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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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.

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 $x \in \mathcal O_K$ ...

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 ...

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 ...

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+...

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

quadratic-reciprocity × 26nt.number-theory × 22

algebraic-number-theory × 5

prime-numbers × 4

quadratic-residues × 4

analytic-number-theory × 3

langlands-conjectures × 2

ag.algebraic-geometry × 1

polynomials × 1

algorithms × 1

ho.history-overview × 1

big-list × 1

qa.quantum-algebra × 1

galois-representations × 1

big-picture × 1

computational-number-theory × 1

quantum-field-theory × 1

class-field-theory × 1

characteristic-p × 1

characters × 1

permutation-groups × 1

sums-of-squares × 1

arithmetic-groups × 1

congruences × 1

theta-functions × 1