Weighted enumeration of number-fields and counting points that take bounded squarefree values along certain polynomials using Pseudo and Sudo maximal orders

Technical Summary: Weighted Enumeration of Number-Fields and Counting Points that Take Bounded Squarefree Values

Problem Statement
The paper addresses the problem of establishing asymptotic lower bounds for the number of degree-$n$ number fields $K$ with discriminant bounded by $X$, denoted as $N(n, X)$. While Malle's conjecture predicts $N(n, X) \sim c_n X$, this remains unproven for $n \geq 6$. Previous work has established lower bounds using various geometric parametrizations: monic polynomials ($A^1(\mathbb{Z})$), binary forms ($P^1(\mathbb{Z})$), and rings associated with specific quadratic constraints ($P^2(\mathbb{Z})$). A significant limitation in prior research, specifically in the author's previous work [12], was that while a lower bound for the count of certain rings (weakly divisible rings) was established, the corresponding count of distinct number fields fell short. This discrepancy arose because the sieves used to distinguish number fields via local conditions were not effective in the "highly skewed" parameter spaces required to parametrize these rings.

Methodology
The paper introduces a new sieve tailored to these skewed regions, building upon the work of Ekedahl, Poonen, Stoll, and Bhargava. The core methodology involves:

1. Definitions of Special Orders:

   • Pseudo-maximal orders: Irreducible orders $O$ where the local invariant $ef(O) = 2$.
   • Sudo-maximal orders: Orders whose irreducible decomposition consists entirely of pseudo-maximal orders.
   • Restricted Sudo-maximal orders (ReSMO): Sudo-maximal orders where, for every rational prime $p$, at most one prime ideal in the singular locus $S(O)$ lies above $p$.

2. Weakly Divisible Forms: The paper utilizes "ultra weakly divisible" binary forms. These are forms $f$ such that for any prime $p$, if $p^2 | \text{disc}(f)$, then $f$ is not "strongly divisible" by $p$ (i.e., $f$ does not have a triple root or two double roots modulo $p$). The author establishes a bijection between ultra weakly divisible forms and ReSMOs. Specifically, if $f$ is ultra weakly divisible, the associated ring $R'_{(f, m_f, l_f)}$ is the ring of integers of the field generated by $f$, and any intermediate ring is a ReSMO.

3. Polynomial Construction ($\eth_n$): To facilitate counting, the author defines a multivariate polynomial $\eth_n$ whose variables parametrize the coefficients of binary forms that are weakly divisible by a squarefree integer $m$. The squarefree locus of $\eth_n$ corresponds one-to-one with the locally non-singular locus of the discriminant form (specifically, ultra weakly divisible forms).

4. Sieving Strategy:

   • The paper employs a version of the Ekedahl sieve adapted to the discriminant polynomial $\eth_n$.
   • It calculates local densities of binary forms that are not strongly divisible modulo $p$.
   • It counts the number of such forms in specific boxes (parameterized by height and divisibility conditions) and sums over the modulus $m$.
   • Crucially, the sieve is applied to enforce modular conditions within error terms, allowing the count to operate effectively in the skewed regions where previous sieves failed.

Key Contributions and Results

1. Improved Lower Bound for Number Fields:
   The paper establishes a new asymptotic lower bound for $N(n, X)$ (the count of degree-$n$ fields with Galois group $S_n$ and discriminant $< X$).

   • Theorem 1: For $n \geq 5$, $N(n, X) \gg_n X^{\frac{1}{2} + \frac{1}{n} - \frac{4}{3} - \epsilon}$ infinitely often.
   • This improves upon previous bounds (e.g., Bhargava-Shankar-Wang's $X^{\frac{1}{2} + \frac{1}{n-1}}$) by utilizing the geometry of rings associated with the variety defined by $aXY - bZ^2$.

2. Weighted Counts:
   The paper provides weighted lower bounds for the number of fields, defined as $N^*(n, X) = \sum_{K \in S(n, X)} \frac{1}{\sqrt{\text{disc}(K)}}$.

   • Theorem 3: For $n \geq 5$, $N^*(n, X) \gg_n X^{\frac{1}{n} - \frac{4}{3}} (\log \log X)^{\dots} (\log X)^{\dots}$.
   • This implies a weighted lower bound for the number of fields with bounded discriminant of $\gg X^{\frac{1}{2} + \frac{1}{n} - \frac{4}{3} - \epsilon}$.

3. Counting Squarefree Values of $\eth_n$:

   • Theorem 5: The paper proves a lower bound for the number of points $P$ where the polynomial $\eth_n(P)$ takes a squarefree value bounded by $X$, modulo symmetries. The count is $\gg_n X^{\frac{1}{2} + \frac{1}{n} - 1.5} (\log \log X)^{\dots} (\log X)^{\dots}$.
   • This result relies on the bijection between the squarefree locus of $\eth_n$ and the set of ultra weakly divisible forms.

4. Specific Ring Structures:

   • The results apply to number fields whose rings of integers admit specific geometric presentations, such as $\mathbb{Z}[\alpha, \beta]$ where $\alpha\beta \in \mathbb{Z}$ (Theorem 6) or rings of the form $\mathbb{P}(\mathbb{Z}[X, Y, Z]/I)$ where $I$ contains a quadratic form $aXY - bZ^2$ (Theorem 50).
   • The paper shows that these specific structures yield the improved exponent in the lower bound.

5. Unramified Extensions:

   • Theorem 7: The results are applied to count unramified $A_n$-extensions of quadratic fields $\mathbb{Q}(\sqrt{s})$. The paper derives a lower bound for $\sum_{s < X} \frac{\log |\text{Cl}_{\mathbb{Q}(\sqrt{s})}[A_n]|}{\sqrt{s}}$, improving upon previous estimates derived from monogenic fields.

Significance and Claims
The paper claims that by developing a sieve effective in the "skewed" parameter spaces of weakly divisible rings, it overcomes the technical difficulties that previously prevented the translation of ring counts into number field counts. The primary significance lies in:

• Advancing the Lower Bound: Pushing the known lower bound exponent for $N(n, X)$ closer to the conjectured value of 1 (Malle's conjecture) for $n \geq 5$.
• Geometric Diversity: Demonstrating that number fields with rings of integers associated with specific geometric varieties (specifically those related to $P^2(\mathbb{Z})$ and the equation $aXY - bZ^2$) contribute significantly to the total count of number fields.
• Methodological Innovation: The successful application of the Ekedahl sieve to the polynomial $\eth_n$ in regions where standard sieves fail, allowing for the enumeration of squarefree values in a non-standard parameter space.

The author notes that while the results are robust for $n \geq 5$, the case $n=4$ presents challenges regarding double-counting in the translation orbits, which are not fully resolved in this work. The paper concludes that these findings provide a "weighted lower bound" that is strictly stronger than previous results for the specified range of $n$.