arXiv subjects

Juan Pablo De Rasis

Publications and source records attributed to Juan Pablo De Rasis.

4 recordsLinked to original sources

arXiv2026

On the efficient computation of Fourier coefficients of eta-quotients

The Fourier coefficients of a negative weight eta-quotient, in many particular cases, and after Sussman in general, are known to be expressible by Hardy-Ramanujan-Rademacher type series. We show that the central terms of the coefficients of these series can be efficiently computed, showing that they can be expressed in terms of twisted Kloosterman sums, and that they satisfy multiplicativity relations; this extends the results from Lehmer for the partition function. We also give explicit bounds for the tails of these series, needed for effectively computing the aforementioned Fourier coefficients.

math.NT↗

arXiv2026

First-order definability of Campana Points and Darmon Points in algebraic function fields in one variable over number fields

We give first-order definitions of Campana and Darmon points in algebraic function fields in one variable over number fields. These sets are geometric generalizations of $n$-full integers (integers whose nonzero valuations are at least $n$) and perfect $n$th powers, respectively, to more general algebraic varieties. For this we exploit the theory of quadratic Pfister forms, which were used by Becher, Daans & Dittmann to extend to the case of algebraic function fields in one variable the methods used by Koenigsmann when proving that $\mathbb{Z}$ is universally defined in $\mathbb{Q}$. These methods had already been generalized to arbitrary global fields by Park (2013) and Eisenträger & Morrison (2018), and the author had already exploited these methods to find first-order definitions of Campana points (2024) and Darmon points (2024, with Handley) in the context of number fields. With the newly expanded version of these methods, we now transfer those results to the new context of algebraic function fields in one variable over number fields.

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arXiv2026

First-order definability of Darmon points in number fields

For a given number field $K$, we give a $\forall\exists\forall$-first order description of affine Darmon points over $\mathbb{P}^1_K$, and show that this can be improved to a $\forall\exists$-definition in a remarkable particular case. Darmon points, which are a geometric generalization of perfect powers, constitute a non-linear set-theoretical filtration between $K$ and its ring of $S$-integers, the latter of which can be defined with universal formulas, as has been progressively proven by Koenigsmann, Park, and Eisenträger & Morrison. We also show that our formulas are uniform with respect to all possible $S$, with a parameter-free uniformity, and we compute the number of quantifiers and a bound for the degree of the defining polynomial.

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arXiv2025

First-order definability of affine Campana points in the projective line over a number field

We offer a $\forall\exists$-definition for (affine) Campana points over $\mathbb{P}^1_K$ (where $K$ is a number field), which constitute a set-theoretical filtration between $K$ and $\mathcal{O}_{K,S}$ ($S$-integers), which are well-known to be universally defined (Koenigsmann 2010, Park 2012, Eisentraeger & Morrison 2016). We also show that our formulas are uniform with respect to all possible $S$, are parameter-free as such, and we count the number of involved quantifiers and offer a bound for the degree of the defining polynomial.

math.NT↗