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Jesse Jääsaari

Publications and source records attributed to Jesse Jääsaari.

11 recordsLinked to original sources
arXiv2026

On the Number of Hecke Eigenvalues of the Same Sign on $\mathrm{GL}_n$

We study the distribution of signs of the (real-valued) Hecke eigenvalues $A(m,1,...,1)$ of self-dual Hecke--Maass cusp forms for the group $\mathrm{SL}_n(\mathbb Z)$, where $n\geq 2$ is an integer. Our main result establishes, under the Generalised Ramanujan--Petersson Conjecture, that for all $X\leq x\leq 2X$ outside a set of measure $o(X)$ the short interval $(x,x+H]$ contains at least $H(\log X)^{1/n^2-1}$ indices with both positive and negative Hecke eigenvalues, provided that $(\log X)^{1-1/n^2+\varepsilon}\leq H\leq X/10$. For $\mathrm{GL}_2$ Hecke--Maass cusp forms we obtain a stronger bound unconditionally, and for self-dual $\mathrm{GL}_3$ forms we obtain the same logarithmic exponent unconditionally. We also establish results under weaker bounds towards the Generalised Ramanujan--Petersson Conjecture together with the Generalised Lindelöf Hypothesis, and obtain lower bounds for the number of sign changes and for the number of Hecke eigenvalues of each sign in long intervals.

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arXiv2026

On the Real Zeroes of Half-integral Weight Hecke Cusp Forms, II

We show that for $\gg K^2$ of the half-integral weight Hecke cusp forms in Kohnen plus subspaces with weight bounded by a large parameter $K$, the number of "real" zeroes grows at the expected rate. A key technical step in the proof is to obtain sharp bounds for the mollified first and second moments of quadratic twists of modular $L$-functions.

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arXiv2026

On the Real Zeroes of Half-integral Weight Hecke Cusp Forms

We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold $Γ_0(4)\backslash\mathbb H$ near a cusp at infinity. In analogue of the Ghosh-Sarnak conjecture for classical holomorphic Hecke cusp forms, one expects that almost all of the zeroes sufficiently close to this cusp lie on two vertical geodesics $\Re(s)=-1/2$ and $\Re(s)=0$ as the weight tends to infinity. We show that, for $\gg_\varepsilon K^2/(\log K)^{3/2+\varepsilon}$ of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large constant $K$, the number of such "real" zeroes grows almost at the expected rate. We also obtain a weaker lower bound for the number of real zeroes that holds for a positive proportion of forms. One of the key ingredients is the asymptotic evaluation of averaged first and second moments of quadratic twists of modular $L$-functions.

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arXiv2026

$Ω$-Results for Exponential Sums Related to Maass Cusp Forms for $\mathrm{SL}_3(\mathbb Z)$

We obtain $Ω$-results for linear exponential sums with rational additive twists of small prime denominators weighted by Hecke eigenvalues of Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. In particular, our $Ω$-results match the expected conjectural upper bounds when the denominator of the twist is sufficiently small compared to the length of the sum. Non-trivial $Ω$-results for sums over short segments are also obtained. Along the way we produce lower bounds for mean squares of the exponential sums in question and also improve the best known upper bound for these sums in some ranges of parameters.

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arXiv2024

Mass equidistribution for Saito-Kurokawa lifts

Let $F$ be a holomorphic cuspidal Hecke eigenform for $\mathrm{Sp}_4(\mathbb{Z})$ of weight $k$ that is a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis (GRH), we prove that the mass of $F$ equidistributes on the Siegel modular variety as $k\longrightarrow \infty$. As a corollary, we show under GRH that the zero divisors of Saito--Kurokawa lifts equidistribute as their weights tend to infinity.

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arXiv2022

On Signs of Fourier Coefficients of Hecke-Maass Cusp Forms on $\mathrm{GL}_3$

We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. When the underlying form is self-dual, we show that there are $\gg_\varepsilon X^{5/6-\varepsilon}$ sign changes among the coefficients $\{A(m,1)\}_{m\leq X}$ and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients $A(m,m)$ for generic $\mathrm{GL}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of $\mathrm{GL}_3$ cusp forms we establish. In addition, non-vanishing of the Fourier coefficients is studied under the Ramanujan-Petersson conjecture.

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arXiv2021

On fundamental Fourier coefficients of Siegel cusp forms of degree 2

Let $F$ be a Siegel cusp form of degree 2, even weight $k \geq 2$ and odd squarefree level $N$. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of $F$ at fundamental matrices $S$ (i.e., with $-4 det(S)$ equal to a fundamental discriminant). We prove that as $S$ varies along the equivalence classes of fundamental matrices with $det(S) \asymp X$, the sequence $a(F,S)$ has at least $X^{1-ε}$ sign changes, and takes at least $X^{1-ε}$ "large values". Furthermore, assuming the Generalized Riemann Hypothesis as well as the refined Gan--Gross--Prasad conjecture, we prove the bound $|a(F,S)| \ll_{F, ε} \frac{\det(S)^{\frac{k}2 - \frac{1}{2}}}{ (\log |\det(S)|)^{\frac18 - ε}}$ for fundamental matrices $S$.

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arXiv2018

On Short Sums Involving Fourier Coefficients of Maass Forms

We study sums of Hecke eigenvalues of Hecke-Maass cusp forms for the group $\mathrm{SL}(n,\mathbb Z)$, with general $n\geq 3$, over certain short intervals under the assumption of the generalised Lindelöf hypothesis and a slightly stronger upper bound concerning the exponent towards the Ramanujan-Petersson conjecture that is currently known. In particular, in this case we evaluate the second moment of the sums in question asymptotically.

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arXiv2015

Exponential Sums Related to Maass Forms

We estimate short exponential sums weighted by the Fourier coefficients of a Maass form. This requires working out a certain transformation formula for non-linear exponential sums, which is of independent interest. We also discuss how the results depend on the growth of the Fourier coefficients in question. As a byproduct of these considerations, we can slightly extend the range of validity of a short exponential sum estimate for holomorphic cusp forms. The short estimates allow us to reduce smoothing errors. In particular, we prove an analogue of an approximate functional equation previously proven for holomorphic cusp form coefficients. As an application of these, we remove the logarithm from the classical upper bound for long linear sums weighted by Fourier coefficients of Maass forms, the resulting estimate being the best possible. This also involves improving the upper bounds for long linear sums with rational additive twists, the gains again allowed by the estimates for the short sums. Finally, we shall use the approximate functional equation to bound somewhat longer short exponential sums.

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