On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

We show that if a modular cuspidal eigenform $f$ of weight $2k$ is $2$-adically close to an elliptic curve $E/\mathbb{Q}$, which has a cyclic rational $4$-isogeny, then $n$-th Fourier coefficient of $f$ is non-zero in the short interval $(X, X + cX^{\frac{1}{4}})$ for all $X \gg 0$ and for some $c > 0$. We use this fact to produce non-CM cuspidal eigenforms $f$ of level $N>1$ and weight $k > 2$ such that $i_f(n) \ll n^{\frac{1}{4}}$ for all $n \gg 0$.Comment: To appear in The Ramanujan Journa

On sign changes of q-exponents of generalized modular functions

Let f be a generalized modular function of weight 0 of level N such that its q-exponents c(n)(n>0) are all real, and div(f) is zero. In this note, we show the equidistribution of signs for c(p)(p prime) by using equidistribution theorems for normalized cuspidal eigenforms of integral weight.Comment: to appear in the Journal of Number Theor

On non-vanishing and sign changes of the Fourier coefficients of Hilbert cusp forms

In this article, we study (simultaneous) non-vanishing, (simultaneous) sign changes of Fourier coefficients of (two) Hilbert cusp forms, respectively

Equidistribution of signs for Hilbert modular forms of half-integral weight

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 fieldsComment: To appear in Research in Number Theor

Some topics on the Fourier coefficients of modular forms

This thesis consists of three parts. In the first part, we study the gaps between non-zero Fourier coefficients of cuspdial CM eigenforms in the short intervals. In the second part, we study the sign changes for the Fourier coefficients of Hilbert modular forms of half-integral weight. In the third part, we study the simultaneous behaviour of Fourier coefficients of two different Hilbert modular cusp forms of integral weight. In Chapter 1, we present the definitions and some preliminaries on classical modular forms. We shall also recall some relevant results from the literature, which are useful in the subsequent chapters. In Chapter 2, we show that for an elliptic curve E over Q of conductor N with complex multiplication (CM) by Q(i), the n-th Fourier coefficient of fE is non-zero in the short interval (X;X + cX 1 4 ) for all X � 0 and for some c > 0, where fE is the corresponding cuspidal Hecke eigenform in S2

On the solutions of x2=Byp+Czpx^2= By^p+Cz^p and 2x2=Byp+Czp2x^2= By^p+Cz^p over totally real fields

In this article, we study the solutions of certain type over $K$ of the Diophantine equation $x^2= By^p+Cz^p$ with prime exponent $p$, where $B$ is an odd integer and $C$ is either an odd integer or $C=2^r$ for $r \in \mathbb{N}$. Further, we study the non-trivial primitive solutions of the Diophantine equation $x^2= By^p+2^rz^p$ ($r\in {1,2,4,5}$) (resp., $2x^2= By^p+2^rz^p$ with $r \in \mathbb{N}$) with prime exponent $p$, over $K$. We also present several purely local criteria of $K$.Comment: Submitted for publication; Any comments are welcome. arXiv admin note: text overlap with arXiv:2207.1093