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Cameron Franc

Publications and source records attributed to Cameron Franc.

25 records · Page 2Linked to original sources
arXiv2017

Vector bundles and modular forms for Fuchsian groups of genus zero

This article lays the foundations for the study of modular forms transforming with respect to representations of Fuchsian groups of genus zero. More precisely, we define geometrically weighted graded modules of such modular forms, where the graded structure comes from twisting with all isomorphism classes of line bundles on the corresponding compactified modular curve, and we study their structure by relating it to the structure of vector bundles over orbifold curves of genus zero. We prove that these modules are free whenever the Fuchsian group has at most two elliptic points. For three or more elliptic points, we give explicit constructions of indecomposable vector bundles of rank two over modular orbifold curves, which give rise to non-free modules of geometrically weighted modular forms.

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arXiv2016

Vector valued modular forms and the modular orbifold of elliptic curves

This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of SL_2(Z) play in the holomorphic theory of vector valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line P(4, 6).

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arXiv2015

On the structure of modules of vector valued modular forms

If $ρ$ denotes a finite dimensional complex representation of $\textbf{SL}_2(\textbf{Z})$, then it is known that the module $M(ρ)$ of vector valued modular forms for $ρ$ is free and of finite rank over the ring $M$ of scalar modular forms of level one. This paper initiates a general study of the structure of $M(ρ)$. Among our results are absolute upper and lower bounds, depending only on the dimension of $ρ$, on the weights of generators for $M(ρ)$, as well as upper bounds on the multiplicities of weights of generators of $M(ρ)$. We provide evidence, both computational and theoretical, that a stronger three-term multiplicity bound might hold. An important step in establishing the multiplicity bounds is to show that there exists a free-basis for $M(ρ)$ in which the matrix of the modular derivative operator does not contain any copies of the Eisenstein series $E_6$ of weight six.

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arXiv2015

Three-dimensional imprimitive representations of the modular group and their associated modular forms

This paper uses previous results of the authors on vector-valued modular forms to study certain non-congruence modular forms. We prove that these forms have unbounded denominators, and in certain cases we verify congruences of Atkin--Swinnerton-Dyer type satisfied by the Fourier coefficients of these forms. Our results rest on group-theoretic facts about the modular group, a detailed study of its imprimitive three-dimensional representations, and the theory of their associated vector-valued modular forms. For the proof of the congruences we also make essential use of a result of Katz.

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arXiv2013

Fourier coefficients of vector-valued modular forms of dimension 2

We prove the following theorem. Suppose that $F=(f_1, f_2)$ is a 2-dimensional vector-valued modular form on $SL_2(Z)$ whose component functions $f_1, f_2$ have rational Fourier coefficients with bounded denominators. Then $f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the modular group.

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