Can't you just feel the moonshine? - Oregon State...

Can’t you just feel the moonshine?


Ken Ono (Emory University)



This talk

I. History of Moonshine: Tale of Two Research Programs

II. Distribution of Monstrous Moonshine (aka Witten’s Question)

III. New Moonshine

Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs

Finite Groups

History of Finite Simple Groups

(1832) Galois finds An (n ≥ 5) and PSL2(Fp) (p ≥ 5).

(1861-1873) Mathieu finds M11,M12,M22,M23 and M24.

(1893) Cole classifies all simple groups with order ≤ 660.

(1890s-1972)Brauer, Burnside, Feit, Frobenius, Dickson, Hall, Thompson,.....

(1972-1983: Gorenstein Program)Aschbacher, Fischer, Glauberman, Gorenstein, Greiss, Tits,.....

(1983) “Classification” announced with much fanfare.


Finite Groups









Finite Groups









Finite Groups









Finite Groups









Finite Groups









Finite Groups

The Monster

Conjecture (Fischer and Griess (1973))

There is a huge simple group M with order

246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71.

Theorem (Griess (1982))

The Monster group M exists.In particular, it is the automorphism gp of an explicit commutativenon-associative algebra on a 196884 dim R-vector space.


Finite Groups

The Monster



246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71.


The Monster group M exists.

In particular, it is the automorphism gp of an explicit commutativenon-associative algebra on a 196884 dim R-vector space.


Finite Groups

The Monster



246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71.


The Monster group M exists.In particular, it is the automorphism gp of an explicit commutativenon-associative algebra on a 196884 dim R-vector space.


Finite Groups

Classification of Finite Simple Groups

Theorem (Classification of Finite Simple Groups (1983))

Finite simple groups live in natural infinite families, apart from 26sporadic groups.


Finite Groups

The Monster’s Happy Family

The subgroups and sub-quotients of M are a Happy Family.


Modular curves

Modular curves

Facts1 SL2(Z) acts on the upper-half complex plane H by

γ =(a bc d

)∈ SL2(Z) ←→ γτ 7→ aτ + b

cτ + d

2 For congruence subgroups Γ ⊂ SL2(Z), number theorists areinterested in the quotients

Y (Γ) := Γ\H.

3 These may be compactified by “adding cusps” to obtaincompact Riemann surfaces, the modular curves X (Γ).


Modular curves

Modular curves


γ =(a bc d

)∈ SL2(Z) ←→ γτ 7→ aτ + b

cτ + d


Y (Γ) := Γ\H.



Modular curves

Modular curves


γ =(a bc d

)∈ SL2(Z) ←→ γτ 7→ aτ + b

cτ + d


Y (Γ) := Γ\H.



Modular curves

Modular curves


γ =(a bc d

)∈ SL2(Z) ←→ γτ 7→ aτ + b

cτ + d


Y (Γ) := Γ\H.



Modular curves

Γ = SL2(Z)

RemarkX0(1) has genus 0, which implies that its field of modularfunctions is C(j(τ)) with a Hauptmodul j(τ).


Modular curves

Γ = SL2(Z)

RemarkX0(1) has genus 0, which implies that its field of modularfunctions is C(j(τ)) with a Hauptmodul j(τ).


Modular curves

Modular functions

DefinitionA meromorphic function f : H 7→ C is a Γ-modular function if forevery γ ∈ Γ we have

f (γτ) = f (τ).

Example (Γ = SL2(Z))

The Hauptmodul is Klein’s j-function (q := e2πiτ )

j(τ)− 744 =∞∑

n=−1c(n)qn

= q−1 + 196884q + 21493760q2 + 864299970q3 + . . .


Modular curves

Modular functions

DefinitionA meromorphic function f : H 7→ C is a Γ-modular function if forevery γ ∈ Γ we have

f (γτ) = f (τ).

Example (Γ = SL2(Z))

The Hauptmodul is Klein’s j-function (q := e2πiτ )

j(τ)− 744 =∞∑

n=−1c(n)qn

= q−1 + 196884q + 21493760q2 + 864299970q3 + . . .


Modular curves

Elliptic functions and curves

Fundamental Facts (Classical)1 If Λ is a rank 2 lattice in C, then C/Λ is an elliptic curve.

2 If Λτ := Z⊕ Zτ , then j(τ) is an invariant of the elliptic curve.

Remarks1 i.e. X0(1) encode isomorphism classes of elliptic curves.

2 Congruence modular curves X (Γ) encodes isomorphism classesof elliptic curves with extra structure (cf. Mazur’s Thm).


Modular curves







Modular curves







Modular curves







Modular curves

Modularity of Elliptic curves

Theorem (Taylor-Wiles, et. al. (1990s))

Every elliptic curve over Q has an X0(N) parametrization.


Modular curves

Hyperelliptic Modular Curves

Theorem (Ogg (1974))

X0(N) est hyperelliptique pour exactement dix-neuf valuers de N.

Problem

An elliptic curve E over an algebraic closure Fp is supersingular ifit has no p-torsion.

For which p is it true that every supersingular E lives over Fp?


Modular curves




Problem




Modular curves




Problem




Modular curves




Problem




Modular curves

First Hint of Moonshine

Corollary (Ogg (1974))

Toutes les valuers supersingulières de j sont Fp si, et seulement si

p ∈ Oggss := {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}.


Modular curves

First Hint of Moonshine

Corollary (Ogg (1974))

Toutes les valuers supersingulières de j sont Fp si, et seulement si

p ∈ Oggss := {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}.


Monstrous Moonshine

Second hint of moonshine

John McKay observed that

196884 = 1 + 196883


Monstrous Moonshine

John Thompson’s generalizations

Thompson further observed:

196884 = 1 + 19688321493760 = 1 + 196883 + 21296876

864299970 = 1 + 1 + 196883 + 196883 + 21296876 + 842609326864299970︸︷︷︸

Coefficients of j(τ)

1 + 1 + 196883 + 196883 + 21296876 + 842609326︸︷︷︸Dimensions of irreducible representations of the Monster M

RemarkKlein’s j-function

j(τ)− 744 =∞∑

n=−1c(n)qn

= q−1 + 196884q + 21493760q2 + 864299970q3 + . . . .


Monstrous Moonshine

John Thompson’s generalizations

Thompson further observed:

196884 = 1 + 19688321493760 = 1 + 196883 + 21296876

864299970 = 1 + 1 + 196883 + 196883 + 21296876 + 842609326864299970︸︷︷︸

Coefficients of j(τ)

1 + 1 + 196883 + 196883 + 21296876 + 842609326︸︷︷︸Dimensions of irreducible representations of the Monster M

RemarkKlein’s j-function

j(τ)− 744 =∞∑

n=−1c(n)qn

= q−1 + 196884q + 21493760q2 + 864299970q3 + . . . .


Monstrous Moonshine

The Monster characters

The character table for M (ordered by size) gives dimensions:

χ1(e) = 1χ2(e) = 196883χ3(e) = 21296876χ4(e) = 842609326

...χ194(e) = 258823477531055064045234375.


Monstrous Moonshine

McKay and Thompson

Conjecture (Thompson)

There is an infinite-dimensional graded module V \ =⊕∞

n=−1 V\n

for which dim(V \n) = c(n).

Definition (Thompson)

Assuming the conjecture, if g ∈M, then define theMcKay–Thompson series

Tg (τ) :=∞∑

n=−1Tr(g |V \

n)qn.


Monstrous Moonshine

McKay and Thompson

Conjecture (Thompson)

There is an infinite-dimensional graded module V \ =⊕∞

n=−1 V\n

for which dim(V \n) = c(n).

Definition (Thompson)

Assuming the conjecture, if g ∈M, then define theMcKay–Thompson series

Tg (τ) :=∞∑

n=−1Tr(g |V \

n)qn.


Monstrous Moonshine

Conway and Norton

Conjecture (Monstrous Moonshine, 1979)

For each g ∈M there is an explicit genus 0 congruence subgroupΓg ⊂ SL2(R) for which Tg (τ) is the Hauptmodul.


Monstrous Moonshine

Borcherds’ work

Theorem (Frenkel–Lepowsky–Meurman (1980s))

If it exists, then the moonshine module V \ =⊕∞

n=−1 V\n is a

specific vertex operator algebra whose automorphism group is M.

Theorem (Borcherds (1998 Fields Medal))

The Monstrous Moonshine Conjecture is true.


Monstrous Moonshine

Borcherds’ work

Theorem (Frenkel–Lepowsky–Meurman (1980s))

If it exists, then the moonshine module V \ =⊕∞

n=−1 V\n is a

specific vertex operator algebra whose automorphism group is M.

Theorem (Borcherds (1998 Fields Medal))

The Monstrous Moonshine Conjecture is true.


The Jack Daniels Problem


Question A

Do order p elements in M know the Fp supersingular j-invariants?

Question BIf p 6∈ Oggss , then why is it true that p - #M?

Question CIf p ∈ Oggss , then why do we know (a priori) that p | #M?




Question A







Question A






Questions A and B

Question A is answered by combining the “group lawinterpreted in moonshine” with work of Dwork.

Question B is answered by the proof of Monstrous Moonshine.



Questions A and B

Question A is answered by combining the “group lawinterpreted in moonshine” with work of Dwork.

Question B is answered by the proof of Monstrous Moonshine.



Ogg’s Problem


Theorem 1 (Duncan-O (2016))

The following are true:

1 If p ∈ Oggss , then the Hauptmodul hp(τ) is spanned p-adicallyby elementary theta functions.

2 These spaces mod p are spanned by elementary thetafunctions iff p | #M.



Ogg’s Problem








Ogg’s Problem








Ogg’s Problem






Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine

Distribution of Monstrous Moonshine


Witten’s Conjecture (2007)

Conjecture (Witten, Li-Song-Strominger)

The vertex operator algebra V \ is dual to a 3d quantum gravitytheory. Thus, there are 194 “black hole states".

Question (Witten)

How are these different kinds of black hole states distributed?


Witten’s Conjecture (2007)

Conjecture (Witten, Li-Song-Strominger)

The vertex operator algebra V \ is dual to a 3d quantum gravitytheory. Thus, there are 194 “black hole states".

Question (Witten)

How are these different kinds of black hole states distributed?


Open Problem

QuestionConsider the moonshine expressions

196884 = 1 + 19688321493760 = 1 + 196883 + 21296876

864299970 = 1 + 1 + 196883 + 196883 + 21296876 + 842609326...

c(n) =194∑i=1

mi (n)χi (e)

How many ‘1’s, ‘196883’s, etc. show up in these equations?


Some Proportions

n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))

1 1/2 1/2 · · · 0

......

......

...40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .


Some Proportions

n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))

1 1/2 1/2 · · · 0...

......

......

40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .

60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .


Some Proportions

n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))

1 1/2 1/2 · · · 0...

......

......

40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .

80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .


Some Proportions

n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))

1 1/2 1/2 · · · 0...

......

......

40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .

160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .


Some Proportions

n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))

1 1/2 1/2 · · · 0...

......

......

40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .

220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .


Some Proportions

n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))

1 1/2 1/2 · · · 0...

......

......

40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .


Distribution of Moonshine

Theorem 2 (Duncan, Griffin, O (2015))

If 1 ≤ i ≤ 194, then as n→ +∞ we have

mi (n) ∼ dim(χi )√2|n|3/4|M|

· e4π√|n|

Corollary (Duncan, Griffin, O)

The Moonshine module is asymptotically regular.In other words, we have

δ (mi ) := limn→+∞

mi (n)∑194i=1mi (n)

=dim(χi )

5844076785304502808013602136.


Distribution of Moonshine


If 1 ≤ i ≤ 194, then as n→ +∞ we have

mi (n) ∼ dim(χi )√2|n|3/4|M|

· e4π√|n|

Corollary (Duncan, Griffin, O)

The Moonshine module is asymptotically regular.In other words, we have

δ (mi ) := limn→+∞

mi (n)∑194i=1mi (n)

=dim(χi )

5844076785304502808013602136.


Proof

Orthogonality

FactIf G is a group and g , h ∈ G , then

∑χi

χi (g)χi (h) =

{|CG (g)| If g and h are conjugate0 otherwise,

where CG (g) is the centralizer of g in G .

From this we can work out that

Tχi (τ) =∞∑

n=−1mi (n)qn.


Proof

Orthogonality

FactIf G is a group and g , h ∈ G , then

∑χi

χi (g)χi (h) =

{|CG (g)| If g and h are conjugate0 otherwise,

where CG (g) is the centralizer of g in G .

From this we can work out that

Tχi (τ) =∞∑

n=−1mi (n)qn.


Proof

Exact Formulas Imply Distributions

Theorem 3 (Duncan, Griffin,O (2015))

We have (ugly) exact formulas for the coefficients of the Tχi (τ).

Sketch of ProofReduces to a “Kloostermania” problem in the spectral theory ofautomorphic forms.


Proof

Exact Formulas Imply Distributions

Theorem 3 (Duncan, Griffin,O (2015))

We have (ugly) exact formulas for the coefficients of the Tχi (τ).

Sketch of ProofReduces to a “Kloostermania” problem in the spectral theory ofautomorphic forms.

Can’t you just feel the moonshine?III. New Moonshines

Umbral Moonshine

Umbral (shadow) Moonshine


Umbral Moonshine

An unexpected observation

Observation (Eguchi, Ooguri, Tachikawa (2010))

Using the K3 surface elliptic genus, there is a mock modular form

H(τ) = q−18(−2 + 45q + 231q2 + 770q3 + 2277q4 + 5796q5 + ...

).

The degrees of the irreducible repn’s of the Mathieu group M24 are:

1, 23,45, 231, 252, 253, 483, 770, 990, 1035,1265, 1771, 2024, 2277, 3312, 3520, 5313, 5544, 5796, 10395.


Umbral Moonshine

An unexpected observation

Observation (Eguchi, Ooguri, Tachikawa (2010))

Using the K3 surface elliptic genus, there is a mock modular form

H(τ) = q−18(−2 + 45q + 231q2 + 770q3 + 2277q4 + 5796q5 + ...

).

The degrees of the irreducible repn’s of the Mathieu group M24 are:

1, 23,45, 231, 252, 253, 483, 770, 990, 1035,1265, 1771, 2024, 2277, 3312, 3520, 5313, 5544, 5796, 10395.


Umbral Moonshine

Mathieu Moonshine

Theorem (Gannon (2013))

There is an infinite dimensional graded M24-module whoseMcKay-Thompson series are specific mock modular forms.


Umbral Moonshine

What are mock modular forms?

Notation. Throughout, let

τ = x + iy ∈ H with x , y ∈ R.

Hyperbolic Laplacian.

∆k := −y2(∂2

∂x2+

∂2

∂y2

)+ iky

(∂

∂x+ i

∂

∂y

).


Umbral Moonshine

What are mock modular forms?

Notation. Throughout, let

τ = x + iy ∈ H with x , y ∈ R.

Hyperbolic Laplacian.

∆k := −y2(∂2

∂x2+

∂2

∂y2

)+ iky

(∂

∂x+ i

∂

∂y

).


Umbral Moonshine

Harmonic Maass forms

DefinitionA harmonic Maass form of weight k on a subgroupΓ ⊂ SL2(Z) is any smooth function M : H→ C satisfying:

1 For all A =(a bc d

)∈ Γ and τ ∈ H, we have

M

(aτ + b

cτ + d

)= (cτ + d)k M(τ).

2 We have that ∆kM = 0.

RemarkModular forms are a density 0 subset of harmonic Maass forms.


Umbral Moonshine





M

(aτ + b

cτ + d

)= (cτ + d)k M(τ).




Umbral Moonshine





M

(aτ + b

cτ + d

)= (cτ + d)k M(τ).




Umbral Moonshine





M

(aτ + b

cτ + d

)= (cτ + d)k M(τ).




Umbral Moonshine

Coming in 2017...


Umbral Moonshine

Fourier expansions (q := e2πiτ )

Fundamental LemmaIf M ∈ H2−k and Γ(a, x) is the incomplete Γ-function, then

M(τ) =∑

n�−∞c+(n)qn +

∑n<0

c−(n)Γ(k − 1, 4π|n|y)qn.

l lHolomorphic part M+ Nonholomorphic part M−

Remark

We call M+ a mock modular form if M− 6= 0.

If ξ2−k := 2iy2−k ∂∂τ , then the shadow of M is ξ2−k(M−).


Umbral Moonshine



M(τ) =∑

n�−∞c+(n)qn +

∑n<0

c−(n)Γ(k − 1, 4π|n|y)qn.


Remark




Umbral Moonshine



M(τ) =∑

n�−∞c+(n)qn +

∑n<0

c−(n)Γ(k − 1, 4π|n|y)qn.


Remark




Umbral Moonshine

Ramanujan’s Deathbed Letter


Umbral Moonshine

Larger Framework of Moonshine?

RemarkThere are well known connections with even unimodular positivedefinite rank 24 lattices:

M ←→ “Leech lattice”

M24 ←→ A241 lattice.


Umbral Moonshine

Umbral Moonshine Conjecture

Conjecture (Cheng, Duncan, Harvey (2013))

Let LX be an even unimodular positive-def rank 24 lattice, and let :X be its root system and W X its Weyl group.

The umbral group GX := Aut(LX )/W X .Then there is moonshine for GX .

RemarkApart from the Leech case, the McKay-Thompson series are weight1/2 mock modular forms with theta function shadows.


Umbral Moonshine




The umbral group GX := Aut(LX )/W X .

Then there is moonshine for GX .



Umbral Moonshine







Umbral Moonshine







Umbral Moonshine

Our results.


The Umbral Moonshine Conjecture is true.

ExampleFor M12 the MT series include Ramanujan’s deathbed functions:

f (q) = 1 +∞∑n=1

qn2

(1 + q)2(1 + q2)2 · · · (1 + qn)2,

φ(q) = 1 +∞∑n=1

qn2

(1 + q2)(1 + q4) · · · (1 + q2n),

χ(q) = 1 +∞∑n=1

qn2

(1− q + q2)(1− q2 + q4) · · · (1− qn + q2n)


Umbral Moonshine

Our results.



ExampleFor M12 the MT series include Ramanujan’s deathbed functions:

f (q) = 1 +∞∑n=1

qn2

(1 + q)2(1 + q2)2 · · · (1 + qn)2,

φ(q) = 1 +∞∑n=1

qn2

(1 + q2)(1 + q4) · · · (1 + q2n),

χ(q) = 1 +∞∑n=1

qn2

(1− q + q2)(1− q2 + q4) · · · (1− qn + q2n)


Pariah Moonshine

Can’t we feel the Moonshine?

QuestionMoonshine is now “largely” understood in the following settings:

The Monster M and its Happy Family.

Umbral Groups (automorphisms of nice rank 24 lattices).

Is there more out there?

Question (Conway-Norton, 1979)

“We ask whether the sporadic simple groups that may not beinvolved with M have moonshine properties.”


Pariah Moonshine









Pariah Moonshine









Pariah Moonshine









Pariah Moonshine









Pariah Moonshine

Pariah Groups

Theorem (O’Nan (1976))

There is a sporadic finite group ON with order29 · 34 · 5 · 73 · 11 · 19 · 31 outside the Happy Family.


Pariah Moonshine

Pariah Groups

Theorem (O’Nan (1976))

There is a sporadic finite group ON with order29 · 34 · 5 · 73 · 11 · 19 · 31 outside the Happy Family.


Pariah Moonshine

O’Nan Moonshine

Theorem 5 (Duncan, Mertens, O (2017))

There is an infinite dimensional graded ON moonshine module.

Its MT series are explicit weight 3/2 mock modular forms.

Remarks (Graded Dimensions)1 If we let W := ⊕nWn, then

dimWn = “traces of CM disc −n values of J2”.

2 We have

dimW163 = deπ√163e2 + deπ

√163e − 393768,

in terms of Ramanujan’s integerdeπ√163e = d262537412640768743.99999999999925...e,


Pariah Moonshine

O’Nan Moonshine


There is an infinite dimensional graded ON moonshine module.Its MT series are explicit weight 3/2 mock modular forms.



2 We have


√163e − 393768,



Pariah Moonshine

O’Nan Moonshine





2 We have


√163e − 393768,



Pariah Moonshine

O’Nan Moonshine





2 We have


√163e − 393768,



Pariah Moonshine

Extraordinary MT Series

Are linear combinations of generating functions for:

Gauss’ class numbers.

Singular moduli on X0(d) (generate ray class fields).

Central L-values of twists of elliptic curvesX0(11),X0(14),X0(15),X0(19).

Central L-values of twists of the X0(31) abelian surface.


Pariah Moonshine








Pariah Moonshine








Pariah Moonshine








Pariah Moonshine








Pariah Moonshine

The Module Sees Class Groups


Suppose that −D < 0 is a fund disc. Then the following are true:

1 If −D < −8 is even, then

dimWD ≡ −24H(D) (mod 24).

2 If p ∈ {3, 5, 7},(−Dp

)= −1 then

dimWD ≡

{−24H(D) (mod 32) if p = 3,−24H(D) (mod p) if p = 5, 7.


Pariah Moonshine



Suppose that −D < 0 is a fund disc. Then the following are true:1 If −D < −8 is even, then


2 If p ∈ {3, 5, 7},(−Dp

)= −1 then

dimWD ≡



Pariah Moonshine



Suppose that −D < 0 is a fund disc. Then the following are true:1 If −D < −8 is even, then


2 If p ∈ {3, 5, 7},(−Dp

)= −1 then

dimWD ≡



Pariah Moonshine

The Module Sees Selmer and Tate-Shafarevich-Groups


Suppose that E14 = X0(14) and E15 = X0(15).

If pp′ = 14 or 15 and(−Dp

)= −1 and

(−Dp′

)= 1, then we have:

1 We have that Sel(EN(−D))[p] 6= {0} if and only if

Tr(gp′ |WD) ≡ δp · (H(D)− δpH(p′)(D)) (mod p).

2 Suppose further that L(EN(−D), 1) 6= 0. Thenp | #X(EN(−D)) if and only if



Pariah Moonshine





)= −1 and

(−Dp′

)= 1, then we have:






Pariah Moonshine





)= −1 and

(−Dp′

)= 1, then we have:






Pariah Moonshine





)= −1 and

(−Dp′

)= 1, then we have:






Pariah Moonshine

What is Moonshine?

AnswerMoonshine organizes special divisors on products of modular curves!

EvidenceMonstrous identifies Hauptmoduln (i.e. Divisors −∞+ a).

Umbral cuts out individual CM (Heegner) divisors.

Pariah sums vertically and packages (all) Heegner divisors.


Pariah Moonshine

What is Moonshine?






Pariah Moonshine

What is Moonshine?






Pariah Moonshine

What is Moonshine?


Evidence

Monstrous identifies Hauptmoduln (i.e. Divisors −∞+ a).




Pariah Moonshine

What is Moonshine?






Pariah Moonshine

What is Moonshine?






Pariah Moonshine

What is Moonshine?





Can’t you just feel the moonshine?Executive Summary

Our Results

Theorem 1 (Duncan, O (2016))

The Jack Daniels problem has been resolved.


The Monstrous Moonshine module is asymptotically regular.In other words, we have that

δ (mi ) =dim(χi )∑194j=1 dim(χj)

=dim(χi )

5844076785304502808013602136.


Our Results

Theorem 1 (Duncan, O (2016))

The Jack Daniels problem has been resolved.


The Monstrous Moonshine module is asymptotically regular.In other words, we have that

δ (mi ) =dim(χi )∑194j=1 dim(χj)

=dim(χi )

5844076785304502808013602136.


New Moonshines




Moonshine exists for the O’Nan pariah sporadic group.

Theorems 6 and 7 (Duncan, Mertens, O (2017))

O’Nan Moonshine informs number theory of class groups, Selmergroups and Tate-Shafarevich groups of elliptic curves.


New Moonshines








New Moonshines







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