27 votes

nonabelian reciprocity law

Will Sawin's answer is perfectly correct, but I wanted to add some further perspective. What Peter Scholze has proven is a profound generalization of (one half) of Class Field Theory. The particular ...
Somerville Scholar's user avatar
13 votes

nonabelian reciprocity law

Peter Scholze explains the same example in more detail in this talk https://youtu.be/cFdm0B9KLcQ?t=48m and gives even more detail here https://youtu.be/gI3Ta73yVuo?t=23m55s. In brief, the ...
Will Sawin's user avatar
  • 137k
13 votes
Accepted

Quadratic reciprocity for three primes?

Turns out the details are easy so I worked them out myself :) The highlighted statement is true. Let me assume $4\mid M$. Pick $p_2,p_3$ arbitrary satisfying the congruence modulo $M$ (they exist by ...
Wojowu's user avatar
  • 27.4k
11 votes
Accepted

Jacobi symbols for two-square sums of primes

The first observation follows from the law of quadratic reciprocity. Indeed, assume that $p\equiv 1\pmod{8}$ and $p=A^2+B^2$. Let $A'$ denote the odd part of $A$. Then $p\equiv B^2\pmod{A'}$, and ...
GH from MO's user avatar
7 votes

Could a nice principle be extracted from this lemma of Gauss

In response to your second query "An alternative proof would be also welcome": A (nearly) one-page proof of a stronger version ($q<\sqrt p$ instead of $q<p$)$^\ast$ is lemma 5.6 in ...
Carlo Beenakker's user avatar
5 votes
Accepted

Density of "simultaneous squares"

This should be asymptotic to an expression of the form $$\frac{cX^2}{\log X}$$ as $X \to \infty$, for some $c > 0$ (this should be interpreted as $(X/\sqrt{\log X}\,)^2$). Proving an upper bound of ...
Daniel Loughran's user avatar
5 votes

Could a nice principle be extracted from this lemma of Gauss

See L. Carlitz, "A Note on Gauss' First Proof of the Quadratic Reciprocity Theorem" Proc. Amer. Math. Soc. 11 (1960), 563-565.
KConrad's user avatar
  • 49.6k
4 votes
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Rational prime factors in the components of powers of Gaussian primes

We have the following generalization: Let $K$ be a quadratic extension of $\mathbb Q$, $\mathcal O_K$ the ring of integers, $q$ an odd prime of $\mathbb Q$ unramified in $K$, and $x \in \mathcal O_K$ ...
Will Sawin's user avatar
  • 137k
2 votes

Conceptual explanation for extra/missing $p$ solutions to $x^2+y^2=a \pmod p$ at $a=0$

I'll begin with a disclaimer that this response doesn't actually resolve the numerical coincidence; I don't provide any direct argument that points gained must equal points lost. But I do show a ...
Jonathan Love's user avatar
2 votes

What do theta functions have to do with quadratic reciprocity?

This is not an answer but a comment concerning the Landsberg-Schaar relation (LS). It admits not only analytic proof. The article A proof of the Landsberg-Schaar relation by finite methods by Ben ...
Alexey Ustinov's user avatar
1 vote

Jacobi symbols for two-square sums of primes

It is not an answer but a comment to GH from MO's answer. As I understand, Jacobstahl proved Gauss's congruences in a more direct way. He didn't use Jacobi sums. In Jacobstahl's representation $p=A^2+...
Alexey Ustinov's user avatar

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