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7
votes
1answer
110 views

Hilbert symbol averages

Let me call a pair of integers $a, b$ acceptable if the equation $ax^2 + by^2 = z^2$ has a non-trivial rational solution. Theorem 4.5.4 of Cojocaru-Murty's book on Sieves says that the number of ...
0
votes
1answer
52 views

Norm Residue Symbol refinement?

From Wikipedia: given $a\in K^\times$, (a,b)=1 for all b [in K*] if and only if a is in K*ⁿ So suppose that $(\frac{a\ ,\ K^\times\!}{p})\neq 1$ [assume $n$ above ...
1
vote
0answers
115 views

Skew symmetry for the Hilbert symbol

Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ ...
9
votes
1answer
934 views

Weil reciprocity vs Artin reciprocity

This is probably an easy question for the experts: Given two rational functions $f$, $g$ on a non-singular projective algebraic curve X (over an algebraically closed field $k$) and $p \in X$, one ...
8
votes
0answers
438 views

Hilbert symbol and Weil index, beyond the quadratic case?

Let $F$ be a local nonarchimedean field. Let $n$ be a positive integer for which the group $\mu_n(F)$ of $n^{th}$ roots of unity in $F$ has order $n$. Let $\epsilon: \mu_n(F) \rightarrow C^\times$ ...