Tagged Questions

0
votes
0answers
22 views

Does 2^m = 3^n + r have finitely many solutions for every r?

Is it true that for every integer $r$, the equation $2^m = 3^n + r$ has at most a finite number of integer solutions? I understand that this is a special case of Pillai's conjectur …
12
votes
5answers
313 views

Where can I find a comprehensive list of equations for small genus modular curves?

Does there exist anywhere a comprehensive list of small genus modular curves $X_G$, for G a subgroup of GL(2,Z/(n))$? (say genus <= 2), together with equations? I'm particularly …
3
votes
1answer
117 views

additive structure in a small multiplicative group of a finite field?

Let $p$ be a prime. Given a positive integer $n$, does there exist a $\beta$ in an extension of $F_p$ such that 1) If $F_p[\beta] = F_{p^N}$, then $N > n^n$; ( $\beta$ lies in a …
1
vote
1answer
207 views

Strengthening of Dirichlet’s theorem on arithmetic progressions

Hello all, Dirichlet's famous theorem asserts that any arithmetic progression $\lbrace ax+b | x \in {\mathbb N}\rbrace$ contains infinitely many primes if a and b are relatively pr …
7
votes
3answers
253 views

Binary Quadratic Forms in Characteristic 2

One of the reasons why the classical theory of binary quadratic forms is hardly known anymore is that it is roughly equivalent to the theory of ideals in quadratic orders. There i …
5
votes
3answers
427 views

When is the Galois representation on the étale cohomology unramified/Hodge-Tate/de Rham/crystalline/semistable?

Let $X/K$ be a variety over a global field $K$. When (and why) is the Galois representation $H^i_{et}(X \times_K \bar{K}, \mathbf{Q}_\ell)$ unramified at a place $v$ of $K$? I gue …
2
votes
3answers
112 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 …
2
votes
3answers
106 views

Given N points on a number line and m total distances between those points, are there efficent ways to optimize for particular values in m?

Given a set of distances S, choose N unique points P on a number line such that the distances between the N points occur in S as much as possible. That is, maximize the occurence i …
7
votes
4answers
541 views

An elementary number theoretic infinite series

For a positive integer k, let d(k) be the number of divisors of k. So d(1)=1, d(p) =2 if p is a prime, d(6)=4, and d(12)=6. What is the precise asymptotics of SUM_{k=1}^n 1/(kd(k …
0
votes
0answers
56 views

Uniformly computable classes of graphs

[Follow-up to Can every finite graph be represented by one prescribed sequence of natural numbers?, reformulated thanks to a hint from Jacques Carette] Let $V(n,\nu)$ and $E(n, …
3
votes
1answer
217 views

Density results for equality of Galois/automorphic representations

I have two "vague questions" which are the following: if you have two $n$-dimensional $\ell$-adic Galois representations of a number field $K$ (with the standard ramification condi …
0
votes
1answer
311 views

Can every finite graph be represented by one prescribed sequence of natural numbers?

(This is a follow-up to my previous question Can every finite graph be represented by an arithmetic sequence of natural numbers?) Since it is obviously false that every finite gra …
5
votes
2answers
313 views

Forms over finite fields and Chevalley’s theorem

Chevalley's theorem says that if $k$ is a finite field and $f(X_1,...,X_n)$ is a form (homogeneous polynomial) of degree $d < n$, then the equation $f(X_1,...,X_n) = 0$ has a no …
3
votes
1answer
170 views

Locally profinite fields ?

This is a question about terminology and should be taken lightly. The expression local field is used in at least three different senses : 1) For a locally compact totally discon …
7
votes
2answers
245 views

Is there a non-trivial topological group structure of $\mathbb{Z}$?

More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?

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