30
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 ...
14
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 ...
- 148k
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 ...
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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 ...
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7
votes
Accepted
About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $
Values for $a$ and $b$ as polynomials in $x,y$, when they exist, correspond to factorization of $\frac{x^p+y^2}{x+y}$ over the imaginary field $K_p:=\mathbb Q(\sqrt{-p})$ of the form:
$$\frac{x^p+y^p}{...
- 34.3k
7
votes
Could a nice principle be extracted from this lemma of Gauss
In response to your second query "An alternative proof would be also welcome":
A (nearly) one-page proof of a stronger version ($q<\sqrt p$ instead of $q<p$)$^\ast$ is lemma 5.6 in ...
- 188k
5
votes
Could a nice principle be extracted from this lemma of Gauss
See L. Carlitz, "A Note on Gauss' First Proof of the Quadratic Reciprocity Theorem" Proc. Amer. Math. Soc. 11 (1960), 563-565.
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5
votes
Accepted
Density of "simultaneous squares"
This should be asymptotic to an expression of the form
$$\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 ...
- 21.3k
5
votes
What do theta functions have to do with quadratic reciprocity?
I think you're looking for the work of Tomio Kubota.
The square of the theta function is a modular form. For a while, and still today, the theta function itself is sometimes considered a modular form ...
5
votes
About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $
Friday, June 28. I found a nice exposition by David Savitt
https://pi.math.cornell.edu/~web401/steve.gauss17gon.pdf
from which this is page 32
David A. Cox, in Galois Theory, gives an account of ...
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4
votes
Accepted
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 $x \in \mathcal O_K$ ...
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3
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 ...
- 12.3k
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 ...
- 1,343
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+...
- 12.3k
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