arXiv subjects
Publications and source records attributed to Narasimha Kumar.
Let $\mathbb{F}_{q}$ be a finite field, and $A:=\mathbb{F}_{q}[T]$. In this article, we give explicit criteria, involving concrete valuations, on the coefficients of the Drinfeld $A$-modules of rank $r$ for $r=2,3$, which ensure the surjectivity of the associated $(T)$-adic Galois representation. As a result, we shall calculate the densities of such Drinfeld $A$-modules.
Let $q = p^e \geq 7$ be an odd prime power, and set $A := \mathbb{F}_q[T]$. In this article, we construct an infinite two-parameter family of Drinfeld $A$-modules of rank $3$ such that, for every non-zero prime ideal $\mathfrak{l}$ of $A$, the associated mod-$\mathfrak{l}$, $\mathfrak{l}$-adic, and adelic Galois representations are surjective. These results generalise the specific example, constructed only for primes $p\equiv 1\pmod{3}$, in~\cite{Che22}.
Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is a prime power and let $A:= \mathbb{F}_{q}[T]$. By~\cite{PR09}, the adelic image of the Galois representation attached to a rank $2$ Drinfeld $A$-module $\varphi$ is open, and determining when it is surjective remains a subtle problem. To resolve this question, in this article, we study the $\mathfrak{p}$-adic surjectivity of the Galois representations attached to $\varphi$, where $\mathfrak{p} \in \Omega_A:= \mathrm{Spec}(A) \setminus \{ (0) \}$. There are two directions to investigate this problem: one by fixing the prime $\mathfrak{p}$, and the other by fixing $\varphi$. In the horizontal direction, for a fixed prime $\mathfrak{p} \in \Omega_A$, we give explicit and easily verifiable conditions on Drinfeld $ A$-modules $\varphi$ of rank $2$ which ensure the surjectivity of the $\mathfrak{p}$-adic Galois representation $\rho_{\varphi,\mathfrak{p}}$. This work not only extends the work of~\cite{Ray24} for $\mathfrak{p}=(T)$, but also obtains a variant of~\cite{Ray24} under comparatively simpler conditions in the case $\mathfrak{p}=(T)$. In the vertical direction, we show that for a fixed rank $2$ Drinfeld $A$-module $\varphi$, whose coefficients satisfy certain congruence and valuation conditions, the $\mathfrak{p}$-adic Galois representation $\rho_{\varphi,\mathfrak{p}}$ is surjective for all primes $\mathfrak{p} \in \Omega_A$. This recovers the example of \cite{Zyw11} and yields new examples beyond those considered in \cite{Zyw25}. As a consequence, we obtain the surjectivity of the associated adelic Galois representation.
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 $K$.
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.
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 $K$.
In this article, we settle a part of the Conjecture by Bandini and Valentino (\cite{BV19a}) for $S_{k,l}(Γ_0(T))$ 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 $S_{k,l}^{\mathrm{old}}(Γ_0(\mathfrak{n}))$, newforms $S_{k,l}^{\mathrm{new}}(Γ_0(\mathfrak{n}))$, 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 $S_{k,l}^{\mathrm{new}}(Γ_0(\mathfrak{n}))$.
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 \cite{Cho09}, Choi studied congruences of coefficients (modulo $T^q-T$) for Drinfeld modular forms of level $Γ_0(T)$, trivial type and the linear relations between the initial coefficients of those. In this article, we generalize these results for level $Γ_0(T)$, arbitrary type.
In this article, we describe the structure of the $R$-algebra of Drinfeld modular forms $M(Γ_0(T))_R$ (resp., $M^0(Γ_0(T))_R$) of level $Γ_0(T)$ and the structure of mod-$\p$ reduction of $M_{\mfp}^0(Γ_0(T))$ for $\p \neq (T)$. As a result, we are able to study the properties of the weight filtration for $M_{k,l}(Γ_0(T))$. Finally, we prove a result on mod-$\p$ congruences for Drinfeld modular forms of level $Γ_0(\p T)$ for $\p \neq (T)$.
In this paper, we are interested in the generalization of Ramanujan-like Eisenstein congruences (congruences between cusp forms and Eisenstein series) for congruence subgroups of the form $\Gamma_0(N)$ with $N \in \mathbb{N}$. We determine the possible primes that can produce Eisenstein congruences. We provide several examples of Eisenstein congruences to substantiate our method. Ribet conjectured (\cite[p. 360]{MR3540618}) about these congruences for the square-free level $N$. Yoo proved the conjecture. For general $N$, Yoo proved a generalization of the conjecture, under some hypotheses, provided that those ideals are {\it rational}. We show that the generalization of Ribet's conjecture for certain non-square-free levels $N$ is true even for {\it non-rational} Eisenstein maximal ideals.
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.
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 $S_k(Γ_0(p))$ 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].
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.
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