arXiv subjects

Victor Manuel Aricheta

Publications and source records attributed to Victor Manuel Aricheta.

5 recordsLinked to original sources

arXiv2025

Ramanujan's continued fractions of order $10$ as modular functions

We explore the modularity of the continued fractions $I(τ), J(τ), T_1(τ), T_2(τ)$ and $U(τ)=I(τ)/J(τ)$ of order $10$, where $I(τ)$ and $J(τ)$ are introduced by Rajkhowa and Saikia, which are special cases of certain identities of Ramanujan. In particular, we show that these fractions can be expressed in terms of an $η$-quotient $g(τ)$ that generates the field of all modular functions on the congruence subgroup $Γ_0(10)$. Consequently, we prove that modular equations for $g(τ)$ and $U(τ)$ exist at any level and derive these equations of prime levels $p\leq 11$. We also show that the continued fractions of order $10$ can be explicitly evaluated using a singular value of $g(τ)$, which under certain conditions, generates the Hilbert class field of an imaginary quadratic field. We employ the methods of Lee and Park to establish our results.

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arXiv2024

Analogue of Ramanujan's function $k(τ)$ for the continued fraction $X(τ)$ of order six

Motivated by the recent work of Park on the analogue of the Ramanujan's function $k(τ)=r(τ)r^2(2τ)$ for the Ramanujan's cubic continued fraction, where $r(τ)$ is the Rogers-Ramanujan continued fraction, we use the methods of Lee and Park to study the modularity and arithmetic of the function $w(τ) = X(τ)X(3τ)$, which may be considered as an analogue of $k(τ)$ for the continued fraction $X(τ)$ of order six introduced by Vasuki, Bhaskar and Sharath. In particular, we show that $w(τ)$ can be written in terms of the normalized generator $u(τ)$ of the field of all modular functions on $Γ_0(18)$, and derive modular equations for $u(τ)$ of smaller prime levels. We also express $j(dτ)$ for $d\in\{1,2,3,6,9,18\}$ in terms of $u(τ)$, where $j$ is the modular $j$-invariant.

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arXiv2023

Density results for the parity of $(4k,k)$-singular overpartitions

The $(k,i)$-singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to $6k$. In this paper we consider the parity of the number $\overline{C}_{k,i}(n)$ of $(k,i)$-singular overpartitions of $n$. In particular, we give a sufficient condition on even values of $k$ so that the values of $\overline{C}_{4k,k}(n)$ are almost always even. Furthermore, we show that for odd values of $k \leq 23$, $k\neq 19$, certain subsequences of $\overline{C}_{4k,k}(n)$ are almost always even.

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arXiv2019

Supersingular Elliptic Curves and Moonshine

We generalize a theorem of Ogg on supersingular $j$-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g., baby monster, Fischer's largest group). This situates Ogg's theorem in a broader setting. More generally, we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. We also present a connection between supersingular elliptic curves and umbral moonshine. Finally, we present a procedure for explicitly computing invariants of supersingular elliptic curves with level structure.

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arXiv2018

Moonshine modules and a question of Griess

We consider the situation in which a finite group acts on an infinite-dimensional graded module in such a way that the graded trace functions are weakly holomorphic modular forms. Under a mild hypothesis we completely describe the asymptotic module structure of the homogeneous subspaces. As a consequence we find that moonshine for a group gives rise to partial orderings on its irreducible representations. This serves as a first answer to a question posed by Griess. In particular, we show that our hypothesis holds for umbral moonshine and for automorphism groups of certain vertex operator algebras.

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