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Here's a quick and dirty version of George Lowther's calculation that I learned from Bjorn Poonen. It is presented in a bit more generality in the next-to-last slide of this talk, so it is in a sense …

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 …

If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves o …

The standard reason why $e^{\pi\sqrt{N}}$ is a near integer for some $N$ is that there is some modular function $f$ with $q$-expansion $q^{-1} + O(q)$, such that substituting $\tau = \frac{1 + i\sqrt …

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 …

If you just want a classical picture over the complex numbers, the objects lying over the cusp points are Néron polygons equipped with some extra structure. To make a Néron $n$-gon, you take $\mathbb …

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 …

There are many equivalent ways to define supersingularity for an elliptic curve over a characteristic $p$ field. One of them is that the $p$-torsion of the curve is connected, i.e., it is a purely in …

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 $ …

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 …

A bit of mastication of Katz-Mazur Theorem 13.7.6 and the surrounding text seems to yield the following description of the special fiber of $Y(p)$: It is fundamentally $p+1$ copies of $\mathbb{P}^1$ …

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 …

Here is a long article by Hazewinkel and a discussion on the nLab. The functor of taking big Witt vectors is right adjoint to the forgetful functor from lambda-rings to commutative rings. Lambda rin …

From the Wikipedia page on Bertrand's postulate, Dusart (1998) showed that for all $x > 3275$, there exists a prime between $x$ and $x \left( 1 + \frac{1}{2 \ln^2 x} \right)$. You are looking for the …

In addition to the power-series methods using Eisenstein series and $\Delta$, and the modular equation methods using, e.g., $h_5$ given in the other answers and comments, there are transcendental meth …