arXiv subjects

Changningphaabi Namoijam

Publications and source records attributed to Changningphaabi Namoijam.

10 recordsLinked to original sources

arXiv2026

Algebraic independence of periods of Anderson modules and their hyperderivatives

Transcendence questions of periods and quasi-periods of Anderson modules, as well as the hyperderivatives thereof, are of major interest in number theory over function fields. In the case of Drinfeld modules, this question was answered by the second author. In this paper, we show that these results hold for general Anderson $t$-modules under certain assumptions on their Galois representations. The proofs use the explicit description of the $\mathfrak{p}$-adic Galois representation of the first author by means of a rigid analytic trivialization of the associated $t$-motive, as well as prolongations of $t$-modules and $t$-motives. The general result is applied to determine algebraic relations between periods and quasi-periods, and their hyperderivatives, of finitely many Drinfeld modules. Along the way, our results provide more evidence for the Mumford-Tate conjecture for Anderson modules.

math.NT↗

arXiv2026

On the transcendence of special values of Goss $L$-functions attached to Drinfeld modules

Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the rational function field $K:=\mathbb{F}_q(θ)$. For a Drinfeld module $ϕ$ defined over $K$, we study the transcendence of special values of the Goss $L$-function attached to the abelian $t$-motive $M_ϕ$ of $ϕ$. Moreover, when $ϕ$ is a Drinfeld module of rank $r\geq 2$ defined over $K$ which has everywhere good reduction, we prove that the value of the Goss $L$-function attached to the $(r-1)$-st exterior power of $M_ϕ$ at any positive integer is transcendental over $K$.

math.NT↗

arXiv2026

Linear equations on $t$-modules

Let $F$ be a number field. Given finitely many $F$-valued points on a commutative algebraic group defined over $F$, a question of interest to number theorists is the determination of the group of their linear relations. In this article, we investigate an analogous problem in the $t$-module setting. Let $L$ be a global function field, and $E$ be a $d$-dimensional $t$-module defined over $L$. Given finitely many points on $E$ with entries in $L$, we establish the connection between their $\mathbb{F}_q[t]$-linear relations and polynomial solutions of Frobenius difference equations. Consequently, we deduce an algorithm to compute the module of their $\mathbb{F}_q[t]$-linear relations.

math.NT↗

arXiv2025

On the algebraic independence of logarithms of Anderson $t$-modules

In the present paper, we determine the algebraic relations among the tractable coordinates of logarithms of Anderson $t$-modules constructed by taking the tensor product of Drinfeld modules of rank $r$ defined over the algebraic closure of the rational function field and their $(r-1)$-st exterior powers with the Carlitz tensor powers. Our results, in the case of the tensor powers of the Carlitz module, generalize the work of Chang and Yu on the algebraic independence of polylogarithms.

math.NT↗

arXiv2025

Computing supersingular endomorphism rings using inseparable endomorphisms

We give an algorithm for computing an inseparable endomorphism of a supersingular elliptic curve $E$ defined over $\mathbb F_{p^2}$, which, conditional on GRH, runs in expected $O(p^{1/2}(\log p)^2(\log\log p)^3)$ bit operations and requires $O((\log p)^2)$ storage. This matches the time and storage complexity of the best conditional algorithms for computing a nontrivial supersingular endomorphism, such as those of Eisenträger-Hallgren-Leonardi-Morrison-Park and Delfs-Galbraith. Unlike these prior algorithms, which require two paths from $E$ to a curve defined over $\mathbb F_p$, the algorithm we introduce only requires one; thus when combined with the algorithm of Corte-Real Santos-Costello-Shi, our algorithm will be faster in practice. Moreover, our algorithm produces endomorphisms with predictable discriminants, enabling us to prove properties about the orders they generate. With two calls to our algorithm, we can provably compute a Bass suborder of $\operatorname{End}(E)$. This result is then used in an algorithm for computing a basis for $\operatorname{End}(E)$ with the same time complexity, assuming GRH. We also argue that $\operatorname{End}(E)$ can be computed using $O(1)$ calls to our algorithm along with polynomial overhead, conditional on a heuristic assumption about the distribution of the discriminants of these endomorphisms. Conditional on GRH and this additional heuristic, this yields a $O(p^{1/2}(\log p)^2(\log\log p)^3)$ algorithm for computing $\operatorname{End}(E)$ requiring $O((\log p)^2)$ storage.

math.NT↗

arXiv2024

Towards a classification of $p^2$-discriminant ideal twins over number fields

Isogenous elliptic curves have the same conductor but not necessarily the same minimal discriminant ideal. In this article, we explicitly classify all $p^2$-isogenous elliptic curves defined over a number field with the same minimal discriminant ideal for odd prime $p$ where $X_0(p^2)$ has genus $0$, i.e., $p = 3$ or $5$. As a consequence, we give a list of all $p^2$-isogenous discriminant (ideal) twins over $\mathbb{Q}$ for such $p$.

math.NT↗

arXiv2024

Elated Numbers

For a base $b \geq 2$, the $b$-elated function, $E_{2,b}$, maps a positive integer written in base $b$ to the product of its leading digit and the sum of the squares of its digits. A $b$-elated number is a positive integer that maps to $1$ under iteration of $E_{2,b}$. The height of a $b$-elated number is the number of iterations required to map it to $1$. We determine the fixed points and cycles of $E_{2,b}$ and prove a range of results concerning sequences of $b$-elated numbers and $b$-elated numbers of minimal heights. Although the $b$-elated function is closely related to the $b$-happy function, the behaviors of the two are notably different, as demonstrated by the results in this work.

math.NT↗

arXiv2024

Algebraic relations among hyperderivatives of periods and logarithms of Drinfeld modules

We determine all algebraic relations among all hyperderivatives of the periods, quasi-periods, logarithms, and quasi-logarithms of Drinfeld modules defined over a separable closure of the rational function field. In particular, for periods and logarithms that are linearly independent over the endomorphism ring of the Drinfeld module, we prove the algebraic independence of their hyperderivatives and the hyperderivatives of the corresponding quasi-periods and quasi-logarithms.

math.NT↗

arXiv2024

On the third kind periods for abelian $t$-modules

Inspired by the relations between periods of elliptic integrals of the third kind and the periods of the extensions of the corresponding elliptic curves by the multiplicative group, we introduce the notion of the third kind periods for abelian $t$-modules and establish an evaluation for these periods that is parallel to the classical setting. When we specialize our result to the case of Drinfeld modules, an explicit formula for these third kind periods is established. We also prove the algebraic independence of periods of the first, the second, and the third kind for Drinfeld modules of arbitrary rank. This generalizes prior results of Chang for rank $2$ Drinfeld modules.

math.NT↗

arXiv2022

Hyperderivatives of periods and quasi-periods for Anderson $t$-modules

We investigate periods, quasi-periods, logarithms, and quasi-logarithms of Anderson $t$-modules, as well as their hyperderivatives. We develop a comprehensive account of how these values can be obtained through rigid analytic trivializations of abelian and $\mathbf{A}$-finite $t$-modules. To do this we build on the exponentiation theorem of Anderson and investigate quasi-periodic extensions of $t$-modules through Anderson generating functions. By applying these results to prolongation $t$-modules of Maurischat, we integrate hyperderivatives of these values together with previous work of Brownawell and Denis in this framework.

math.NT↗