arXiv subjects

Boaz Moerman

Publications and source records attributed to Boaz Moerman.

4 recordsLinked to original sources

arXiv2025

$\mathcal{M}$-points of bounded height on toric varieties

We establish an asymptotic formula for the number of $\mathcal{M}$-points of bounded height on split toric varieties, for the height induced by any big and nef divisor class. This formula establishes new cases of the extension of Manin's conjecture to $\mathcal{M}$-points, as introduced by the author. As a special case of our result, we strengthen the results obtained by Pieropan and Schindler on Campana points of bounded height on toric varieties. As another special case, we obtain an asymptotic for the number of weak Campana points of bounded height, which is novel even for projective space. We illustrate this result by giving an asymptotic for the number of points on projective space of bounded height for which the product of coordinates is powerful. Finally, we derive an asymptotic for the number of rational points in the image of a toric rational map, in the spirit of the Loughran-Smeets conjecture.

math.NT↗

arXiv2025

Manin's conjecture for $\mathcal{M}$-points

We initiate a general quantitative study of sets of $\mathcal{M}$-points, which are special subsets of rational points, generalizing Campana points, Darmon points, and squarefree solutions of Diophantine equations. We propose an asymptotic formula for the number of $\mathcal{M}$-points of bounded height on rationally connected varieties, extending Manin's conjecture as well as its generalization to Campana points by Pieropan, Smeets, Tanimoto and V\'arilly-Alvarado. Finally, we show that the conjecture explains several previously established results in arithmetic statistics.

math.NT↗

arXiv2024

Generalized Campana points and adelic approximation on toric varieties

We introduce a general framework for studying special subsets of rational points on an algebraic variety, termed $\mathcal{M}$-points. The notion of $\mathcal{M}$-points generalizes the concepts of integral points, Campana points and Darmon points. We introduce and study $M$-approximation over number fields and function fields, which is a notion that generalizes weak and strong approximation. We show that this property implies that the set of $\mathcal{M}$-points is not thin. We then give a simple characterisation of when a split toric variety satisfies $M$-approximation, generalizing work of Nakahara and Streeter. Further, we determine when the set of $\mathcal{M}$-points on a split toric variety is thin.

math.AG↗

arXiv2020

$L$-values for conductor $32$

In recent years, Rogers and Zudilin developed a method to write $L$-values attached to elliptic curves as periods. In order to apply this method to a broader collection of $L$-values, we study Eisenstein series and determine their Fourier series at cusps. Subsequently, we write the $L$-values of an elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate the value at $k>1$ explicitly as a period. As a side result, we give simple integral expressions for the generating functions of $L(E,k)$ when even (or odd) $k$ runs over positive integers.

math.NT↗