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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 …
Lev Borisov's user avatar
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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 …
Lev Borisov's user avatar
  • 5,186
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 …
Lev Borisov's user avatar
  • 5,186
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 …
Lev Borisov's user avatar
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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 …
Lev Borisov's user avatar
  • 5,186
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 …
Lev Borisov's user avatar
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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 …
Lev Borisov's user avatar
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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) = …
Lev Borisov's user avatar
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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 …
Lev Borisov's user avatar
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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.
Lev Borisov's user avatar
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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 …
Lev Borisov's user avatar
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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 …
Lev Borisov's user avatar
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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 …
Lev Borisov's user avatar
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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 …
Lev Borisov's user avatar
  • 5,186
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 …
Lev Borisov's user avatar
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