arXiv subjects
Publications and source records attributed to Surjeet Kaushik.
In this article, we study the simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different integral weights. We also study the simultaneous non-vanishing of Fourier coefficients, of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights, at powers of a fixed prime ideal.
We prove an equidistribution of signs for the Fourier coefficients of Hilbert modular forms of half-integral weight. Our study focuses on certain subfamilies of coefficients that are accessible via the Shimura correspondence. This is a generalization of the result of Inam and Wiese to the setting of totally real number fields
Suppose $E$ is an elliptic curve over $\mathbb{Q}$ of conductor $N$ with complex multiplication (CM) by $\mathbb{Q}(i)$, and $f_E$ is the corresponding cuspidal Hecke eigenform in $S^{\mathrm{new}}_2(Γ_0(N))$. Then $n$-th Fourier coefficient of $f_E$ is non-zero in the short interval $(X, X + cX^{\frac{1}{4}})$ for all $X \gg 0$ and for some $c > 0$. As a consequence, we produce infinitely many cuspidal CM eigenforms $f$ level $N>1$ and weight $k > 2$ for which $i_f(n) \ll n^{\frac{1}{4}}$ holds, for all $n \gg 0$.