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In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields $K$, i.e., $Ax^p+By^p=Cz^3$ with prime exponent $p$ and $A,B,C \in \mathcal{O}_K \setminus \{0\}$. For certain class of fields $K$, we prove that $Ax^p+By^p=Cz^3$ has no asymptotic solutions over $K$ (resp., solutions of certain type over $K$) with restrictions on $A,B,C$ (resp., for all $A,B,C \in \mathcal{O}_K \setminus \{0\}$). Finally, we present several local criteria over $K$., Comment: arXiv admin note: substantial text overlap with arXiv:2301.09263

In this article, we study Lehmer-type bounds for the N\'eron-Tate height of $\bar{K}$-points on abelian varieties $A$ over number fields $K$. Then, we estimate the number of $K$-rational points on $A$ with N\'eron-Tate height $\leq \log B$ for $B\gg 0$. This estimate involves a constant $C$, which is not explicit. However, for elliptic curves and the product of elliptic curves over $K$, we make the constant explicitly computable., Comment: To appear in the International Journal of Number Theory

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: To appear in the Ramanujan Journal

In this article, we study the non-trivial primitive solutions of a certain type for the Diophantine equations $x^p+y^p=2^rz^p$ and $x^p+y^p=z^2$ of prime exponent $p$, $r \in \mathbb{N}$, over a totally real field $K$. Then for $r=2,3$, we study the non-trivial primitive solutions over $\mathcal{O}_K$ for the equation $x^p+y^p=2^rz^p$ of prime exponent $p$. Finally, we give several purely local criteria for $K$ such that the equation $x^p+y^p=2^rz^p$ has no non-trivial primitive solutions over $\mathcal{O}_K$.

In this article, we settle a part of the Conjecture by Bandini and Valentino (\cite{BV19a}) for $S_{k,l}(\Gamma_0(T))$ when $\mathrm{dim}\ S_{k,l}(\mathrm{GL}_2(A))\leq 2$. Then, we frame this conjecture for prime, higher levels, and provide some evidence in favour of it. For any square-free level $\mathfrak{n}$, we define oldforms $S_{k,l}^{\mathrm{old}}(\Gamma_0(\mathfrak{n}))$, newforms $S_{k,l}^{\mathrm{new}}(\Gamma_0(\mathfrak{n}))$, and investigate their properties. These properties depend on the commutativity of the (partial) Atkin-Lehner operators with the $U_\mathfrak{p}$-operators. Finally, we show that the set of all $U_\mathfrak{p}$-operators are simultaneously diagonalizable on $S_{k,l}^{\mathrm{new}}(\Gamma_0(\mathfrak{n}))$.

In \cite{Cho09}, Choi studied congruences of coefficients (modulo $T^q-T$) for Drinfeld modular forms of level $\Gamma_0(T)$, trivial type and the linear relations between the initial coefficients of those. In this article, we generalize these results for level $\Gamma_0(T)$, arbitrary type.

In this article, we describe the structure of the $R$-algebra of Drinfeld modular forms $M(\Gamma_0(T))_R$ (resp., $M^0(\Gamma_0(T))_R$) of level $\Gamma_0(T)$ and the structure of mod-$\p$ reduction of $M_{\mfp}^0(\Gamma_0(T))$ for $\p \neq (T)$. As a result, we are able to study the properties of the weight filtration for $M_{k,l}(\Gamma_0(T))$. Finally, we prove a result on mod-$\p$ congruences for Drinfeld modular forms of level $\Gamma_0(\p T)$ for $\p \neq (T)$.

According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form $X_0(N)$ for a {\it prime} number $N$. There is a recent interest to generalize the conjecture for arbitrary $N$ by Ribet, Ohta and Yoo. In this direction, Ribet conjectured that all the Eisenstein maximal ideals are "cuspidal". Hwajong Yoo proved the conjecture ( under certain hypothesis) provided that those ideals are {\it rational}. In this article, we show that ( under certain hypothesis), Ribet's conjecture is true for {\it non-rational} Eisenstein maximal ideals.

For a primitive Hilbert modular form $f$ over $F$ of weight $k$, under certain assumptions on image of $\bar{\rho}_{f,\lambda}$, we calculate the Dirichlet density of primes $\mathfrak{p}$ for which the $\mathfrak{p}$-th Fourier coefficient $C(\mathfrak{p}, f)$ generates the coefficient field $E_f$. If $k=2$, then we show that the assumption on the image of $\bar{\rho}_{f,\lambda}$ is satisfied when the degrees of $E_f, F$ are equal and odd prime. We also compute the density of primes $\mathfrak{p}$ for which $C^*(\mathfrak{p}, f)$ generates $F_f$. Then, we provide some examples of $f$ to support our results. Finally, we calculate the density of primes $\mathfrak{p}$ for which $C(\mathfrak{p}, f) \in K$ for any field $K$ with $F_f \subseteq K \subseteq E_f$. This density is completely determined by the inner twists of $f$ associated with $K$. This work can be thought of as a generalization of~\cite{KSW08} to primitive Hilbert modular forms.

In~\cite{CS04}, Calegari and Stein studied the congruences between classical cusp forms $S_k(\Gamma_0(p))$ of prime level and made several conjectures about them. In~\cite{AB07} (resp., ~\cite{BP11}) the authors proved one of those conjectures (resp., their generalizations). In this article, we study the analogous conjecture and its generalizations for Drinfeld modular forms.

In this semi-expository article, we discuss about the non-vanishing of the Fourier coefficients of primitive forms. Also, we shall make a note of a discrepancy in the statement of [KRW07, Lemma 2.2]., Comment: To appear in the "special issue of the Proceedings of Telangana Academy of Sciences", 2020

In this article, we prove a generalization of a theorem (Ogg's conjecture) due to Bary Mazur for arbitrary $N\in \N$ and for {\it number fields}. The main new observation is a modification of a theorem due to Glenn Stevens for the congruence subgroups of the form $\Ga_0(N)$ for any $N \in \N$. This in turn help us to determine the relevant part of the cuspidal subgroups without dependence on Shimura subgroups., Comment: fatal error

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

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., Comment: Submitted for publication

We show that signs of Fourier coefficients, on certain sub-families, determine the half-integral weight cuspidal eigenform uniquely, up to a positive constant. We also study sign change results for the product of the Fourier coefficients of two distinct half-integral weight eigenforms.

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, Comment: To appear in Research in Number Theory

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 $S^{\mathrm{new}}_2(\Gamma_0(N))$. 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 $n \gg 0$.

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 Journal

We study the behaviour of ordinary parts of the homology modules of modular curves, associated to a decreasing sequence of congruence subgroups ${\Gamma}_1(N2^r)$ for $r \geq 2$, and prove a control theorem for these homology modules.

In this note, we prove multiplicity one theorems for generalized modular functions (GMF), in terms of their q-exponents, and make a general statement about the nature of values that the prime q-exponents of a GMF can take. We shall also study the integrality of general q-exponents of a GMF and give an upper bound on the first sign change of these q-exponents., Comment: Comments are welcome

In this note, we show that the algebraicity of the Fourier coefficients of half-integral weight modular forms can be determined by checking the algebraicity of the first few of them. We also give a necessary and sufficient condition for a half-integral weight modular form to be in Kohnen's +-subspace by considering only finitely many terms., Comment: To appear in Archiv der Mathematik

Let F be a Siegel cusp form of weight k and genus n>1 with Fourier-Jacobi coefficients f_m. In this article, we estimate the growth of the Petersson norms of f_m, where m runs over an arithmetic progression. This result sharpens a recent result of Kohnen in [5]., Comment: to appear in Archiv der Mathematik

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 Theory