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For any finite abelian group $A$, there is a discrete Fourier transform that takes in complex-valued functions $f: A \to \mathbb{C}$. The transformed function is a complex-valued function on the dual …

Following Agol's answer, you can find a list of genus zero groups of $n|h$-type together with a discussion of fundamental domain computations in Ferenbaugh's paper: The Genus-zero problem for n|h-type …

The condition that the pullback $e^*(\mathcal{F})$ be naturally identified with the pushforward $p_*(\mathcal{F})$ can be tautologically interpreted as saying that for any open set $U$ in $X$, any sec …

I'm not an expert in this area, but I've heard that algebraic topologists run into congruences quite often. For example, the stable homotopy groups of spheres are almost always finite abelian groups, …

I'm not an expert, but the content of the article Artin's primitive root conjecture -a survey - (modified December 2004) by Pieter Moree suggests the Wikipedia article is reasonably up-to-date.

If you don't specify more about the structure of the field $K$, then we can't say much about the structure of the group $E(K)$. There are special cases (described in the Wikipedia article): If $K$ i …

The previous version of this question was rather badly broken, and I hope this version makes some sense. There have been a few questions on MathOverflow about how much representation-theoretic inform …

The answer to your question is yes, but it is a stretch to claim that the topology is due to Furstenberg. There is an extended discussion on Furstenberg's proof in the comments to this answer. The s …

As I mentioned in a comment, Question 2 (in its revised form) has a negative answer, because odd natural numbers have unbounded abundancy index, while the Odd Order Theorem implies all groups of odd o …

One historical reason for considering $\ell$-adic cohomology, not completely disconnected from the example you introduce, is that for a curve over a field, we get a natural Galois representation by ta …

The logo has a piece of the complex upper half plane divided into fundamental domains for the action of $SL_2(\mathbb{Z})$ by Möbius transformations (which are hyperbolic isometries - see the approp …

I think there are two phenomena at work, and often one can separate behaviors based on whether they are "caused by''one or the other (or both). One phenomenon is the smallness of $2$, i.e., the expres …

Last fall, there was a conference in Nagoya about precisely this question (oddly enough, funded by a "Riemann Hypothesis" DARPA grant). Since I was attending a different conference at the same univer …

Suppose I have a prime $p$ and a finite group $G$ together with representations $\sigma: G \to GL_n(\mathbb{Q}_p)$ and $\pi: G \to GL_n(\mathbb{F}_p)$. My question is: When does there exist a rep …

Let $p$ be a prime, let $\mathbb{Z}_p$ be the ring of $p$-adic integers, and let $G$ be a cyclic group of order $p$. It is rather well-known that finite rank $\mathbb{Z}_p$-free representations of $G …