# Tag Info

### 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 ...

### 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 ...
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### 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 ...
• 27.4k
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### 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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### 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 ...
• 178k
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### 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 ...
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### 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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### 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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### 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 ...
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