**Monstrous Moonshine for integral group rings**, with S. Urano. IMRN doi:10.1093/imrn/rnad028 (ArXiv).- We formulate a conjecture that unifies and generalizes the main claims of Monstrous Moonshine and Modular Moonshine, and prove it in a few new cases. This conjecture came out of our attempts to generalize Modular Moonshine to the situation of elements of composite order. We had found experimentally that the older interpretation of modular moonshine in terms of Tate cohomology did not seem to generalize well, because Tate cohomology is not so well-behaved when the group is not cyclic of prime order. Instead, the property of Tate cohomology in the prime order case that was truly fruitful was the fact that one could produce genus zero modular functions from ring homomorphisms out of representation rings. Thus, we conjecture that for any subring R of the complex numbers, any subgroup G of the Monster, and any ring homomorphism φ from the Green ring Rep
_{R}(G) of stable isomorphism classes of finitely generated R-free RG-modules to the complex numbers, the power series obtained by evaluating φ on the weight spaces of a self-dual R-form of the Monster vertex operator algebra is the q-expansion of a hauptmodul. We apply the integral no-ghost theorem and Hodge theory to the cohomology of the monster Lie algebra to show that any power series obtained from such a ring homomorphism satisfies an infinite collection of recursion relations, giving a weak form of replicability. **Building vertex algebras from parts**Commun. Math. Phys.**373**(2020) 1-43. doi:10.1007/s00220-019-03607-0 (ArXiv version).- This paper gives sufficient conditions for assembly of a vertex algebra from a collection of modules of a smaller vertex algebra parametrized by an abelian group A. We identify an obstruction to locality in H
^{4}(K(A,2),**C**^{×}), and show that when this canonical element vanishes, we obtain a vertex algebra. We reduce this obstruction problem to the question of vanishing of a ± 1-valued quadratic form on A/2A. For simple C_{2}-cofinite Möbius vertex algebras, we reduce the question of existence of simple current extensions to a hypothesis concerning the symmetry of contragradient bilinear forms. For a simple regular vertex operator algebra V this extension is a simple regular vertex operator algebra that is unique up to isomorphism. **A self-dual integral form of the Moonshine Module**SIGMA**15**(2019), 030 (ArXiv version).- The Modular Moonshine conjecture was originally proposed by Ryba in 1994 following some computations involving modular representations of sporadic simple groups. The conjecture asserts the existence of vertex algebras over finite prime fields attached to certain elements of prime order in the Monster, equipped with well-behaved actions of the corresponding centralizers. Work of Borcherds and Ryba in the mid-1990s essentially resolved this conjecture, but their proof made a few assumptions that were unresolved at the time. The final open assumption, which is resolved in this paper, is the existence of a self-dual integral form of the Monster Vertex Operator Algebra that has Monster symmetry. The construction combines the cyclic orbifold theory of van Ekeren, Möller, and Scheithauer with faithfully flat descent for vertex operator algebras over commutative rings. As a corollary, we find that Griess's original 196884-dimensional representation of the monster has a positive-definite self-dual integral form with monster symmetry.
**Characterizing moonshine functions by vertex-operator-algebraic conditions**(with T. Komuro, S. Urano) SIGMA**14**(2018), 114 (ArXiv).- Borcherds's proof of the Monstrous Moonshine conjecture gives as an intermediate result the complete replicability of the 171 monstrous moonshine functions. This was later shown by Cummins and Gannon to imply the functions are genus zero modular functions. In fact, Borcherds's argument yields completely replicable functions for all traces of finite order automorphisms of vertex operator algebras with character J and a self-dual invariant bilinear form. Using some new developments in vertex operator algebra theory, we show that 154 of the 157 non-monstrous completely replicable functions cannot possibly occur as trace functions, under the additional assumption that the vertex operator algebra is regular.
**Monstrous moonshine over Z?**RIMS Proceedings: Research on algebraic combinatorics and representation theory of finite groups and vertex operator algebras 2017/12/11~2017/12/14, ed. T. Abe**2086**(2018) 91-98 (ArXiv).- This is a short introduction to the recent solutions to the generalized moonshine and modular moonshine conjectures, together with a lot of speculative conjectures concerning possible mixed-characteristic unifications of the two phenomena.
**51 constructions of the Moonshine module**Communications in Number Theory and Physics**12**:2 (2018) 305-334. (ArXiv).- Tuite conjectured in 1993 that the cyclic orbifold construction applied to the Leech lattice vertex operator algebra along any fixed-point free automorphism satisfying a "no massless states" condition yields the Monster vertex operator algebra, and furthermore, that the cyclic orbifold correspondence gives a natural bijection between algebraic conjugacy classes of such automorphisms of the Leech lattice and algebraic conjugacy classes of non-Fricke elements in the monster simple group. I show that this conjecture is a straightforward consequence of the recent cyclic orbifold work of van Ekeren, Möller, and Scheithauer. We apply the orbifold correspondence to show that all Monstrous Lie algebras are Borcherds-Kac-Moody Lie algebras, and use the resulting twisted denominator identities to refine the ambiguous constants relating Hauptmoduln in the Generalized Moonshine conjecture to roots of unity.
**Fricke Lie algebras and the genus zero property in Moonshine**Journal of Physics A**50**:40 (2017) (ArXiv).- This paper gives a new proof of the Hecke-monic property (hence the genus zero property) of characters from an action of a finite group on a rank 2 Borcherds-Kac-Moody Lie algebra that "comes from string theory" in a precise sense. The new proof employs a decomposition due to Jurisich, where instead of the usual triangular decomposition using a torus, we employ a copy of gl
_{2}, and find that the remaining parts are free. The freeness eliminates the need for the BGG-type arguments that were used in the original proofs of Monstrous Moonshine and Generalized Monstrous Moonshine. We consider the group G of automorphisms of the Monster Lie algebra that are compatible with a groupoid structure coming from string quantization: G contains the monster, G embeds in the finite dimensional Lie group GL_{334088631214}(**C**), and the prime numbers that are the orders of elements of G are precisely those that divide the order of the monster. In particular, we find that G does not contain any copies of U(1). **Equivariant intertwining operators for twisted modules**RIMS Proceedings: Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics 2014/12/16~2014/12/19, ed. H. Tanaka**1965**(2015) 21-29.- This is an expository account of my work developing a geometric theory describing fusion of twisted modules of a vertex algebra (with Möbius or conformal structure) using orbifold conformal blocks on moduli spaces of log-smooth twisted curves.
**Monstrous Lie Algebras**RIMS Proceedings: Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics 2013/01/07~2013/01/10, ed. Y. Takegahara**1872**(2014) 83-93.- This is an expository account of my work on generalized moonshine up to 2012, focusing on the role of infinite-dimensional Lie algebras, and it is an expansion of my talk at the RIMS conference. I outline a pair of constructions that give, for each element g of the monster, Borcherds-Kac-Moody Lie algebras L
_{g}and m_{g}. Each L_{g}is built from an automorphic infinite product, and has well-behaved simple root structure. Each m_{g}is built from twisted modules of the monster vertex operator algebra, and has a canonical projective action of the centralizer of g. When g is Fricke (a property that holds for about 3/4 of the conjugacy classes), the existence of an isomorphism between these Lie algebras implies all g-twisted generalized moonshine characters are Hauptmoduln, as conjectured by Norton. **Generalized Moonshine II: Borcherds products**Duke Mathematical Journal**161**:5 (2012) 893-950. (arXiv)- This is an expanded version of chapters 4 and 5 in my dissertation. It contains two proofs of a family of infinite product identities that generalize the Koike-Norton-Zagier j-function identity. The first proof uses Hecke operators on spaces of elliptic curves with G-torsors together with analytic continuation, while the second uses the Borcherds-Harvey-Moore singular theta correspondence. I show that the product identities attached to moonshine functions are totally non-negative, and use the identities to construct 171 infinite dimensional Lie algebras with automorphic Weyl denominator functions. Certain structural features of these Lie algebras allow us to reduce Norton's Generalized Moonshine conjecture to some more tractable hypotheses concerning the actions of centralizers on twisted modules of the monster vertex operator algebra.
**Generalized Moonshine I: Genus zero functions**, Algebra and Number Theory**4**:6 (2010) 649-679 (arXiv)- This paper generalizes a 1997 paper of Cummins and Gannon on modular equations and genus zero modular functions, and sets up connections to generalized moonshine. I introduced a notion of Hecke-monicity for functions on moduli spaces of multiplicatively uniformized elliptic curves with G-torsors, and showed that weakly Hecke-monic functions with algebraic integer coefficients are either degenerate in a specified way, or are modular functions invariant under a genus zero congruence group, and generate the function field of the corresponding upper-half-plane quotient. As a corollary, I showed that replicable functions of finite order are weakly Hecke-monic, and are therefore degenerate or Hauptmoduln (this is a converse to a 1979 conjecture of Conway and Norton). The paper ends with an application to moonshine and generalized moonshine: given a group acting on a certain type of graded Lie algebra, one obtains weakly Hecke-monic functions as characters, and they are therefore Hauptmoduln. This eliminates the necessity of some explicit computation at the end of Borcherds's proof of the Monstrous Moonshine Conjecture, and Hoehn's proof of generalized moonshine for the Baby Monster.
**Counting Hopf Galois structures on non-abelian Galois field extensions.**(with L. Childs), J. Algebra**218**:2 (1999) 81-92. (pdf)- This was an undergraduate research project in Hopf Galois theory. Greither and Pareigis introduced a notion of Hopf Galois structure for a separable field extension, generalizing the group ring of a Galois group, and Byott discovered a formula for enumerating such structures that, for Galois extensions, only depended on the structures of groups of the same order. This paper was the first attempt to do explicit enumerations for non-abelian Galois groups. For nonabelian simple groups, we found that there are exactly two such structures, and for symmetric groups, we found that the number of such structures grows very roughly like the square root of the order of the group.

**Generalized Moonshine IV: Monstrous Lie algebras**(ArXiv - new version available Aug. 2016).- This paper completes the proof of the Generalized Moonshine Conjecture. We employ the recent results of van Ekeren, Möller, and Scheithauer to construct abelian intertwining algebras from twisted modules of the monster vertex algebra. We then apply a string-theoretic "add a spacetime torus and quantize" functor to these abelian intertwining algebras to construct infinite dimensional Borcherds-Kac-Moody Lie algebras with canonical actions of large finite groups by automorphisms. We identify these Lie algebras with those constructed in Generalized Moonshine II, and from this we show that all Fricke-twisted twining characters are Hauptmoduln. The rest of the cases then follow from SL
_{2}(**Z**)-compatibility. **Regularity of fixed-point vertex operator subalgebras**, with M. Miyamoto (ArXiv - new version available February 2018).- Suppose we are given a simple regular non-negatively graded vertex operator algebra V with a nonsingular invariant bilinear form. If we are given a finite order automorphism g, then the vertex operator subalgebra V
^{g}of fixed points is also simple, regular, non-negatively graded, with a nonsingular invariant bilinear form. Induction then implies the same properties for fixed point subalgebras for all finite solvable groups of automorphisms. We show along the way that any irreducible V^{g}-module is contained in some irreducible twisted V-module. As a corollary, we find that the category of g-twisted V-modules is semisimple. When V is holomorphic, we obtain an SL_{2}(**Z**)-compatibility result for twisted twining characters, and this resolves one of the claims in the Generalized Moonshine Conjecture. **Why do the symmetries of the monster vertex algebra form a finite simple group?**(ArXiv).- We prove some properties of the automorphism group of the monster vertex algebra, most notably that it is is finite and simple, using recent developments in the theory of vertex operator algebras, but mostly 19th century group theory. In particular, if we take modern vertex algebra theory as a black box, then it is rather easy to prove that the monster group is finite and simple.

**Generalized Moonshine III: Equivariant intertwining operators**, In preparation.- This is a treatment of equivariant conformal blocks on twisted genus zero curves with smooth logarithmic structures, and is an expansion of chapter 2 in my dissertation. Genus zero conformal blocks are related to intertwining operators between twisted modules, and the main result is that irreducible twisted modules of a holomorphic C
_{2}-cofinite vertex operator algebra have the fusion rules you would expect. Bonus: one obtains a canonical element of degree three cohomology of the monster with coefficients in U(1). Thanks to the work of van Ekeren, Möller, and Scheithauer, this paper is no longer necessary for the results of part IV. - Assorted papers building up to Generalized Moonshine III
- The geometric foundations we needed turned out to require a large number of pages, so sections of part III are being split up and reorganized into publishable pieces.