arXiv subjects

Liam Baker

Publications and source records attributed to Liam Baker.

7 recordsLinked to original sources

arXiv2026

Direct-product rigidity and factor reconstruc- tion for monoids with zero

We study direct products of monoids with zero and give a criterion under which their factors can be recovered from the multiplicative structure alone. While classical decomposition theory encodes direct products through factor congruences, central elements, and refinement properties, we give a concrete multiplicative reconstruction mechanism. If the factors have no nontrivial complemented central idempotents, then the coordinate idempotents are precisely the atoms and coatoms of the complemented-central-idempotent poset, and their multiplicative stabilizers are precisely the coordinate factors and cofactors. It follows that every isomorphism between such products is monomial, yielding the corresponding wreath-product description of automorphism groups. More generally, every product decomposition is obtained by grouping the original factors; in particular, strict refinement follows. We apply these results to multiplicative monoids of directly indecomposable unital rings, including connected commutative rings, and to arithmetic examples arising from residue-class rings.

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arXiv2026

Drinfeld modular forms of higher rank from a lattice-oriented point of view

A space $\mathcal{L}_N^r$ of Drinfeld modules of rank $r \geq 1$ with level structure, or equivalently lattices of rank $r$ with level structure, is introduced, and its irreducible components and group actions on it are investigated. A metric is defined on this space, its completion $\overleftarrow{\mathcal{L}_{N}^{r}}$ is established and the aforementioned group actions are extended to the completion. A decomposition of the completion into multiple smaller spaces $\mathcal{L}_N^s$ is proven. Drinfeld modular forms are defined as homogeneous holomorphic functions on $\mathcal{L}_{N}^{r}$ which are continuous on the completion $\overleftarrow{\mathcal{L}_{N}^{r}}$, and the group actions above are extended to actions on the spaces of modular forms. Finally, the modular forms defined here are compared with those of Basson, Breuer, and Pink and with those of Gekeler, and it is shown that the cusp forms (those which are zero on the boundary) coincide.

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arXiv2026

Drinfeld modules in rank 2 with CM and S-unit j-invariants

We prove the finiteness of the set of $j$-invariants of Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$ which are CM and $S$-units, for $S$ the infinite set of primes with even degrees. The proof is based on the study of ordinary reduction and supersingular reduction of Drinfeld modules, and on the splitting behaviour of primes dividing the difference of two Drinfeld singular moduli. We also provide an algorithm to compute a polynomial with coefficients in $\mathbb{F}_q[T]$ and roots the $j$-invariants having CM by a given order, and use it to compute some explicit examples, providing for instance counterexamples to a conjecture of Dorman. For a maximal order $\mathcal{O}$, we prove by a universality argument that our algorithm computes the Hilbert modular polynomial $H_\mathcal{O}$.

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arXiv2026

Automorphism groups of direct products of multiplicative monoids of certain rings

In this paper, we establish a rigidity result for automorphisms of multiplicative direct products of $D$-rings which are total ring of fraction that have pairwise distinct cardinalities. Under these assumptions, every automorphism acts independently on each factor, so that no interaction between distinct components occurs; in particular, the automorphism group decomposes canonically as the direct product of the automorphism groups of the factors. As a consequence, the automorphism group of the multiplicative monoid of integers modulo $n$ is entirely determined by its $p$-power components.

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arXiv2025

A parallelogram height inequality for Drinfeld modules

We prove inequalities relating the Taguchi heights, respectively the graded heights, of four Drinfeld modules arranged in a ``parallelogram of isogenies''. This inequality is the analogue for Drinfeld modules of the parallelogram inequality of R\'emond (2022) for abelian varieties over number fields and of Griffon--Le Fourn--Pazuki (2025) for abelian varieties over function fields.

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arXiv2024

On the automorphism group of the monoid of the integers modulo a prime power

This paper determines the structure of the automorphism group of the unit group $(U_{p^e}, \cdot)$ and the monoid $(\mathbb{Z}/p^e \mathbb{Z}, \cdot)$. For $ e \geq 5 $, we establish that the automorphism group $ \Aut(U_{2^e}, \cdot) $ is the direct product of $ \mathbb{Z}/2\mathbb{Z} $ with the central product of a dihedral group of order 8 and the cyclic group $ \mathbb{Z}/2^{e-3}\mathbb{Z} $. Moreover, we show that the automorphism group $ \Aut(\mathbb{Z}/p^e \mathbb{Z}, \cdot) $ is isomorphic to a canonical semidirect product of $ U_{p^{e-1}} $ and the subgroup of $ \Aut(U_{p^e}, \cdot) $ consisting of automorphisms that induce an automorphism of $ (U_{p^f}, \cdot) $ for any integer $ f $ such that $ 0 \leq f \leq e $.

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arXiv2015

Erdős-Surányi sequences and trigonometric integrals

We study representations of integers as sums of the form $\pm a_1\pm a_2\pm \dotsb \pm a_n$, where $a_1,a_2,\ldots$ is a prescribed sequence of integers. Such a sequence is called an Erdős-Surányi sequence if every integer can be written in this form for some $n\in\mathbb{N}$ and choices of signs in infinitely many ways. We study the number of representations of a fixed integer, which can be written as a trigonometric integral, and obtain an asymptotic formula under a rather general scheme due to Roth and Szekeres. Our approach, which is based on Laplace's method for approximating integrals, can also be easily extended to find higher-order expansions. As a corollary, we settle a conjecture of Andrica and Ionaşcu on the number of solutions to the signum equation $\pm 1^k \pm 2^k \pm \dotsb \pm n^k = 0$.

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