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Taylor Dupuy

Publications and source records attributed to Taylor Dupuy.

At least 19 recordsLinked to original sources
arXiv2026

Ordinary abelian varieties: isogeny graphs and polarizations

Given an integer $D$ and an ordinary isogeny class of abelian varieties defined over a finite field $\mathbb{F}_q$ with commutative $\mathbb{F}_q$-endomorphism algebra, we provide algorithms for computing all isogenies of degree dividing $D$ and polarizations of degree dividing $D$. We discuss phenomena that arise for higher dimension abelian varieties but not elliptic curves, bounds on the diameter of the graph of minimal isogenies, and decompositions of isogeny graphs into orbits for the Picard group of the Frobenius order.

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arXiv2025

Labeling abelian varieties over finite fields

We describe a deterministic process to associate a practical, permanent label to isomorphism classes of abelian varieties defined over finite fields with commutative endomorphism algebra as long as they are ordinary or defined over a prime field. In the ordinary case, we also provide labels for the polarizations they admit.

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arXiv2024

Ford Spheres in the Clifford-Bianchi Setting

We define Ford Spheres $\mathcal{P}$ in hyperbolic $n$-space associated to Clifford-Bianchi groups $PSL_2(O)$ for $O$ orders in rational Clifford algebras associated to positive definite, integral, primitive quadratic forms. For $\mathcal{H}^2$ and $\mathcal{H}^3$ these spheres correspond to the classical Ford circles and Ford spheres (these are non-maximal subsets of classical Apollonian packings). We prove the Ford spheres are integral, have disjoint interiors, and intersect tangentially when they do intersect. If we assume that $O$ is Clifford-Euclidean then $\mathcal{P}$ is also connected. We also give connections to Dirichlet's Theorem and Farey fractions. In a discussion section, we pose some questions related to existing packings in the literature.

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arXiv2024

The Basic Theory of Clifford-Bianchi Groups for Hyperbolic n-Space

Let $K$ be a $\mathbb{Q}$-Clifford algebra associated to an $(n-1)$-ary positive definite quadratic form and let $\mathcal{O}$ be a maximal order in $K$. A Clifford-Bianchi group is a group of the form $\operatorname{SL}_2(\mathcal{O})$ with $\mathcal{O}$ as above. The present paper is about the actions of $\operatorname{SL}_2(\mathcal{O})$ acting on hyperbolic space $\mathcal{H}^{n+1}$ via Möbius transformations $x\mapsto (ax+b)(cx+d)^{-1}$. We develop the general theory of orders exhibiting explicit orders in low dimensions of interest. These include, for example, higher-dimensional analogs of the Hurwitz order. We develop the abstract and computational theory for determining their fundamental domains and generators and relations (higher-dimensional Bianchi-Humbert Theory). We make connections to the classical literature on symmetric spaces and arithmetic groups and provide a proof that these groups are $\mathbb{Z}$-points of a $\mathbb{Z}$-group scheme and are arithmetic subgroups of $\operatorname{SO}_{1,n+1}(\mathbb{R})^{\circ}$ with their Möbius action. We report on our findings concerning certain Clifford-Bianchi groups acting on $\mathcal{H}^4$, $\mathcal{H}^5$, and $\mathcal{H}^6$ .

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arXiv2023

The Theory of the Entire Algebraic Functions

Let A be the integral closure of the ring of polynomials CC[t], within the field of algebraic functions in one variable. We show that A interprets the ring of integers. This contrasts with the analogue for finite fields, proved to have a decidable theory (see Prestel-Schmid and van den Dries-A. Macintyre).

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arXiv2023

Order one differential equations on nonisotrivial algebraic curves

In this paper we provide new examples of geometrically trivial strongly minimal differential algebraic varieties living on nonisotrivial curves over differentially closed fields of characteristic zero. These are systems whose solutions only have binary algebraic relations between them. Our technique involves developing a theory of $τ$-forms, and building connections to deformation theory. This builds on previous work of Buium and Rosen. In our development, we answer several open questions posed by Rosen and Hrushovski-Itai.

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arXiv2023

Order one differential equations on nonisotrivial algebraic curves

In this paper we provide new examples of geometrically trivial strongly minimal differential algebraic varieties living on nonisotrivial curves over differentially closed fields of characteristic zero. Our technique involves developing a theory of Kodaira-Spencer forms and building connections to deformation theory. In our development, we answer several open questions posed by Rosen and some natural questions about Manin kernels.

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arXiv2023

Angle ranks of abelian varieties

Using the formalism of Newton hyperplane arrangements, we resolve the open questions regarding angle rank left over from [DKRV20]. As a consequence we end up generalizing theorems of Lenstra--Zarhin and Tankeev proving several new cases of the Tate conjecture for abelian varieties over finite fields. We also obtain an effective version of a recent theorem of Zarhin bounding the heights of coefficients in multiplicative relations among Frobenius eigenvalues.

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arXiv2020

Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB

This document is intended to summarize the theory and methods behind fq_isog collection inside the ab_var database in the LMFDB as well as some observations gleaned from these databases. This collection consists of tables of Weil q-polynomials, which by the Honda-Tate theorem are in bijection with isogeny classes of abelian varieties over finite fields.

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arXiv2020

Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12

In \cite{Dupuy2020a} we gave some explicit formulas for the "indeterminacies" Ind1,Ind2,Ind3 in Mochizuki's Inequality as well as a new presentation of initial theta data. In the present paper we use these explicit formulas, together with our probabilistic formulation of \cite[Corollary 3.12]{IUT3} to derive variants of Szpiro's inequality (in the spirit of \cite{IUT4}). In particular, for an elliptic curve in initial theta data we show how to derive uniform Szpiro (with explicit numerical constants). The inequalities we get will be strictly weaker than \cite[Theorem 1.10]{IUT4} but the proofs are more transparent, modifiable, and user friendly. All of these inequalities are derived from an probabilistic version of \cite[Corollary 3.12]{IUT3} formulated in \cite{Dupuy2020a} based on the notion of random measurable sets.

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arXiv2020

Counterexamples to a Conjecture of Ahmadi and Shparlinski

Ahmadi-Shparlinski conjectured that every ordinary, geometrically simple Jacobian over a finite field has maximal angle rank. Using the L-Functions and Modular Forms Database, we provide two counterexamples to this conjecture in dimension 4.

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arXiv2018

Deligne--Illusie Classes as Arithmetic Kodaira--Spencer Classes

Faltings showed that "arithmetic Kodaira--Spencer classes" satisfying a certain compatibility axiom cannot exist. By modifying his definitions slightly, we show that the Deligne--Illusie classes satisfy what could be considered an "arithmetic Kodaira--Spencer" compatibility condition. Afterwards we discuss a "wittfinitesimal Torelli problem" and its relation to CM Jacobians.

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arXiv2017

Total $p$-differentials on schemes over $Z/p^2$

For a scheme $X$ defined over the length $2$ $p$-typical Witt vectors $W_2(k)$ of a characteristic $p$ field, we introduce total $p$-differentials which interpolate between Frobenius-twisted differentials and Buium's $p$-differentials. They form a sheaf over the reduction $X_0$, and behave as if they were the sheaf of differentials of $X$ over a deeper base below $W_2(k)$. This allows us to construct the analogues of Gauss-Manin connections and Kodaira-Spencer classes as in the Katz-Oda formalism. We make connections to Frobenius lifts, Borger-Weiland's biring formalism, and Deligne--Illusie classes.

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arXiv2017

A rigid analytic proof that the Abel-Jacobi map extends to compact-type models

Let $K$ be a non-Archimedean valued field with valuation ring $R$. Let $C_η$ be a $K$-curve with compact type reduction, so its Jacobian $J_η$ extends to an abelian $R$-scheme $J$. We prove that an Abel-Jacobi map $ι\colon C_η\to J_η$ extends to a morphism $C\to J$, where $C$ is a compact-type $R$-model of $J$, and we show this is a closed immersion when the special fiber of $C$ has no rational components. To do so, we apply a rigid-analytic "fiberwise" criterion for a finite morphism to extend to integral models, and geometric results of Bosch and Lütkebohmert on the analytic structure of $J_η$.

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arXiv2015

Arithmetic differential equations on $GL_n$, II: arithmetic Lie theory

Motivated by the search of a concept of linearity in the theory of arithmetic differential equations we introduce here an arithmetic analogue of Lie algebras and a concept of skew arithmetic differential cocycles. We will then construct such skew cocycles, based on certain remarkable lifts of Frobenius for the classical groups $GL_n, SL_n, SO_n, Sp_{n}$. The theory for $GL_n$, especially on the Galois side, will be further developed in a sequel to this paper.

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