As initially planned several weeks ago, let’s meet this afternoon and try to adapt the ideas of Cassels’s isotropy theorem to apply to quadratic forms over k[t] (as has been done in a Crelle paper of Prestel, but the idea is to try to do it ourselves rather than closely read his paper). We should probably also spend some time discussing future directions and talks in the seminar. Hope to see you at 2:30 pm today!

Hi All,

For this week’s talk, Hans Parshall will be talking about the “Cassels Isotropy Theorem.”

Hope to see you all there!
-Danny

Hi QForms folks,

This week, Lee Troupe will be giving a talk titled “Towards a Function Field Cochrane-Mitchell Theorem.” The abstract is below — hope you see you all there!

-Danny

Abstract: In the initial discussion of Legendre’s Theorem, it was made clear that may be viewed as the n = 3, K = Q case of the celebrated Hasse-Minkowski Theorem.  In general, it is a natural (and wide open!) problem to find other cases of Hasse-Minkowski which can be proved by GoN methods.  One advantage of the GoN approach is that there is also the prospect for an explicit upper bound on the size of an isotropic vector.  (This is also largely an open problem: when explicit bounds are known, they are generally known not to be sharp in any reasonable sense.)

In this talk we pursue the case n = 3, K = F_q(t), where q is an odd prime power.  Both linear forms and Hermite constants have proven analogues in this context.  Perhaps surprisingly, linear forms methods seem more auspicious; in particular, absent a meaningful notion of “majorization” in this context, an approach via Hermite constants doesn’t yield explicit bounds.  We will present a linear forms theorem due to L. Tornheim and try to apply it to prove a Cochrane-Mitchell type result.  This is work (jointly with Pete L. Clark) in progress: as of this writing, it seems that we can prove the result when deg(a), deg(b), deg(c) are neither all even nor all odd.  Help from the audience will be warmly appreciated.

Hi All,

Coming up next, we’ll have a talk by Jacob Hicks continuing on the “geometry of numbers” theme. More detailed information below!

Hope to see you there,
-Danny

Title: The Cochrane-Mitchell Theorem

Abstract: We present Cochrane-Mitchell’s strengthening of the GoN proof of Legendre’s Theorem which simultaneously achieves the existence of a solution and the Holzer bound on the size of a solution. The key idea is to choose a “better lattice”: one with congruence
properties modulo 2abc rather than modulo abc. This turns out to use significantly more GoN input, namely the sharp value of the three dimensional Hermite constant and the classification of extremal (a theorem of Gauss). Time permitting, we will discuss a natural followup question: what are the “best lattices” one can find in this context?

This Friday, we’ll be starting off the quadratic forms seminar with a talk by Pete Clark. He’ll be giving a proof of Legendre’s Theorem on rational points on conics using the Geometry of Numbers.

We’ll be meeting in Boyd 326 at 2:30pm. This will likely be our room and time for the remainder of the semester.

All are welcome!

-Danny