arXiv subjects

Bumkyu Cho

Publications and source records attributed to Bumkyu Cho.

5 recordsLinked to original sources

arXiv2020

Recurrence relations satisfied by the traces of singular moduli for $Γ_0(N)$

We compute the divisor of the modular equation on the modular curve $Γ_0(N) \backslash \mathbb H^*$ and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup $Γ_0(N)$ of genus zero. We also introduce the notions and properties of $Γ$-equivalence and $Γ$-reduced forms about binary quadratic forms. Using these, we can explicitly compute the recurrence relations for $N = 2, 3, 4, 5$.

math.NT↗

arXiv2017

On the $Γ$-equivalence of binary quadratic forms

For a congruence subgroup $Γ$, we define the notion of $Γ$-equivalence on binary quadratic forms which is the same as proper equivalence if $Γ= \mathrm{SL}_2(\mathbb Z)$. We develop a theory on $Γ$-equivalence such as the finiteness of $Γ$-reduced forms, the isomorphism between $Γ_0(N)$-form class group and the ideal class group, $N$-representation of integers, and $N$-genus of binary quadratic forms. As an application, we deal with representations of integers by binary quadratic forms under certain congruence condition on variables.

math.NT↗

arXiv2014

On the number of representations of integers by quadratic forms with congruence conditions

We characterize the generating function of the number of representations described in the title in terms of the theory of modular forms. Appealing to this characterization we obtain explicit formulas for the representation numbers as examples.

math.NT↗

arXiv2011

On harmonic weak Maass forms of half integral weight

We show that certain space of vector valued harmonic weak Maass forms of half integral weight is isomorphic to a space of scalar valued ones whose Fourier coefficients are supported on suitable progressions. This kind of result for holomorphic modular forms was proved by Eichler and Zagier.

math.NT↗

arXiv2011

Zagier duality for harmonic weak Maass forms of integral weight

We show the existence of "Zagier duality" between vector valued harmonic weak Maass forms and vector valued weakly holomorphic modular forms of integral weight. This duality phenomenon arises naturally in the context of harmonic weak Maass forms as developed in recent works by Bruinier, Funke, Ono, and Rhoades. Concerning the isomorphism between the spaces of scalar and vector valued harmonic weak Maass forms of integral weight, "Zagier duality" between scalar valued ones is derived.

math.NT↗