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I: Genus-zero functions
mathematical sciences publishers
ALGEBRA AND NUMBER THEORY 4:6(2010)
I: Genus-zero functions
We introduce a notion of Hecke-monicity for functions on certain moduli spaces
associated to torsors of ﬁnite groups over elliptic curves, and show that it implies
strong invariance properties under linear fractional transformations. Speciﬁcally, if a weakly Hecke-monic function has algebraic integer coefﬁcients and a pole at inﬁnity, then it is either a holomorphic genus-zero function invariant under a congruence group or of a certain degenerate type. As a special case, we prove the same conclusion for replicable functions of ﬁnite order, which were introduced by Conway and Norton in the context of monstrous moonshine. As an applica- tion, we introduce a class of Lie algebras with group actions, and show that the characters derived from them are weakly Hecke-monic. When the Lie algebras come from chiral conformal ﬁeld theory in a certain sense, then the characters form holomorphic genus-zero functions invariant under a congruence group.
1. Equivariant Hecke operators 653
2. Hecke-monicity 659
3. Modular equations 661
4. Finite level 665
5. Replicability 669
6. twisted denominator formulas 673 Acknowledgments 677 References 677 Introduction
We deﬁne a holomorphic genus-zero function to be a holomorphic function f :
H → on the complex upper half-plane, with ﬁnite-order poles at cusps, such that there exists a discrete group f ⊂ SL2 ( ) for which f is invariant under the action of f by M¨ bius transformations, inducing a dominant injection H/ f →.
o Keywords: moonshine, replicable function, Hecke operator, generalized moonshine.
This material is partly based upon work supported by the National Science Foundation under grant DMS-0354321.
A holomorphic genus-zero function f therefore generates the ﬁeld of meromorphic functions on the quotient of H by its invariance group. In this paper, we are interested primarily in holomorphic congruence genus-zero functions, especially those f for which (N ) ⊂ f for some N 0. These functions are often called Hauptmoduln.
The theory of holomorphic genus-zero modular functions began with Jacobi’s work on elliptic and modular functions in the early 1800’s, but did not receive much attention until the 1970’s, when Conway and Norton found numerical relationships between the Fourier coefﬁcients of a distinguished class of these functions and the representation theory of the largest sporadic ﬁnite simple group, called the monster. Using their own computations together with work of Thompson and McKay, they formulated the monstrous moonshine conjecture, which asserts the existence of a graded representation V = n≥−1 Vn of such that for each g ∈, the graded character Tg (τ ) := n≥−1 Tr(g|Vn )q n is a normalized holomorphic genus-zero function invariant under some congruence group 0 (N ), where the normalization indicates a q-expansion of the form q −1 + O(q). More precisely, they gave a list of holomorphic genus-zero functions f g as candidates for Tg, whose ﬁrst several coefﬁcients arise from characters of the monster, and whose invariance groups f g contain some 0 (N ) [Conway and Norton 1979]. By unpublished work of Koike, the power series expansions of f g satisfy a condition known as complete replicability, given by a family of recurrence relations, and the relations determine the full expansion of f g from only the ﬁrst seven coefﬁcients of f gn for n ranging over powers of two.
Borcherds  proved this conjecture using a combination of techniques from the theory of vertex algebras and inﬁnite-dimensional Lie algebras: V was constructed by Frenkel, Lepowsky, and Meurman  as a vertex operator algebra, and Borcherds used it to construct the monster Lie algebra, which inherits an action of the monster. Since the monster Lie algebra is a generalized Kac–Moody algebra with a homogeneous action of, it admits twisted denominator formulas, which relate the coefﬁcients of Tg to characters of powers of g acting on the root spaces.
In particular, each Tg is completely replicable, and Borcherds completed the proof by checking that the ﬁrst seven coefﬁcients matched the expected values.
Knowing this theorem and some additional data, one can ask at least two natural
(1) The explicit checking of coefﬁcients at the end of the proof has been called a “conceptual gap” in [Cummins and Gannon 1997], and this problem has been
rectiﬁed in some sense by replacing that step with noncomputational theorems:
Borcherds  pointed out that the twisted denominator formulas imply • that the functions Tg are completely replicable.
Generalized moonshine, I: Genus-zero functions 651 Kozlov  showed that completely replicable functions satisfy lots of • modular equations.
Cummins and Gannon  showed that power series satisfying enough • modular equations are either holomorphic genus-zero and invariant under 0 (N ), or of a particular degenerate type resembling trigonometric functions.
One can eliminate the degenerate types, either by appealing to a result of • Martin  asserting that completely replicable series that are “J -ﬁnal” (a condition that holds for all Tg, since T1 = J ) are invariant under 0 (N ) for some N, or by using a result of Dong, Li, and Mason  that restricts the form of the q-expansions at other cusps.
Since modular functions live on moduli spaces of structured elliptic curves, one might ask how these recursion relations and replicability relate to group actions and moduli of elliptic curves.
(2) One might wonder if similar behavior applies to groups other than the monster.
Conway and Norton  suggested that other sporadic groups may exhibit properties resembling moonshine, and [Queen 1981] produced strong computational evidence for this. Norton [Mason 1987, Appendix] organized this data into the generalized moonshine conjecture, which asserts the existence of a generalized character Z that associates a holomorphic function on H to each commuting pair
of elements of the monster, satisfying the following conditions:
Z (g, h, τ ) is invariant under simultaneous conjugation of g and h.
Z (g, h, τ ) = j (τ ) − 744 = q −1 + 196884q + 21493760q 2 + · · · if and only if • g = h = 1.
This conjecture is still open, but if we ﬁx g = 1, it reduces to the original moonshine conjecture. One might hope that techniques similar to those used in [Borcherds 1992] can be applied to attack this conjecture in other cases, and the answer seems to be afﬁrmative. For example, H¨ hn  has proved it for the case when g o is an involution in conjugacy class 2A, using a construction of a vertex algebra with baby monster symmetry, and roughly following the outline of Borcherds’ proof. However, there are obstructions to making this technique work in general, 652 Scott Carnahan since there are many elements of the monster for which we do not know character tables of centralizers or their central extensions. One might ask whether there is a reasonably uniform way of generating holomorphic congruence genus-zero functions from actions of groups on certain Lie algebras.
This paper is an attempt to unify the two questions, and set the stage for a more detailed study of the inﬁnite-dimensional algebraic structures involved. The main result is that modular functions (and more generally, singular q-expansions with algebraic integer coefﬁcients) that are holomorphic on H and satisfy a certain Hecke-theoretic property are holomorphic congruence genus-zero or degenerate in a speciﬁed way. As a special case, we ﬁnd that ﬁnite-order replicable functions with algebraic integer coefﬁcients, as deﬁned in Section 4, satisfy the same property. The algebraic integer condition is sufﬁcient for our purposes, since we intend to use this theorem in the context of representations of ﬁnite groups. Since modular functions with algebraic coefﬁcients that are holomorphic on H and invariant under a congruence group have bounded denominators (see [Shimura 1971, Theorem 3.52] and divide by a suitable power of ), it is reasonable to conjecture that all holomorphic congruence genus-zero functions whose poles have integral residue and constant term have algebraic integer coefﬁcients.
We apply the theory to show that when a group acts on an inﬁnite-dimensional Lie algebra with a special form, the character functions are holomorphic congruence genus-zero. We call these algebras Fricke compatible because they have the form we expect from elements g ∈ for which the function Tg is invariant under a Fricke involution τ → −1/N τ. Later papers in this series will focus on constructing these and other (non-Fricke compatible) Lie algebras, ﬁrst by generatorsand-relations, and then by applying a version of the no-ghost theorem to abelian intertwiner algebras. At the time of writing, this strategy does not seem to yield a complete proof of generalized moonshine, because of some subtleties in computing eigenvalue multiplicities for certain cyclic groups of composite order acting on certain irreducible twisted modules of V. It is possible that some straightforward method of controlling these multiplicities has escaped our attention, but for the near future we plan to rest the full result on some precisely stated assumptions.
Most of the general ideas in the proof are not new, but our speciﬁc implementation bears meaningful differences from the existing literature. In fact, Hecke operators have been related to genus-zero questions since the beginning of moonshine, under the guise of replicability, and the question of relating replication to holomorphic genus-zero modular functions was proposed in the original paper [Conway and Norton 1979]. However, the idea of using an interpretation via moduli of elliptic curves with torsors is relatively recent, and arrives from algebraic topology. Equivariant Hecke operators, or more generally, isogenies of (formal) groups, can be used to describe operations on complex-oriented cohomology theories like elliptic Generalized moonshine, I: Genus-zero functions 653 cohomology, and they were introduced in various forms in [Ando 1995; Baker 1998]. More precise connections to generalized moonshine were established in [Ganter 2009].
Summary. In Section 1, we introduce Hecke operators, ﬁrst as operators on modular functions, and then on general power series. In Section 2, we deﬁne Heckemonicity and prove elementary properties of Hecke-monic functions. In Section 3, we relate Hecke-monicity to equivariant modular equations. Most of this step is a minor modiﬁcation of part of Kozlov’s master’s thesis . In Section 4, we prove a holomorphic congruence genus-zero theorem, and our proof borrows heavily from [Cummins and Gannon 1997]. Most of the arguments require minimal alteration from the form given in that paper, so in those cases we simply indicate which changes need to be made. In Section 5, we focus on the special case of replicable functions, and we show that those with ﬁnite order and algebraic integer coefﬁcients are holomorphic congruence genus-zero or of a speciﬁc degenerate type. In Section 6, we conclude with an application to groups acting on Lie algebras, and show that under certain conditions arising from conformal ﬁeld theory, the characters from the action on homology yield holomorphic congruence genuszero functions.
1. Equivariant Hecke operators
The aims of this section are to introduce a combinatorial formula for equivariant Hecke operators for functions that are not necessarily modular, and to prove some elementary properties. The geometric language of stacks and torsors is only used in this section, and only to justify the claim that these Hecke operators occur naturally.
It is not strictly necessary for understanding the formula, and the reader may skip everything in this section except for the statements of the lemmata without missing substantial constituents of the main theorem.
G Let G be a ﬁnite group, and let Ell denote the analytic stack of elliptic curves equipped with G-torsors (also known as the Hom stack Hom( Ell, BG)). Objects in the ﬁbered category are diagrams e P→E S
of complex analytic spaces satisfying:
P → E is a G-torsor (that is, an analytically locally trivial principal G-bundle).
• E → S is a smooth proper morphism, whose ﬁbers are genus-one curves.
Morphisms are ﬁbered diagrams satisfying the condition that the torsor maps are G-equivariant. This is a smooth Deligne–Mumford stack (in the sense of [Behrend 654 Scott Carnahan
E 1 → S and E 2 → S are smooth proper morphisms, whose geometric ﬁbers • are genus-one curves.
e1 and e2 are sections of the corresponding maps.
where the sum is over all degree-n isogenies to E. When G is trivial, this is the usual weight-zero Hecke operator.
We wish to describe these operators in terms of functions on the complex upper half-plane, and this requires an analytic uniformization of the moduli problem.
Generalized moonshine, I: Genus-zero functions 655 Following the unpublished book [Conrad ≥ 2010] and [Deligne 1971], the upper half-plane classiﬁes pairs (π, ψ), where π : E → S is an elliptic curve, and