arXiv subjects

Michael J. Griffin

Publications and source records attributed to Michael J. Griffin.

6 recordsLinked to original sources

arXiv2022

AGM and jellyfish swarms of elliptic curves

The classical $\mathrm{AGM}$ produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. For finite fields $\mathbb{F}_q,$ with $q\equiv 3\pmod 4,$ we introduce a finite field analogue $\mathrm{AGM}_{\mathbb{F}_q}$ that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a $\mathit{jellyfish~swarm},$ as the 3D renderings of the connected components resemble $\mathit{jellyfish}$ (i.e. tentacles connected to a bell head). These swarms turn out to be more than the stuff of child's play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of $\mathbb{F}_q$-points, which can be used to prove that each swarm has at least $(1/2-\varepsilon)\sqrt{q}$ jellyfish. Additionally, this interpretation gives a description of the $\mathit{class~numbers}$ of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish.

math.NT↗

arXiv2020

On p-adic haromonic Maass functions

Modular and mock modular forms possess many striking $p$-adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology of modular curves. In the setting of over-convergent $p$-adic modular forms, Candelori and Castella showed this leads to $p$-adic analogs of harmonic Maass forms. In this paper we take an analytic approach to construct $p$-adic analogs of harmonic Maass forms of weight $0$ %and $1/2$ with square free level. Although our approaches differ, where the two theories intersect the forms constructed are the same. However our analytic construction defines these functions on the full super singular locus as well as on the ordinary locus. As with classical harmonic Maass forms, these $p$-adic analogs are connected to weight $2$ cusp forms and their modular derivatives are weight $2$ weakly holomorphic modular forms. Traces of their CM values also interpolate the coefficients of half integer weight modular and mock modular forms. We demonstrate this through the construction of $p$-adic analogs of two families of theta lifts for these forms.

math.NT↗

arXiv2016

A proof of the Thompson Moonshine Conjecture

In this paper we prove the existence of an infinite dimensional graded super-module for the finite sporadic Thompson group $Th$ whose McKay-Thompson series are weakly holomorphic modular forms of weight $\frac 12$ satisfying properties conjectured by Harvey and Rayhaun.

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arXiv2016

A framework of Rogers-Ramanujan identities and their arithmetic properties

The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+σ)}}{(1-q)\cdots (1-q^n)}, \] where $σ=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers-Ramanujan identities to doubly-infinite families of $q$-series identities. If $a\in\{1,2\}$ and $m,n\geq 1$, then we have \[ \sum_{\substack{λλ_1\leq m}} q^{a|λ|} P_{2λ}(1,q,q^2,\dots;q^n) =\textrm{"infinite product modular function"}, \] where the $P_λ(x_1,x_2,\dots;q)$ are Hall-Littlewood polynomials. These $q$-series are specialized characters of affine Kac--Moody algebras. Generalizing the Rogers-Ramanujan continued fraction, we prove in the case of $\textrm{A}_{2n}^{(2)}$ that the relevant $q$-series quotients are integral units.

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arXiv2015

Proof of the Umbral Moonshine Conjecture

The Umbral Moonshine Conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay-Thompson series are certain distinguished mock modular forms. Gannon has proved this for the special case involving the largest sporadic simple Mathieu group. Here we establish the existence of the umbral moonshine modules in the remaining 22 cases.

math.RT↗

arXiv2015

Moonshine

Monstrous moonshine relates distinguished modular functions to the representation theory of the monster. The celebrated observations that 196884=1+196883 and 21493760=1+196883+21296876, etc., illustrate the case of the modular function j-744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of the monster. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold, and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions.

math.RT↗