Three perspectives on n points in P^{n-2}

Abstract

This thesis is divided into three parts, each focused on extending the work of Bhargava's Higher Composition Laws and his asymptotics of the density of discriminants.

The first part is focused on the gemoetry of numbers methods of . In this part, we prove asymptotics for the class number times the regulator of cubic fields and orders, and for the square of the same for quadratic fields and orders. This requires extending Bhargava's methods to prehomogeneous vector spaces with a positive-dimensional generic isotropy group.

The second part focuses on more traditional analytic methods. In it, we demonstrate that the values $L(1, k)$ for $k$ an $S_3$-cubic or $S_4$-quartic number field behave like random Euler products, extending work of Barban and Eliott from the quadratic case. More generally, we show that the same holds if $k$ is replaced by an order in such a field. Conditional on the Strong Artin Conjecture for Galois representations $\Gal(\Bar{\Q}/\Q) \to \GL_{n-1}(\C)$ with image $S_n$ and what we call the Strong Malle Conjecture for $S_n$ number fields, we prove that $L(1, k)$ behaves like a random Euler product for $k$ an $S_n$ number field for any $n \geq 2$.

In the final part, we turn to putting Bhargava's Higher Composition Laws in a more commutative algebraic context. In particular, taking inspriation from a letter of Deligne to Bhargava and Wood's thesis, we produce a canonical associative, commutaive, differential graded algebra structures on the minimal free resolutions of $n$ general

points in $\bP^{n-2}_\Q$. We believe our method is novel, relying on the representation theory of the symmetric group. In our penultimate chapter, we show that this method extends to produce canonical multiplicative structures on other free resolutions associated to the representation theory of the symmetric group. Of particular interest is that all of the resolutions we produce are pure and do not appear to arise from the so-called tensor complexes of Berkesch, Erman, Kummini, and Sam.

Finally, in the last chapter, we briefly discuss the representation theoretic import of the sextic resolvent of a quintic field. At this time, we have not been able to use this to extend Bhargava's parameterization of quintic rings to arbitrary base schemes in the spirit of Deligne and Wood, but we believe that this chapter forms an important start in that direction.