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I'll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least c …

I was wondering about the distribution of $\sqrt{p}$ mod $1$ this morning, as one does while brushing one's teeth. I remembered the paper of Elkies and McMullen (Duke Math. J. 123 (2004), no. 1, 95–1 …

I don't think that such a survey paper or textbook exists, but the closest thing I know of is "A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture" by Spencer Blo …

Hi David, This is the first example of a multiple zeta identity. Your sum S is just $\zeta(1,2)$, where the multiple zeta value is defined by: $$\zeta(s_1, s_2, \ldots, s_k) = \sum_{0 < n_1 < n_2 < …

There's no more serious name for the topograph, as far as I know. And Conway puts a lot of thought into his names, so I think it's best to keep it. I think it's meant to fit into a larger metaphoric …

I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ple …

In the case $p \equiv 1$ mod $4$, the connection is to the real quadratic field ${\mathbb Q}(\sqrt{p})$, whereas the case $p \equiv 3$ mod $4$ is connected to the imaginary quadratic field ${\mathbb Q …

Upon David Loeffler's request, here is a more fleshed out version of my former comments: In his comment, Nick Ramsey mentioned that the natural L-function for a half-integral weight modular form is re …

I think, although it's dated later than Deligne's paper that you mentioned, that the first written instance of Kazhdan's principle is in the paper "Representations of groups over close local fields", …

This is a common point of confusion, and the OP is on exactly the right track. A good reference for the representation theory is Chapter 1, Section 6, of Jacquet-Langlands book "Automorphic forms on …

First, I'd like to second the reference given by JT: David Vogan, "The local Langlands conjecture", appearing in Representation Theory of Groups and Algebras (J. Adams et al., eds. Contemporary Mathe …

It is known that $H^2(W, C^\times)$ is trivial, when $W$ is the Weil group of a global or local field, with the trivial action on $C^\times$, and the cohomology is taken in the sense of Moore (measura …

In my opinion, instead of a "height" on the Weil-Chatelet group, one should consider a "depth", using the local duality between the points on an elliptic curve and the elements of the Weil-Chatelet gr …

The Langlands correspondence for higher local fields is still at an early stage of development. I haven't really kept up with it, but here's some key points. As the question stated, and Loren commen …

From Exercise 5.14, Ch. V of Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves", I learned that the local duality for elliptic curves over $p$-adic fields can be proven for Tate curves …