arXiv subjects

Erick Ross

Publications and source records attributed to Erick Ross.

14 recordsLinked to original sources

arXiv2026

Asymptotics of Hecke polynomial coefficients on the Atkin-Lehner eigenspaces

Let $S_k^\sigma(N)$ denote the space of cusp forms of level $N$, weight $k$, and Atkin-Lehner sign pattern $\sigma$, and $S_k^{\mathrm{new},\sigma}$ denote its new subspace. In this paper, we study the asymptotic behavior of the coefficients of the $m$-th Hecke polynomial over $S_k^\sigma(N)$ and $S_k^{\mathrm{new},\sigma}(N)$. In particular, we show that in certain settings, all but finitely many of these coefficients take a particular sign. We also study settings in which the coefficients do not tend to any particular sign.

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arXiv2026

Interlacing for zeros of the Serre derivative of Eisenstein series

In 1970, Rankin and Swinnerton-Dyer showed that the non-elliptic zeros of Eisenstein series $E_k$ in the fundamental domain all lie on the lower arc $\{ e^{i\theta}: \frac{\pi}{2} < \theta < \frac{2\pi}{3}\}$. Very recently, Sugibayashi showed that the same property also holds for the Serre derivative $\vartheta_k(E_k)$ of Eisenstein series. In this paper, we first give very precise estimates for where exactly these zeros are located on the lower arc. These location estimates then allow us to prove four main results. First, we show that the zeros of $\vartheta_\ell(E_\ell)$ Stieltjes interlace with the zeros of $\vartheta_k(E_k)$ on the lower arc for all $\ell > k$. Second, we classify precisely when the zeros of $\vartheta_\ell(E_\ell)$ (standard) interlace with the zeros of $\vartheta_k(E_k)$ on the lower arc. Third, we show that the zeros of $\vartheta_k(E_k)$ always (standard) interlace with the zeros of $E_{k+2}$ on the lower arc. Fourth, as an application of the third main result, we show that the zeros of the cuspidal projection of $\vartheta_k(E_k)$ all lie on the lower arc, extending a result of Xue and Zhu.

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arXiv2026

Equidistribution of CM points and RM curves

In 1988, William Duke showed that CM points of fundamental discriminant $D$ are equidistributed in the complex upper half-plane $\mathcal H$ as $D \to -\infty$. He also showed a similar result for RM curves (a positive discriminant analog of CM points). In this paper, we investigate analogous problems concerning the distribution of CM points and RM curves along fixed geodesics in $\mathcal H$, and around fixed points in $\mathcal H$. Specifically, we show that CM points and RM curves are equidistributed along every fixed rational geodesic in $\mathcal H$, and around every fixed CM point in $\mathcal H$. To prove these results, we solve the aggregate Linnik problem for arbitrary binary quadratic forms.

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arXiv2026

Boundary CM points and class groups of small exponent

Let $\mathcal F$ denote the fundamental domain for $\text{SL}_2(\mathbb{Z})$ on the upper half plane $\mathcal H$. William Duke showed that as fundamental discriminants $D \to -\infty$, the sets $\mathrm{CM}_{D}$ (CM points of discriminant $D$) are equidistributed in $\mathcal F$. In this paper, we investigate the behavior of CM points on the boundary of $\mathcal F$. We prove that such CM points are equidistributed on the boundary, and also give a complete characterization of when every $\mathrm{CM}_D$ point lies on the boundary. Along the way, we also (conditionally) give a complete classification of negative discriminants with class group of small exponent.

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arXiv2025

Hecke Eigenvalue Equidistribution over the Newspaces with Nebentypus

Fix a prime $p$, and let $\widehat T_p^{\mathrm{new}}(N,k,\chi) := \chi(p)^{-1/2} p^{-(k-1)/2} T_p^{\mathrm{new}}(N,k,\chi)$ denote the normalized $p$'th Hecke operator over the newspace with nenbentypus $S_k^{\mathrm{new}}(N,\chi)$. In this paper, we determine the distribution of the eigenvalues of $\widehat T_p^{\mathrm{new}}(N,k,\chi)$ as $N+k \to \infty$.

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arXiv2025

Proportion of Atkin-Lehner sign patterns and Hecke Eigenvalue Equidistribution

Let $N \ge 1$, $k \ge 2$ even, and $\sigma$ denote a sign pattern for $N$. In this paper, we first determine the exact proportion of forms in $S_k(N)$ and $S_k^\mathrm{new}(N)$ with a given Atkin-Lehner sign pattern $\sigma$. Then we study the asymptotic behavior of the Hecke operators $T_p$ over the subspaces of $S_k(N)$ and $S_k^{\mathrm{new}}(N)$ with Atkin-Lehner sign pattern $\sigma$. In particular, for the $p$-adic Plancherel measure $\mu_p$, we show that the Hecke eigenvalues for $T_p$ over these subspaces are $\mu_p$-equidistributed as $N+k \to \infty$.

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arXiv2025

Dimension sequences of modular forms

For $N \geq 1$, let $S_{2}^{\text{new}}(N)$ denote the newspace of cuspidal modular forms of weight $2$ and level $N$. In 2004, Greg Martin conjectured that as a sequence in $N$, $\dim S_2^{\text{new}}(N)$ takes on all possible natural numbers. In this paper, we investigate several generalizations and variations of this type of problem. In each case, we provide a complete characterization of when such a property holds.

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arXiv2024

Non-repetition of second coefficients of Hecke polynomials

Let $T_m(N,2k)$ denote the $m$-th Hecke operator on the space $S_{2k}(\Gamma_0(N))$ of cuspidal modular forms of weight $2k$ and level $N$. In this paper, we study the non-repetition of the second coefficient of the characteristic polynomial of $T_m(N,2k)$. We obtain results in the horizontal aspect (where $m$ varies), the vertical aspect (where $k$ varies), and the level aspect (where $N$ varies). Finally, we use these non-repetition results to extend a result of Vilardi and Xue on distinguishing Hecke eigenforms.

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arXiv2024

Asymptotics and sign patterns of Hecke polynomial coefficients

We determine the asymptotic behavior of the coefficients of Hecke polynomials. In particular, this allows us to determine signs of these coefficients when the level or the weight is sufficiently large. In all but finitely many cases, this also verifies a conjecture on the nanvanishing of the coefficients of Hecke polynomials.

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arXiv2024

Zeros of even and odd period polynomials

Let $f \in S_k(\Gamma_0(N))$ be a newform, and let $r_f^{\pm}(X)$ denote its corresponding even and odd period polynomials. For sufficiently large level and weight, we show that the zeros of $r_f^{\pm}(X)$ all lie on the circle $|X| = \frac{1}{\sqrt N}$.

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arXiv2024

On the average size of the eigenvalues of the Hecke operators

We determine the average size of the eigenvalues of the Hecke operators acting on the cuspidal modular forms space $S_k(\Gamma_0(N))$ in both the vertical and the horizontal perspective. The "average size" is measured via the quadratic mean.

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arXiv2024

Nonvanishing of Second Coefficients of Hecke Polynomials on the Newspace

For $m \geq 1$, let $N \geq 1$ be coprime to $m$, $k \geq 2$, and $\chi$ be a Dirichlet character modulo $N$ with $\chi(-1)=(-1)^k$. Then let $T_m^{\text{new}}(N,k,\chi)$ denote the restriction of the $m$-th Hecke operator to the space $S_k^{\text{new}}(\Gamma_0(N), \chi)$. We demonstrate that for fixed $m$ and trivial character $\chi$, the second coefficient of the characteristic polynomial of $T_m^{\text{new}}(N,k)$ vanishes for only finitely many pairs $(N,k)$, and we further determine the sign. To demonstrate our method, for $m=2,4$, we also compute all pairs $(N,k)$ for which the second coefficient vanishes. In the general character case, we also show that excluding an infinite family where $S_k^{\text{new}}(\Gamma_0(N), \chi)$ is trivial, the second coefficient of the characteristic polynomial of $T_m^{\text{new}}(N,k,\chi)$ vanishes for only finitely many triples $(N,k,\chi)$.

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arXiv2024

Signs of the Second Coefficients of Hecke Polynomials

Let $T_m(N, k, \chi)$ be the $m$-th Hecke operator of level $N$, weight $k \ge 2$, and nebentypus $\chi$, where $N$ is coprime to $m$. We first show that for any given $m \ge 1$, the second coefficient of the characteristic polynomial of $T_m(N, k, \chi)$ is nonvanishing for all but finitely many triples $(N,k,\chi)$. Furthermore, for $\chi$ trivial and any fixed $m$, we determine the sign of the second coefficient for all but finitely many pairs $(N,k)$. Finally, for $\chi$ trivial and $m=3,4$, we compute the sign of the second coefficient for all pairs $(N,k)$.

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arXiv2024

Newspaces with Nebentypus: An Explicit Dimension Formula and Classification of Trivial Newspaces

Consider $N \geq 1$, $k \geq 2$, and $\chi$ a Dirichlet character modulo $N$ such that $\chi(-1) = (-1)^k$. For any bound $B$, one can show that $\dim S_k(\Gamma_0(N),\chi) \le B$ for only finitely many triples $(N,k,\chi)$. It turns out that this property does not extend to the newspace; there exists an infinite family of triples $(N,k,\chi)$ for which $\dim S_k^{\text{new}}(\Gamma_0(N),\chi) = 0$. However, we classify this case entirely. We also show that excluding the infinite family for which $\dim S_k^{\text{new}}(\Gamma_0(N),\chi) = 0$, $\dim S_k^{\text{new}}(\Gamma_0(N),\chi) \leq B$ for only finitely many triples $(N,k,\chi)$. In order to show these results, we derive an explicit dimension formula for the newspace $S_k^{\text{new}}(\Gamma_0(N),\chi)$. We also use this explicit dimension formula to prove a character equidistribution property and disprove a conjecture from Greg Martin that $\dim S_2^{\text{new}}(\Gamma_0(N))$ takes on all possible non-negative integers.

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