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Brandon Alberts

Publications and source records attributed to Brandon Alberts.

15 recordsLinked to original sources
arXiv2026

Inertial multiplicity bounds for two dimensional projective representations and bounds for the number of $\operatorname{PSL}_2(\mathbb{F}_q)$ and $\operatorname{PGL}_2(\mathbb{F}_q)$ number fields

We prove upper bounds for certain number field counting functions using Serre's modularity conjecture (now a theorem of Khare--Wintenberger). These results are comparable to sharp upper bounds for the number of abelian extensions with fixed or bounded discriminant, with Serre's modularity conjecture playing the role of class field theory.

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arXiv2026

Counting number fields using multiple Dirichlet series

We provide a method for counting number fields of fixed Galois group ordered by arbitrary inertial invariants using analytic techniques from the study of multiple Dirichlet series. We prove unconditional results for infinitely many new (concentrated and semiconcentrated) groups that were not approachable by previous methods. Conditional on subconvexity bounds bounds for certain Dirichlet series (e.g. the generalized Lindel\"of hypothesis), we use these techniques to prove the existence of an asymptotic growth rate for $G$-extensions for infinitely many new groups $G$ for which the minimum index elements of $G$ are contained in a union of proper abelian normal subgroups. In particular, our conditional results include all groups with nilpotency class $2$. Additionally, when $G$ is nilpotent our results give a power saving error term.

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arXiv2025

An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields

Given a Dirichlet series $L(s) = \sum a_n n^{-s}$, the asymptotic growth rate of $\sum_{n\le X} a_n$ can be determined by a Tauberian theorem. Bounds on the error term are typically controlled by the size of $|L(\sigma+it)|$ for fixed real part $\sigma$. We modify this approach to prove new Tauberian theorems with error terms depending only on the average size of $L(\sigma+it)$ as $t$ varies, and we take care to track explicit dependence on various parameters. This often leads to stronger error bounds, and introduces strong connections between asymptotic counting problems and moments of $L$-functions. We provide self-contained statements of Tauberian theorems in anticipation that these results can be used ``out of the box'' to prove new asymptotic expansions. We demonstrate this by proving square root saving error bounds for the number of $C_n$-extensions of $\mathbb{Q}$ of bounded discriminant when $n=3$, $4$, $8$, $16$, or $2p$ for $p$ an odd prime.

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arXiv2025

Inductive methods for counting number fields

We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle's Conjecture and counterexamples to Malle's Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group $G$. Our method relies on having asymptotic counts for $T$-extensions for some normal subgroup $T$ of $G$, uniform bounds for the number of such $T$-extensions, and possibly weak bounds on the asymptotic number of $G/T$-extensions. However, we do not require that most $T$-extensions of a $G/T$-extension are $G$-extensions. Our new results use $T$ either abelian or $S_3^m$, though our framework is general.

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arXiv2024

Explicit Analytic Continuation of Euler Products

The generating series of a number of different objects studied in arithmetic statistics can be built out of Euler products. Euler products often have very nice analytic properties, and by constructing a meromorphic continuation one can use complex analytic techniques, including Tauberian theorems to prove asymptotic counting theorems for these objects. One standard technique for producing a meromorphic continuation is to factor out copies of the Riemann zeta function, for which a meromorphic continuation is already known. This paper is an exposition of the "Factorization Method" for meromorphic continuation. We provide the following three resources with an eye towards research in arithmetic statistics: (1) an introduction to this technique targeted at new researchers, (2) exposition of existing works, with self-contained proofs, that give a continuation of Euler products with constant or Frobenain coefficients to the right halfplane ${\rm Re}(s)>0$ (away from an isolated set of singularities), and (3) explicit statements on the locations and orders of all singularities for these Euler products.

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arXiv2024

Power Savings for Counting (Twisted) Abelian Extensions of Number Fields

We prove significant power savings for the error term when counting abelian extensions of number fields (as well as the twisted version of these results for nontrivial Galois modules). In some cases over $\mathbb{Q}$, these results reveal lower order terms following the same structure as the main term that were not previously known. Assuming the generalized Lindel\"of hypothesis for Hecke $L$-functions, we prove square root power savings for the error compared to the order of the main term.

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arXiv2023

A Random Group with Local Data Realizing Heuristics for Number Field Counting

We define a group with local data over a number field $K$ as a group $G$ together with homomorphisms from decomposition groups ${\rm Gal}(\overline{K}_p/K_p)\to G$. Such groups resemble Galois groups, just without global information. Motivated by the use of random groups in the study of class group statistics, we use the tools given by Sawin-Wood to construct a random group with local data over $K$ as a model for the absolute Galois group ${\rm Gal}(\overline{K}/K)$ for which representatives of Frobenius are distributed Haar randomly as suggested by Chebotarev density. We utilize Law of Large Numbers results for categories proven by the author to show that this is a random group version of the Malle-Bhargava principle. In particular, it satisfies number field counting conjectures such as Malle's Conjecture under certain notions of probabilistic convergence including convergence in expectation, convergence in probability, and almost sure convergence. These results produce new heuristic justifications for number field counting conjectures, and begin bridging the theoretical gap between heuristics for number field counting and class group statistics.

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arXiv2022

Counting Functions for Random Objects in a Category

In arithmetic statistics and analytic number theory, the asymptotic growth rate of counting functions giving the number of objects with order below $X$ is studied as $X\to \infty$. We define general counting functions which count epimorphisms out of an object on a category under some ordering. Given a probability measure $\mu$ on the isomorphism classes of the category with sufficient respect for a product structure, we prove a version of the Law of Large Numbers to give the asymptotic growth rate as $X$ tends towards $\infty$ of such functions with probability $1$ in terms of the finite moments of $\mu$ and the ordering. Such counting functions are motivated by work in arithmetic statistics, including number field counting as in Malle's conjecture and point counting as in the Batyrev-Manin conjecture. Recent work of Sawin--Wood gives sufficient conditions to construct such a measure $\mu$ from a well-behaved sequence of finite moments in very broad contexts, and we prove our results in this broad context with the added assumption that a product structure in the category is respected. These results allow us to formalize vast heuristic predictions about counting functions in general settings.

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arXiv2021

Harmonic Analysis and Statistics of the First Galois Cohomology Group

We utilize harmonic analytic tools to count the number of elements of the Galois cohomology group $f\in H^1(K,T)$ with discriminant-like invariant ${\rm inv}(f)\le X$ as $X\to\infty$. Specifically, Poisson summation produces a canonical decomposition for the corresponding generating series as a sum of Euler products for a very general counting problem. This type of decomposition is exactly what is needed to compute asymptotic growth rates using a Tauberian theorem. These new techniques allow for the removal of certain obstructions to known results and answer some outstanding questions on the generalized version of Malle's conjecture for the first Galois cohomology group.

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arXiv2020

Statistics of the First Galois Cohomology Group: A Refinement of Malle's Conjecture

Malle proposed a conjecture for counting the number of $G$-extensions $L/K$ with discriminant bounded above by $X$, denoted $N(K,G;X)$, where $G$ is a fixed transitive subgroup $G\subset S_n$ and $X$ tends towards infinity. We introduce a refinement of Malle's conjecture, if $G$ is a group with a nontrivial Galois action then we consider the set of crossed homomorphisms in $Z^1(K,G)$ (or equivalently $1$-coclasses in $H^1(K,G)$) with bounded discriminant. This has a natural interpretation given by counting $G$-extensions $F/L$ for some fixed $L$ and prescribed extension class $F/L/K$. If $T$ is an abelian group with any Galois action, we compute the asymptotic growth rate of this refined counting function for $Z^1(K,T)$ (and equivalently for $H^1(K,T)$) and show that it is a natural generalization of Malle's conjecture. The proof technique is in essence an application of a theorem of Wiles on generalized Selmer groups, and additionally gives the asymptotic main term when restricted to certain local behaviors. As a consequence, whenever the inverse Galois problem is solved for $G\subset S_n$ over $K$ and $G$ has an abelian normal subgroup $T\trianglelefteq G$ we prove a nontrivial lower bound for $N(K,G;X)$ given by a nonzero power of $X$ times a power of $\log X$. For many groups, including many solvable groups, these are the first known nontrivial lower bounds. These bounds prove Malle's predicted lower bounds for a large family of groups, and for an infinite subfamily they generalize Klüners' counter example to Malle's conjecture and verify the corrected lower bounds predicted by Türkelli.

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arXiv2019

The Weak Form of Malle's Conjecture and Solvable Groups

For a fixed finite solvable group $G$ and number field $K$, we prove an upper bound for the number of $G$-extensions $L/K$ with restricted local behavior (at infinitely many places) and ${\rm inv}(L/K)<X$ for a general invariant $"{\rm inv}"$. When the invariant is given by the discriminant for a transitive embedding of a nilpotent group $G\subset S_n$, this realizes the upper bound given in the weak form of Malle's conjecture. For other solvable groups, the upper bound depends on the size of torsion of the class group of number fields with fixed degree. In particular, the bounds we prove realize the upper bound given in the weak form of Malle's conjecture for the transitive embedding of a solvable group $G\subset S_n$ if we assume that for each finite abelian group $A$ the average size of class group torsion $|{\rm Hom}({\rm Cl}(L),A)|$ is smaller than $X^ε$ as $L/K$ varies over certain families of extensions with ${\rm inv}(L/K)<X$.

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arXiv2017

Certain Unramified Metabelian Extensions Using Lemmermeyer Factorizations

We study solutions to the Brauer embedding problem with restricted ramification. Suppose $G$ and $A$ are a abelian groups, $E$ is a central extension of $G$ by $A$, and $f:\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow G$ a continuous homomorphism. We determine conditions on the discriminant of $f$ that are equivalent to the existence of an unramified lift $\widetilde{f}:\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow E$ of $f$. As a consequence of this result, we use conditions on the discriminant of $K$ for $K/\mathbf{Q}$ abelian to classify and count unramified nonabelian extensions $L/K$ normal over $\mathbf{Q}$ where the (nontrivial) commutator subgroup of $\text{Gal}(L/\mathbf{Q})$ is contained in its center. This generalizes a result due to Lemmermeyer, which states that a quadratic field $\mathbf{Q}(\sqrt{d})$ has an unramified extension normal over $\mathbf{Q}$ with Galois group $H_8$ the quaternion group if and only if the discriminant factors $d=d_1 d_2 d_3$ as a product of three coprime discriminants, at most one of which is negative, satisfying the following condition on Legendre symbols: \[ \left(\frac{d_i d_j}{p_k}\right)=1 \] for $\{i,j,k\}=\{1,2,3\}$ and $p_i$ any prime dividing $d_i$.

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arXiv2017

The distribution of $H_{8}$-extensions of quadratic fields

We compute all the moments of a normalization of the function which counts unramified $H_{8}$-extensions of quadratic fields, where $H_{8}$ is the quaternion group of order 8, and show that the values of this function determine a constant distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified G-extensions of quadratic fields for G in a large class of 2-groups, which we conjecture will give finite moments which determine a distribution. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified $H_{8}$-extensions.

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arXiv2016

Cohen-Lenstra Moments for Some Nonabelian Groups

Cohen and Lenstra detailed a heuristic for the distribution of odd p-class groups for imaginary quadratic fields. One such formulation of this distribution is that the expected number of surjections from the class group of an imaginary quadratic field k to a fixed odd abelian group is 1. Class field theory tells us that the class group is also the Galois group of the Hilbert class field, the maximal unramified abelian extension of k, so we could equivalently say the expected number of unramified G-extensions of k is 1/#Aut(G) for a fixed abelian group G. We generalize this to asking for the expected number of unramified G-extensions Galois over $\mathbb{Q}$ for a fixed finite group G, with no restrictions placed on G. We review cases where the answer is known or conjectured by Boston-Wood, Boston-Bush-Hajir, and Bhargava, then answer this question in several new cases. In particular, we show when the expected number is zero and give a nontrivial family of groups realizing this. Additionally, we prove the expected number for the quaternion group $Q_8$ and dihedral group $D_4$ of order 8 is infinite. Lastly, we discuss the special case of groups generated by elements of order 2 and give an argument for an infinite expected number based on Malle's conjecture.

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