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Let $N \in \mathbb{Z}$ be a prime number, and let $B = \left( \dfrac{a, b}{\mathbb{Q}} \right)$ be the unique quaternion algebra over $\mathbb{Q}$ ramified at $N$ and at $\infty$. Then, in section 3 of his paper "Heights and the Special Values of L-series", Gross constructs a genus 0 curve $Y$ over $\mathbb{Q}$ associated to the quaternion algebra $B$ as follows.

For any $\mathbb{Q}$-algebra $E$, the points of $Y$ in $E$ are given by

$$ Y(E) = \{ \alpha \in B \otimes_{\mathbb{Q}} E \mid \operatorname{Tr}{(\alpha)} = 0 = \operatorname{N}(\alpha) \} / E^{\times} $$

where $\operatorname{Tr}(\cdot)$ and $\operatorname{N}(\cdot)$ are the reduced trace and the reduced norm in the quaternion algebra $B$, given as follows. For $h = x + iy + jz + ijw \in B$, with $x, y, z, w \in \mathbb{Q}$,

\begin{align} \operatorname{Tr}(h) &= h + \overline{h} = 2x\\ \operatorname{N}(h) &= h\overline{h} = x^2 - ay^2 - bz^2 + abw^2 \end{align}

Now, my confusion is that then I'm not sure about what it means to take the reduced trace and norm of an element $\alpha \in B \otimes_{\mathbb{Q}} E$.

Finally, I have also seen in another paper describing the same construction, that then combining the two equations $\operatorname{Tr}{(\alpha)} = 0 = \operatorname{N}(\alpha)$, $Y$ is basically the conic $ay^2 + bz^2 = abw^2$.

Questions

  1. How should I interpret taking the trace $\operatorname{Tr}{(\alpha)}$ and norm $\operatorname{N}{(\alpha)}$ for $\alpha \in B \otimes_{\mathbb{Q}} E$ ?

  2. According to the last observation about $Y$ being the conic $ay^2 + bz^2 = abw^2$, does that mean that $Y(E)$ can be thought of as $\{ (y, z, w) \in E^3 \mid ay^2 + bz^2 = abw^2 \}$?

Thank you very much for any help.

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1 Answer 1

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$B\otimes E$ is an $E$-algebra, and an element $\alpha$ induces a linear map $B\otimes E\to B\otimes E$ given by $x \mapsto \alpha x$ and you can think of trace and norm of $\alpha$ as respectively the trace and determinant of this linear transformation. Since you are just extending scalars, you will get the same formulas you had before. That should answer 1.

For 2., you forgot to mod out by $E^{\times}$ so the actual result is the set of points in the projective plane satisfying the equation.

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    $\begingroup$ No, the trace and norm of those maps are respectively twice and the square of the "reduced trace" and "reduced norm". The reduced trace and reduced norm of a central simple algebra $C$ of rank $n^2$ over an arbitrary field $k$ (of any characteristic) are defined as homogeneous polynomial maps (of respective degrees 1 and $n$) by choosing a finite Galois splitting field $k'/k$ and carrying out Galois descent of the matrix trace and norm on $C_{k'} \simeq {\rm{Mat}}_n(k')$, using Skolem-Noether (to handle ambiguity of this latter isomorphism) and conjugation-invariance of matrix trace and norm. $\endgroup$
    – Marguax
    Commented Oct 10, 2013 at 2:20
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    $\begingroup$ Also, the description of $Y(E)$ by Gross is wrong when $E$ isn't local for the same reason that the points in $\mathbf{P}^n(E)$ are not generally given by $(\mathbf{A}^{n+1} - \{0\})(E)/E^{\times}$ when $E$ isn't local. That is, there are obstructions arising from ${\rm{Pic}}(E)$ (invisible if $E$ is a field, but already a serious issue if $E$ is as mild as a Dedekind domain with nontrivial class group). The same applies to the discussion of a projective conic. $\endgroup$
    – Marguax
    Commented Oct 10, 2013 at 2:23
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    $\begingroup$ @Adrian: I told you the definition, so you don't need a reference. :) Again: after choosing such $k'/k$ so we have an isomorphism $f:C_{k'} \simeq {\rm{Mat}}_n(k')$ that is well-defined up to conjugation by ${\rm{GL}}_n(k')$, we can transfer the conjugation-invariant trace and norm over to $C_{k'}$. Independence of $f$ implies that these maps $C_{k'} \rightarrow k'$ are Galois-invariant and so descend to maps $C \rightarrow k$ visibly independent of $k'/k$ and given by homogeneous polynomials with respective degrees 1 and $n$. Use the same polynomials after scalar extension to any $k$-algebra. $\endgroup$
    – Marguax
    Commented Oct 10, 2013 at 3:39
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    $\begingroup$ @Adrian: In the preceding, to be "safe" when $k$ is finite one ought to demand $k'$ is infinite (so take $k' = k_s$, noting that finiteness of $[k':k]$ isn't ever needed, just the Galois condition). Any graduate algebra book discussing central simple algebras ought to discuss the reduced trace and reduced norm built via Galois descent. (I never learned this from a book; I guessed it by osmosis from seeing how the concepts are used.) Perhaps try Jacobson's "Basic Algebra II"? $\endgroup$
    – Marguax
    Commented Oct 10, 2013 at 3:42
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    $\begingroup$ @Adrián: I think the book "Central simple algebras and Galois cohomology" by Gille and Szamuely should certainly answer many of your questions. $\endgroup$
    – Daniel Loughran
    Commented Oct 10, 2013 at 8:35

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