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Can’t you just feel the moonshine?
Can’t you just feel the moonshine?
Ken Ono (Emory University)
Can’t you just feel the moonshine?
Can’t you just feel the moonshine?
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.
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.
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.
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.
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.
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
Finite Groups
The Monster’s Happy Family
The subgroups and sub-quotients of M are a Happy Family.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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).
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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).
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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).
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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).
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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).
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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).
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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 + . . .
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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 + . . .
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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).
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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).
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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).
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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).
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
Modular curves
Modularity of Elliptic curves
Theorem (Taylor-Wiles, et. al. (1990s))
Every elliptic curve over Q has an X0(N) parametrization.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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?
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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?
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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?
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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?
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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}.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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}.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
Monstrous Moonshine
Second hint of moonshine
John McKay observed that
196884 = 1 + 196883
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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 + . . . .
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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 + . . . .
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
The Jack Daniels Problem
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?
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
The Jack Daniels Problem
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?
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
The Jack Daniels Problem
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?
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
The Jack Daniels Problem
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
The Jack Daniels Problem
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
The Jack Daniels Problem
Ogg’s Problem
Question CIf p ∈ Oggss , then why do we know (a priori) that p | #M?
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
The Jack Daniels Problem
Ogg’s Problem
Question CIf p ∈ Oggss , then why do we know (a priori) that p | #M?
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
The Jack Daniels Problem
Ogg’s Problem
Question CIf p ∈ Oggss , then why do we know (a priori) that p | #M?
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.
Can’t you just feel the moonshine?I. History of Moonshine: Tale of Two Research Programs
The Jack Daniels Problem
Ogg’s Problem
Question CIf p ∈ Oggss , then why do we know (a priori) that p | #M?
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.
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
Distribution of Monstrous Moonshine
Can’t you just feel the moonshine?II. 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?
Can’t you just feel the moonshine?II. 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?
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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?
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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. . .
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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. . .
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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. . .
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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. . .
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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. . .
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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. . .
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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.
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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.
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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.
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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.
Can’t you just feel the moonshine?II. Distribution of Monstrous Moonshine
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?II. Distribution of Monstrous Moonshine
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
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
Umbral Moonshine
Mathieu Moonshine
Theorem (Gannon (2013))
There is an infinite dimensional graded M24-module whoseMcKay-Thompson series are specific mock modular forms.
Can’t you just feel the moonshine?III. New Moonshines
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
).
Can’t you just feel the moonshine?III. New Moonshines
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
).
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
Umbral Moonshine
Coming in 2017...
Can’t you just feel the moonshine?III. New Moonshines
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−).
Can’t you just feel the moonshine?III. New Moonshines
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−).
Can’t you just feel the moonshine?III. New Moonshines
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−).
Can’t you just feel the moonshine?III. New Moonshines
Umbral Moonshine
Ramanujan’s Deathbed Letter
Can’t you just feel the moonshine?III. New Moonshines
Umbral Moonshine
Larger Framework of Moonshine?
RemarkThere are well known connections with even unimodular positivedefinite rank 24 lattices:
M ←→ “Leech lattice”
M24 ←→ A241 lattice.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
Umbral Moonshine
Our results.
Theorem 4 (Duncan, Griffin, O (2015))
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)
Can’t you just feel the moonshine?III. New Moonshines
Umbral Moonshine
Our results.
Theorem 4 (Duncan, Griffin, O (2015))
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)
Can’t you just feel the moonshine?III. New Moonshines
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.”
Can’t you just feel the moonshine?III. New Moonshines
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.”
Can’t you just feel the moonshine?III. New Moonshines
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.”
Can’t you just feel the moonshine?III. New Moonshines
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.”
Can’t you just feel the moonshine?III. New Moonshines
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.”
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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,
Can’t you just feel the moonshine?III. New Moonshines
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,
Can’t you just feel the moonshine?III. New Moonshines
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,
Can’t you just feel the moonshine?III. New Moonshines
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,
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
Pariah Moonshine
The Module Sees Class Groups
Theorem 6 (Duncan, Mertens, O (2017))
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.
Can’t you just feel the moonshine?III. New Moonshines
Pariah Moonshine
The Module Sees Class Groups
Theorem 6 (Duncan, Mertens, O (2017))
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.
Can’t you just feel the moonshine?III. New Moonshines
Pariah Moonshine
The Module Sees Class Groups
Theorem 6 (Duncan, Mertens, O (2017))
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.
Can’t you just feel the moonshine?III. New Moonshines
Pariah Moonshine
The Module Sees Selmer and Tate-Shafarevich-Groups
Theorem 7 (Duncan, Mertens, O (2017))
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
Tr(gp′ |WD) ≡ δp · (H(D)− δpH(p′)(D)) (mod p).
Can’t you just feel the moonshine?III. New Moonshines
Pariah Moonshine
The Module Sees Selmer and Tate-Shafarevich-Groups
Theorem 7 (Duncan, Mertens, O (2017))
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
Tr(gp′ |WD) ≡ δp · (H(D)− δpH(p′)(D)) (mod p).
Can’t you just feel the moonshine?III. New Moonshines
Pariah Moonshine
The Module Sees Selmer and Tate-Shafarevich-Groups
Theorem 7 (Duncan, Mertens, O (2017))
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
Tr(gp′ |WD) ≡ δp · (H(D)− δpH(p′)(D)) (mod p).
Can’t you just feel the moonshine?III. New Moonshines
Pariah Moonshine
The Module Sees Selmer and Tate-Shafarevich-Groups
Theorem 7 (Duncan, Mertens, O (2017))
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
Tr(gp′ |WD) ≡ δp · (H(D)− δpH(p′)(D)) (mod p).
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
Pariah Moonshine
What is Moonshine?
AnswerMoonshine organizes special divisors on products of modular curves!
Evidence
Monstrous identifies Hauptmoduln (i.e. Divisors −∞+ a).
Umbral cuts out individual CM (Heegner) divisors.
Pariah sums vertically and packages (all) Heegner divisors.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?III. New Moonshines
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.
Can’t you just feel the moonshine?Executive Summary
Our Results
Theorem 1 (Duncan, O (2016))
The Jack Daniels problem has been resolved.
Theorem 2 (Duncan, Griffin, O (2015))
The Monstrous Moonshine module is asymptotically regular.In other words, we have that
δ (mi ) =dim(χi )∑194j=1 dim(χj)
=dim(χi )
5844076785304502808013602136.
Can’t you just feel the moonshine?Executive Summary
Our Results
Theorem 1 (Duncan, O (2016))
The Jack Daniels problem has been resolved.
Theorem 2 (Duncan, Griffin, O (2015))
The Monstrous Moonshine module is asymptotically regular.In other words, we have that
δ (mi ) =dim(χi )∑194j=1 dim(χj)
=dim(χi )
5844076785304502808013602136.
Can’t you just feel the moonshine?Executive Summary
New Moonshines
Theorem 4 (Duncan, Griffin, O (2015))
The Umbral Moonshine Conjecture is true.
Theorem 5 (Duncan, Mertens, O (2017))
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.
Can’t you just feel the moonshine?Executive Summary
New Moonshines
Theorem 4 (Duncan, Griffin, O (2015))
The Umbral Moonshine Conjecture is true.
Theorem 5 (Duncan, Mertens, O (2017))
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.
Can’t you just feel the moonshine?Executive Summary
New Moonshines
Theorem 4 (Duncan, Griffin, O (2015))
The Umbral Moonshine Conjecture is true.
Theorem 5 (Duncan, Mertens, O (2017))
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.