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Let $K$ be a number field, and $\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the solutions of the generalized Fruit Diophantine equation $ax^d-y^2-z^2 +xyz-b=0$ over $K$, where $a\in \mathcal{O}_K\setminus \{0\}$, $b \in \mathcal{O}_K \setminus \{0\}$ with $b=2^da-3^r$ for positive odd integers $r,d$ with $3 |d$. As an application of this, we construct infinitely many elliptic curves over $K$ having no integral point $(x,y) \in \mathcal{O}_K^2$ with $2|x$. In particular, we study the solutions of $ax^d-y^2-z^2 +xyz-b=0$ over quadratic fields $K=\mathbb{Q}(\sqrt{d})$, for square-free integers $d$. Further, we find explicit values of $d$ for the above result and calculate the density of such square-free integers $d$ with $d \geq 2$., Comment: 5 pages

Let $K$ be a totally real number field, and $ \mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation, i.e., $Ax^p+By^p+Cz^p=0$ over $K$ of prime exponent $p$, where $A,B,C \in \mathcal{O}_K \setminus \{0\}$ with $ABC$ is even (in the sense that $\mathfrak{P}| ABC$, for some prime ideal $\mathfrak{P}$ of $ \mathcal{O}_K$ with $\mathfrak{P} |2$). For certain class of fields $K$, we prove that the equation $Ax^p+By^p+Cz^p=0$ has no asymptotic solution in $K^3$ (resp., of certain type in $K^3$), under some assumptions on $A,B,C$ (resp., for all $A,B,C \in \mathcal{O}_K \setminus \{0\}$ with $ABC$ is even). We also present several purely local criteria of $K$ such that $Ax^p+By^p+Cz^p=0$ has no asymptotic solutions in $K^3$., Comment: 14 pages. arXiv admin note: text overlap with arXiv:2403.14640, arXiv:2301.09263

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 $\mathcal{O}_K$.

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.