fact explicit equations (at least for the prime level) worked out in arXiv:math/0010272.
4
votes
Accepted
Number of solutions of linear homogenous Diophantine equation inside a box
The solution set $L$ is clearly a sublattice in $\mathbb Z^d$. If $a$ is fixed and $N$ grows, the number of lattice elements in a box grows approximately as $cN^{rank}$. So if you have your statement …
1
vote
1
answer
302
views
Weierstrass points on modular curves
What is knowns about Weierstrass points on modular curves? Are there any explicit formulas of them, or any information about Weierstrass gaps? I am interested in (compactifications of) the quotients o …
12
votes
1
answer
885
views
Is the universal elliptic curve $\overline M_{1,2}$ a toric stack?
It is well-known that the compactification $\overline M_{1,1}$ of the moduli space of elliptic curves over $\mathbb C$ is a weighted projective line with
weights $4$ and $6$. As far as I can tell, th …
15
votes
1
answer
1k
views
Derived categories of arithmetic schemes?
Let $X$ be a smooth scheme of finite type over $\mathbb Q$ or $\mathbb Z$. It is natural to consider $D(X)$, the derived category of coherent sheaves on $X$. Beyond the definition, have there been any …
9
votes
1
answer
347
views
Degenerations of modular curves
Has anyone come across anything along the following lines?
Let $X_1(p)$ be the compactification of the quotient of upper half plane by $\Gamma_1(p)$ for some unspecified large prime. Let $X_1(p) \to …