Search Results

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 …

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 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 …

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 …

The interpretation of Hecke operators on modular forms and Jacobi forms in terms of a sum over isogenies is classical - this is the worldsheet interpretation. Their "target space" interpretation in t …

Note that for any $\left(\begin{smallmatrix} a & b \\ cpN & d \end{smallmatrix} \right) \in \Gamma_0(pN)$, we have $$f\left(\frac{a\tau+bN}{cp\tau + d}\right) = \phi\left(\frac{a\tau+bN}{cpN\tau+Nd}\r …

The idea is that you can extract the explicit Kummer extension directly from the Fourier transform of the roots. If you have an element of the Galois group that cyclically permutes the roots, by weig …

Tyler basically answered the question in the comments, but I might as well fill in an answer. Your definition of "meromorphic modular form" is often called "weakly holomorphic modular form" in the li …

Your observation, which was expanded into a concrete conjecture last year by Harvey and Rayhaun, is now a theorem. See M. Griffin, M. Mertens, "A proof of the Thompson Moonshine conjecture". This …

The question of existence of large prime Fermat numbers $2^{2^n}+1$ is open, but the corresponding question for probable primes is straightforward to solve. $2^{2^n} \equiv -1 \mod 2^{2^n}+1$, so by s …

There are infinitely many Mersenne probable primes. Let $p$ be a prime (or even a probable prime). Then I claim that $2^p-1$ is a probable prime. Proof: $2^k \equiv 1 \pmod{2^p-1}$ if and only if $ …

I'm not an expert on black holes, but I can give you a couple pointers. From work of Bekenstein and Hawking in the 1970s, we are pretty sure that macroscopic black holes in our 3+1 dimensional univer …

The coefficients of a modular form of non-positive weight can be given by an explicit formula that depends only on the poles at cusps (and constant terms when the weight is zero). The asymptotics are …