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Daniel Loughran

Publications and source records attributed to Daniel Loughran.

At least 19 recordsLinked to original sources
arXiv2026

The leading constant in Malle's conjecture

We give an overview of a recent conjecture of the authors on the leading constant in Malle's conjecture on number fields of bounded discriminant. This comes from applying the philosophy from Manin's conjecture on rational points of bounded height on Fano varieties to classifying stacks. To make these ideas more accessible we assume no background in algebraic geometry, which requires some new perspectives and alternative approaches to the theory. We also give some new conjectures on multi-heights and Bhargava's heuristics on counting with local conditions imposed.

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arXiv2026

Thin sets in weighted projective stacks

We prove an upper bound for the number of rational points of bounded height in a weighted projective stack which lie in a given thin subset. As a consequence, we show that $100\%$ of hyperelliptic curves do not admit a prescribed on-trivial level structure.

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arXiv2025

Fermat near misses and the integral Hilbert Property

We consider the Diophantine equation $x^4 + y^4 - w^2 = n$ for $n \in \mathbb{Z}$, which is related to near misses for the quartic case of Fermat's Last Theorem. For certain $n$ we show that the set of solutions is infinite, or more generally not thin. Our approach is via the geometry of del Pezzo surfaces of degree $2$, and we prove a more general result on non-thinness of integral points on double conic bundle surfaces.

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arXiv2025

Lower bounds for counting $A_4$-quartic fields

A conjecture of Malle predicts the quantity of number fields with bounded discriminant of given Galois group. We present a lower bound matching this in the case of quartic fields with Galois group $A_4$.

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arXiv2024

Rational points in a family of conics over $\mathbb{F}_2(t)$

Serre famously showed that almost all plane conics over $\mathbb{Q}$ have no rational point. We investigate versions of this over global function fields, focusing on a specific family of conics over $\mathbb{F}_2(t)$ which illustrates new behaviour. We obtain an asymptotic formula using harmonic analysis, which requires a Tauberian theorem over function fields for Dirichlet series with branch point singularities.

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arXiv2024

Malle's conjecture and Brauer groups of stacks

We put forward a conjecture for the leading constant in Malle's conjecture on number fields of bounded discriminant, guided by stacky versions of conjectures of Batyrev-Manin, Batyrev-Tschinkel, and Peyre on rational points of bounded height on Fano varieties. A new framework for Brauer groups of stacks plays a key role in our conjecture, and we define a new notion of the unramified Brauer group of an algebraic stack.

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arXiv2024

Campana points on wonderful compactifications

We prove a variant of Manin's conjecture for Campana points on wonderful compactifications of semi-simple algebraic groups of adjoint type. We use this to provide evidence for a new conjecture on the leading constant in Manin's conjecture for Campana points.

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arXiv2023

Rationality and arithmetic of the moduli of abelian varieties

We study the rationality properties of the moduli space $\mathcal{A}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular we show that any principally polarised abelian threefold over $\mathbb{F}_p$ may be lifted to an abelian variety over $\mathbb{Q}$. This is a phenomenon of low dimension: assuming the Bombieri-Lang conjecture we also show that this is not the case for abelian varieties of dimension at least seven. About moduli spaces, we show that $\mathcal{A}_g$ is unirational over $\mathbb{Q}$ for $g \leq 5$ and stably rational for $g=3$. This also allows us to make unconditional one of the results of Masser and Zannier about the existence of abelian varieties over $\mathbb{Q}$ that are not isogenous to Jacobians.

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arXiv2023

The Hasse norm principle for abelian extensions

We study the distribution of abelian extensions of bounded discriminant of a number field $k$ which fail the Hasse norm principle. For example, we classify those finite abelian groups $G$ for which a positive proportion of $G$-extensions of $k$ fail the Hasse norm principle. We obtain a similar classification for the failure of weak approximation for the associated norm one tori. These results involve counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which we achieve using tools from harmonic analysis, building on work of Wright.

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arXiv2023

Normal distribution of bad reduction

We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, $100\%$ of curves in these families have, in a precise sense, many such primes.

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arXiv2023

Distribution of genus numbers of abelian number fields

We study the quantitative behaviour of genus numbers of abelian extensions of number fields with given Galois group. We prove an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time.

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arXiv2023

The elliptic sieve and Brauer groups

A theorem of Serre states that almost all plane conics over $\mathbb{Q}$ have no rational point. We prove an analogue of this for families of conics parametrised by elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve for elliptic curves. We also give more general results for specialisations of Brauer groups, which yields applications to norm form equations.

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arXiv2022

The leading constant for rational points in families

We prove asymptotics for Serre's problem on the number of diagonal planar conics with a rational point and use this to put forward a new conjecture on counting the number of varieties in a family which are everywhere locally soluble.

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arXiv2022

Frobenian multiplicative functions and rational points in fibrations

We consider the problem of counting the number of varieties in a family over $\mathbb{Q}$ with a rational point. We obtain lower bounds for this counting problem for some families over $\mathbb{P}^1$, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics.

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arXiv2022

Good reduction and cyclic covers

We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double covers of abelian varieties, and reduce the Shafarevich conjecture for hypersurfaces to the case of hypersurfaces of high dimension. These are special cases of a general set-up for integral points on moduli stacks of cyclic covers, and our arithmetic results are achieved via a version of the Chevalley-Weil theorem for stacks.

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arXiv2022

Number fields with prescribed norms

We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100\%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.

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