arXiv subjects

Cruz Castillo

Publications and source records attributed to Cruz Castillo.

4 recordsLinked to original sources

arXiv2026

Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves

We show that a positive proportion of Hecke $L$-functions attached to the quartic residue symbols $\big( \frac{\cdot}{q} \big)_4$ for squarefree $q \in \mathbb{Z}[i]$ do not vanish at the central point. Our method also extends to the Hecke characters associated to quartic twists of the congruent number curve $E : y^2 = x^3 - x$. In particular, we prove that the elliptic curve $E^{(q)} : y^2 = x^3 - qx$ has Mordell-Weil rank $0$ over $\mathbb{Q}(i)$ for a positive proportion of squarefree $q \in \mathbb{Z}[i]$ ordered by norm.

math.NT↗

arXiv2025

Arithmetic properties of generalized Frobenius partitions

Ramanujan proved three famous congruences for the partition function modulo 5, 7, and 11. The first author and Boylan proved that these congruences are the only ones of this type. In 1984 Andrews introduced the $m$-colored Frobenius partition functions $c\phi_m$; these are natural higher-level analogues of the partition function which have attracted a great deal of attention in the ensuing decades. For each $m\in \{5, 7, 11\}$ there are two analogues of Ramanujan's congruences for $c\phi_m$, and for these $m$ we prove there are no congruences like Ramanujan's other than these six. Our methods involve a blend of theory and computation with modular forms.

math.NT↗

arXiv2025

Eisenstein series modulo prime powers

If $p\geq 5$ is prime and $k\geq 4$ is an even integer with $(p-1)\nmid k$ we consider the Eisenstein series $G_k$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo powers of $p$. It is classically known that for such $k$ we have $G_k\equiv G_{k'}\pmod p$ if $k\equiv k'\pmod{p-1}$. Here we obtain a generalization modulo prime powers $p^m$ by giving an expression for $G_k\pmod{p^m}$ in terms of modular forms of weight at most $mp$. As an application we extend a recent result of the first author with Hanson, Raum and Richter by showing that, modulo powers of $E_{p-1}$, every such Eisenstein series is congruent modulo $p^m$ to a modular form of weight at most $mp$. We prove a similar result for the normalized Eisenstein series $E_k$ in the case that $(p-1)\mid k$ and $m<p$.

math.NT↗

arXiv2023

Sign changes of the error term in the Piltz divisor problem

We study the function $\Delta_k(x):=\sum_{n\leq x} d_k(n) - \mbox{Res}_{s=1} ( \zeta^k(s) x^s/s )$, where $k\geq 3$ is an integer, $d_k(n)$ is the $k$-fold divisor function, and $\zeta(s)$ is the Riemann zeta-function. For a large parameter $X$, we show that if the Lindel\"{o}f hypothesis is true, then there exist at least $X^{\frac{1}{k(k-1)}-\varepsilon}$ disjoint subintervals of $[X,2X]$, each of length $X^{1-\frac{1}{k}-\varepsilon}$, such that $|\Delta_k(x)|\gg x^{\frac{1}{2}-\frac{1}{2k}}$ for all $x$ in the subinterval. If the Riemann hypothesis is true, then we can improve the length of the subintervals to $\gg X^{1-\frac{1}{k}} (\log X)^{-k^2-2}$. These results may be viewed as higher-degree analogues of theorems of Heath-Brown and Tsang, who studied the case $k=2$, and Cao, Tanigawa, and Zhai, who studied the case $k=3$. The first main ingredient of our proofs is a bound for the second moment of $\Delta_k(x+h)-\Delta_k(x)$. We prove this bound using a method of Selberg and a general lemma due to Saffari and Vaughan. The second main ingredient is a bound for the fourth moment of $\Delta_k(x)$, which we obtain by combining a method of Tsang with a technique of Lester.

math.NT↗