[Paper Review] Counting $S_5$-fields with a power saving error term

Arul Shankar, Jacob Tsimerman|arXiv (Cornell University)|Oct 8, 2013

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper establishes the first known power-saving error term for counting $S_5$-quintic fields by applying the Selberg $\Lambda^2$-sieve to refine counting methods in the geometry of numbers. It proves that the number of $S_5$-quintic fields with discriminant bounded by $X$ is asymptotically $d_i \prod_p(1 + p^{-2} - p^{-4} - p^{-5})X + O_\epsilon(X^{399/400 + \epsilon})$, where $d_0 = 1/240$, $d_1 = 1/24$, and $d_2 = 1/16$, achieving a non-trivial error saving beyond the main term.

ABSTRACT

We show how the Selberg $Λ^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, we give such an error term for the counting function of $S_5$-quintic fields.

Motivation & Objective

• To establish a power-saving error term for the counting function of $S_5$-quintic fields with bounded discriminant.
• To extend the application of the Selberg $\Lambda^2$-sieve to geometry-of-numbers counting problems with degenerate orbits.
• To refine prior asymptotic counts by obtaining a non-trivial error term of the form $O(X^{a - \delta})$ with $\delta > 0$.
• To demonstrate the method on $S_5$-quintic fields, achieving the first such power-saving result in this context.

Proposed method

• Applies the Selberg $\Lambda^2$-sieve to control the number of degenerate $G_\mathbb{Z}$-orbits in the space of quintic rings.
• Uses uniform density estimates for congruence conditions modulo $m$, ensuring $|L \cap F_0(X)| = \mu(L)cX^a\log^bX + O(X^{a - \delta}m^A)$ for fixed $A$.
• Employs a hybrid sieve strategy: first sieve over primes to control degenerate points via $\bigcap_p S_p(5)$ and $\bigcap_p S_p(1112)$, then apply a second sieve to pass from orders to fields.
• Leverages Bhargava’s parametrization of quintic rings via $G_\mathbb{Z}$-orbits on $V_\mathbb{Z}$, where $V_\mathbb{Z}$ consists of quadruples of $5 \times 5$ alternating matrices.
• Uses the uniformity estimate from [5, Proposition 19] generalized to $N(W_d, X) = O_\epsilon(X/d^{2 - \epsilon})$ for maximal orders.
• Optimizes the sieve parameter $T = X^{1/400}$ to balance error terms in the final asymptotic, yielding $O_\epsilon(X^{399/400 + \epsilon})$.

Experimental results

Research questions

• RQ1 Can the Selberg $\Lambda^2$-sieve be systematically applied to obtain power-saving error terms in geometry-of-numbers counting problems?
• RQ2 What is the optimal error term achievable for the count of $S_5$-quintic fields with bounded discriminant?
• RQ3 How can degenerate $G_\mathbb{Z}$-orbits (non-$S_5$-fields) be controlled with a power-saving error term using sieve methods?
• RQ4 Can the transition from $S_5$-orders to $S_5$-fields be achieved with a power-saving error term via a second sieve?

Key findings

• The number of $S_5$-quintic fields with $i$ complex places and discriminant bounded by $X$ is asymptotically $d_i \prod_p(1 + p^{-2} - p^{-4} - p^{-5})X + O_\epsilon(X^{399/400 + \epsilon})$, with $d_0 = 1/240$, $d_1 = 1/24$, $d_2 = 1/16$.
• The error term $O_\epsilon(X^{399/400 + \epsilon})$ represents a power-saving improvement over the main term, with exponent $399/400 = 1 - 1/400$.
• The method achieves power-saving by applying the Selberg $\Lambda^2$-sieve to control the number of degenerate $S_5$-orders, yielding $N^*(V^{\rm deg,(i)}_\mathbb{Z}, X) \ll_\epsilon X^{199/200 + \epsilon}$.
• The final sieve from orders to fields uses the uniformity $N(W_d, X) = O_\epsilon(X/d^{2 - \epsilon})$, which allows balancing error terms via $T = X^{1/400}$.
• The proof establishes that the density of maximal $S_5$-orders is $\prod_p(1 - c_p)$, with $c_p = O_\epsilon(p^{-2 + \epsilon})$, and the main term matches known asymptotic constants.
• The result is the first known power-saving error term for $S_5$-quintic fields, resolving a key challenge in low-lying zero statistics and arithmetic statistics.