arXiv subjects

Wei Ho

Publications and source records attributed to Wei Ho.

14 recordsLinked to original sources

arXiv2025

Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field

We show that for any quadratic extension of number fields $K/F$, there exists an abelian variety $A/F$ of positive rank whose rank does not grow upon base change to $K$. This result implies that Hilbert's tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring $\mathcal{O}_K$ of integers of any number field $K$, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over $\mathcal{O}_K$ has solutions in $\mathcal{O}_K$.

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arXiv2022

Quadratic enrichment of the logarithmic derivative of the zeta function

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck--Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.

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arXiv2022

On average sizes of Selmer groups and ranks in families of elliptic curves having marked points

We determine average sizes/bounds for the $2$- and $3$-Selmer groups in various families of elliptic curves with marked points, thus confirming several cases of the Poonen--Rains heuristics. As a consequence, we deduce that the average ranks of the elliptic curves in all of these families are bounded. Our proofs are uniform and make use of parametrizations involving various forms of $2 \times 2 \times 2 \times 2$ and $3 \times 3 \times 3$ matrices that we studied in a previous paper. We also deduce that $100\%$ of genus one curves of the form $y^2 = Ax^4 + Bx^2 z^2 + Cz^4$ with $A, B, C \in \mathbb{Z}$, when ordered by $\max\{|B|^2,|AC|\}$, fail the Hasse principle. Other forthcoming applications include proofs that a positive proportion of integers are (respectively, are not) the sum of two rational cubes, and a positive proportion of genus one curves in $\mathbb{P}^1 \times \mathbb{P}^1$ over $\mathbb{Q}$ fail the Hasse principle.

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arXiv2019

Everywhere local solubility for hypersurfaces in products of projective spaces

We prove that a positive proportion of hypersurfaces in products of projective spaces over $\mathbb{Q}$ are everywhere locally soluble, for almost all multidegrees and dimensions, as a generalization of a theorem of Poonen and Voloch. We also study the specific case of genus $1$ curves in $\mathbb{P}^1 \times \mathbb{P}^1$ defined over $\mathbb{Q}$, represented as bidegree $(2,2)$-forms, and show that the proportion of everywhere locally soluble such curves is approximately $87.4\%$. The proportion of these curves in $\mathbb{P}^1 \times \mathbb{P}^1$ soluble over $\mathbb{Q}_p$ is a rational function of $p$ for each finite prime $p$. Finally, we include some experimental data on the Hasse principle for these curves.

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arXiv2018

The second moment of the number of integral points on elliptic curves is bounded

Let $K$ be a number field and $S$ a finite set of places of $K$ containing all archimedean places. In this paper, we show that the second moment of the number of $S$-integral points on elliptic curves over $K$ is bounded. In particular, we prove that, for any positive real number $r\leq \log_2 5 = 2.3219 \ldots$, the $r$-th moment of the number of $S$-integral points is bounded for the family of all integral short Weierstrass curves ordered by height, or for any positive density subfamily thereof. For certain other families of elliptic curves over $\mathbb{Q}$, such as those with one or two marked points, we prove that the average of the number of integral points is bounded; in fact, for the family with one marked point, the $r$-th moment is also bounded for all positive $r\leq \log_2 3$. The essential new ingredient in our proof is an upper bound on the number of $S$-integral points on an affine integral Weierstrass model $\mathcal{E}$ of an elliptic curve $E$ over $K$ depending on the rank of the curve, the class group and degree of $K$, and the number of primes of $K$ whose square divides the discriminant of $\mathcal{E}$. For example, over $\mathbb{Q}$, the bound for integral points on $\mathcal{E}$ is $2^{\mathrm{rank}{E(\mathbb{Q})}} O(1)^s$, where $s$ is the number of prime squares dividing the discriminant of $\mathcal{E}$. The theorems on moments then follow from averaging this new upper bound; crucially, we can bound the average of the $2^{\mathrm{rank}}$ term by using results on the average sizes of Selmer groups in the families. In order to prove the bounds for the $r$-th moment when $r= \log_2 5$ (and the analogous equality cases for the other families), we introduce a method to count orbits of coregular representations with possibly unbounded weights.

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arXiv2018

Galois closures of non-commutative rings and an application to Hermitian representations

Galois closures of commutative rank n ring extensions were introduced by Bhargava and the second author. In this paper, we generalize the construction to the case of non-commutative rings. We show that non-commutative Galois closures commute with base change and satisfy a product formula. As an application, we give a uniform construction of many of the representations arising in arithmetic invariant theory, including many Vinberg representations.

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arXiv2018

Splitting Brauer classes using the universal Albanese

We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class, and there exist many such curves splitting the class. We show that this can be false when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class.

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arXiv2016

Orbit Parametrizations for K3 Surfaces

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose Néron-Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.

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arXiv2016

Odd degree number fields with odd class number

For every odd integer $n \geq 3$, we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_n$ whose class number is odd. To do so, we study the class groups of families of number fields of degree $n$ whose rings of integers arise as the coordinate rings of the subschemes of $\mathbb{P}^1$ cut out by integral binary $n$-ic forms. By obtaining upper bounds on the mean number of $2$-torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to $1$ as $n$ tends to $\infty$) of such fields have trivial $2$-torsion subgroup in their class groups and narrow class groups. Conditional on a tail estimate, we also prove the corresponding lower bounds and obtain the exact values of these averages, which are consistent with the heuristics of Cohen-Lenstra-Martinet-Malle and Dummit-Voight. Additionally, for any order $\mathcal{O}_f$ of degree $n$ arising from an integral binary $n$-ic form $f$, we compare the sizes of $\mathrm{Cl}_2(\mathcal{O}_f)$, the $2$-torsion subgroup of ideal classes in $\mathcal{O}_f$, and $\mathcal{I}_2(\mathcal{O}_f)$, the $2$-torsion subgroup of ideals in $\mathcal{O}_f$. For the family of orders arising from integral binary $n$-ic forms and contained in fields with fixed signature $(r_1,r_2)$, we prove that the mean value of the difference $|\mathrm{Cl}_2(\mathcal{O}_f)| - {2^{1-r_1-r_2}}|\mathcal{I}_2(\mathcal{O}_f)|$ is equal to $1$, generalizing a result of Bhargava and the third-named author for cubic fields. Conditional on certain tail estimates, we also prove that the mean value of $|\mathrm{Cl}_2(\mathcal{O}_f)| - {2^{1-r_1-r_2}}|\mathcal{I}_2(\mathcal{O}_f)|$ remains $1$ for certain families obtained by imposing local splitting and maximality conditions.

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arXiv2016

Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava-Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$ ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$ ordered by height in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon observed in these databases is that the average rank eventually decreases as height increases.

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arXiv2015

Zeta functions of a class of Artin-Schreier curves with many automorphisms

This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.

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arXiv2013

Coregular spaces and genus one curves

A coregular space is a representation of an algebraic group for which the ring of polynomial invariants is free. In this paper, we show that the orbits of many coregular irreducible representations where the number of invariants is at least two, over a (not necessarily algebraically closed) field k, correspond to genus one curves over k together with line bundles, vector bundles, and/or points on their Jacobians. In forthcoming work, we use these orbit parametrizations to determine the average sizes of Selmer groups for various families of elliptic curves.

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arXiv2012

Genus one curves and Brauer-Severi varieties

Let K be a field. Let A be a central simple algebra over K and let X be the associated Brauer-Severi variety over K. It has recently been asked if there exists a genus 1 curve C over K such that K(C) splits A. In other words, is there a genus one curve C over K with a morphism to X? In this short note, we explicitly construct such a curve in the case where X has dimension at most 4 (equivalently, when A has degree at most 5).

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arXiv2012

Moduli of products of stable varieties

We study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties, (b) this map is always finite \'etale, and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension 1. The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.

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