Pariah moonshine

Our main results (Theorems 2, 3 and 4) reveal a role for the O’Nan pariah group as a provider of hidden symmetry to quadratic forms and elliptic curves. They also represent the intersection of moonshine theory with the Langlands program, which, since its inception in the 1960s, has become a driving force for research in number theory, geometry and mathematical physics.

Hidden symmetry

The Langlands program predicts an expansive system of hidden symmetry in algebraic statistics^{19, 20}. For an example of this consider the Riemann zeta function

This product converges for s = σ + it a complex number with σ > 1, but at s = 1 equates to the harmonic series, and therefore diverges there. This is one proof that there are infinitely many primes. Setting \(\Gamma (s): = {\int}_0^\infty {{v^{s - 1}}{e^{ - v}}{\rm{d}}v}\) and \(\theta (z): = \mathop {\sum}\nolimits_{n = - \infty }^{\infty} {{e^{\pi iz{n^2}}}}\) we may write

Riemann used (2) and the identity \({v^{ - \frac{1}{2}}}\theta \left( {i{v^{ - 1}}} \right) = \theta (iv)\) to show²¹ that ζ(s) can be defined for all complex numbers except s = 1, in such a way that the completed zeta function \(\xi (s): = {\pi ^{ - \frac{s}{2}}}\Gamma \left( {\frac{s}{2}} \right)\zeta (s)\) is invariant under the two-fold symmetry that swaps s with 1 − s.

For the Langlands program ζ(s) is the first in a family arising from Diophantine analysis, which is the classical art²² of finding rational solutions to polynomials with integer coefficients. Linear equations are controlled by ζ(s), and the next examples relate quadratic forms Q(x, y): = Ax ² + Bxy + Cy ² to the Dirichlet L-series

Here A, B and C are integers, D: = B ² − 4AC is the discriminant of Q, and χ _D (p) is a certain function depending on D. If D = 1 then L _D (s) = ζ(s). Dirichlet L-series admit completions which have the same two-fold symmetry as ζ(s). The as yet unresolved generalised Riemann hypothesis²³ predicts that if L _D (s) = 0 for s = σ + it with 0 < σ < 1 then \(\sigma = \frac{1}{2}\).

Just as for ζ(s), the behaviour of L _D (s) near s = 1 is important. Call a discriminant D fundamental if it cannot be written as d ² D′ where d is an integer greater than 1 and D′ is the discriminant of another quadratic form. If we focus on the values Q(x, y) for x and y integers then it is the same to consider Q′(x, y): = Q(ax + by,cx + dy), for any integers a, b, c and d, so long as ad − bc = 1. Such a modular transformation preserves discriminants, so we may consider the forms with given discriminant D, and count them modulo modular transformations. For D fundamental this is the class number h(D). For example, h(−7) = 1 because Q(x, y): = x ² + xy + 2y ² has discriminant −7, and if Q′(x, y) is another such form then Q′(x, y) = Q(ax + by, cx + dy) for some integers a, b, c and d with ad − bc = 1. Gauss conjectured²⁴ that h(D) takes any given value only finitely many times for negative D. Dirichlet proved²⁵

for fundamental D < −4. Using this, Siegel solved the Gauss conjecture in a weak sense in 1944 by showing²⁶ that for any \(\epsilon\) > 0 there is a constant c such that \(h(D) >c{\left| D \right|^{\frac{1}{2} - \epsilon }}\) for D < 0. But the method is not effective because the constant c it produces depends upon the validity of the generalised Riemann hypothesis. The best known effective lower bound^{27, 28} takes the form \(h(D) >c{\left( {{\rm{log}}\left| D \right|} \right)^{1 - \epsilon }}\) for D < 0.

Elliptic curves

It is a familiar fact that 3² + 4² = 5². Fermat’s last theorem claims that if x, y and z are integers such that x ⁿ + y ⁿ = z ⁿ with n > 2 then at least one of them is zero. Wiles famously proved this²⁹ by establishing hidden symmetry in the Diophantine analysis of

If 4A ³ + 27B ² ≠ 0 then this cubic equation defines an elliptic curve E, an integer N called the conductor of E, and a Hasse–Weil L-series

Here a _p is an integer depending on E and p, and \(\varepsilon\)(p) is 0 or 1 according as p divides N or not. Modularity for E is the existence³⁰ of a modular form f _E (z), (complex) differentiable for z = u + iv with v > 0, such that \({(Ncz + d)^{ - 2}}{f_E}\left( {\frac{{az + b}}{{Ncz + d}}} \right) = {f_E}(z)\) for any integers a, b, c and d with ad − Nbc = 1, and

For L _E (s) the behaviour near s = 1 is again important, but not yet understood, and a focus of current research. The Birch–Swinnerton–Dyer (BSD) conjecture predicts^{31, 32} that it is a key that unlocks the rational solutions to the equation defining E. Specifically, if r _E is the rank of E, representing the number of independent infinite families of rational solutions, then \({\rm{li}}{{\rm{m}}_{s \to 1}}{(s - 1)^{ - {r_E}}}{L_E}(s)\) should be finite and non-zero. In particular, L _E (1) should vanish if and only if there are infinitely many rational solutions. The Tate–Shafarevich group Ш (E) measures the extent to which computations with E can be carried out modulo primes. A stronger form of the BSD conjecture asserts that

for certain computable constants Ω _E and c _E , where |Ш(E)| is the cardinality of Ш(E).

The BSD conjecture is known only in its weak form^{28, 33} for r _E  = 0 and r _E  = 1. The strong form gives us complete control once we know Ш(E) and r _E . For each prime p there is a Selmer group Sel _p (E), which ties together the p-fold symmetries in III(E) with the infinite families of rational solutions to E, so Selmer groups and Tate–Shafarevich groups are of primary importance.

The moonshine that we establish for the O’Nan group (see Theorem 1) enables us to prove new constraints on class numbers h(D) (see Theorem 2), and empowers us to relate certain Selmer groups Sel _p (E) and Tate–Shafarevich groups Ш(E) to effectively computable series a _g (D) (see Theorems 3 and 4).

Moonshine

The elliptic modular invariant J is the unique (complex) differentiable function of z = u + iv for v > 0 that satisfies \(J\left( {\frac{{az + b}}{{cz + d}}} \right) = J(z)\) when a, b, c and d are integers such that ad − bc = 1, and \({\lim _{v \to \infty }}\left( {J(iv) - {e^{2\pi v}}} \right) = 0\). The connection between the monster and J is that the dimension of \(V_n^\natural\) is the coefficient of e ^2nπiz in the Fourier expansion

Recursion relations³⁴ compute these Fourier coefficients effectively.

Let F(z) be the unique (complex) differentiable function of z = u + iv for v > 0 such that

1. 1.

   \({(4cz + d)^{ - 2}}\tilde F\left( {\frac{{az + b}}{{4cz + d}}} \right) = \tilde F(z)\) when \(\tilde F(z)\!\!: = F(z)\theta (2z)\) and a, b, c and d are integers satisfying ad − 4bc = 1,

2. 2.

   \({F^ + }\left( {z + \frac{1}{4}} \right) = {F^ + }(z)\) and \({F^ - }\left( {z + {\textstyle{1 \over 4}}} \right) = - i{F^ - }(z)\) for \({F^ \pm }(z)\!\!: = \frac{1}{2}\left( {F(z) \pm F\left( {z + \frac{1}{2}} \right)} \right)\),

3. 3.

   \({\rm{li}}{{\rm{m}}_{v \to \infty }}\left( {F(iv) + {e^{8\pi v}}} \right)\) is finite.

Under these conditions F is related to J by \(F\left( {z + {\textstyle{1 \over 2}}} \right)\theta (8z) + {F^ - }(z)\theta (2z) = \frac{1}{{8\pi i}}\frac{{\rm{d}}}{{{\rm{d}}z}}J(4z)\), and recursion relations³⁵ compute the Fourier coefficients of F effectively.

Theorem 1

There exists a sequence of spaces W _D for discriminants D < 0 that admit the O’Nan group as symmetry, the dimension of W _D being the coefficient of e ^2|D|πiz in the Fourier expansion

We sketch the proof of Theorem 1. Let a(D) denote the coefficient of e ^2|D|πiz in the Fourier expansion of F, so that we have \(F(z)\!\!: = - {e^{ - 8\pi iz}} + 2 + {\sum_{D\, < \, 0}} {a(D){e^{2|D|\pi iz}}}\). Theorem 1 amounts to the association of a(D) × a(D) matrices to symmetries in the O’Nan group, for each D < 0. Taking a _g (D) to be the trace (i.e., sum of diagonal entries) of the matrix corresponding to a symmetry g we obtain a new function \(F_{g}(z)\!\!: = - {e^{ - 8\pi iz}} + 2 + {\sum_{D\, < \, 0}} {a_{g}(D){e^{2|D|\pi iz}}}\). The O’Nan group has a character table¹⁷, which encodes the possible traces a _g (D) that can arise, and thereby equips the F _g with special properties. Conversely, to specify functions F _g that are compatible with the character table is enough to prove that corresponding matrices exist, and enough to confirm moonshine for the O’Nan group. This is how we prove Theorem 1. We first realise the F _g as modular forms satisfying conditions analogous to those defining F. Then we use those conditions to prove growth estimates and congruences for the a _g (D). For example, if g is a seven-fold symmetry then it develops that a(D) is congruent to a _g (D) modulo 343, for every D < 0. Finally, we use the growth estimates and congruences to prove compatibility with the character table.

Define J _O′N (z): = J(z)² − J(z) − 393768. For Q(x, y) = Ax ² + Bxy + Cy ² a quadratic form with D = B ² − 4AC < 0 set \({z_Q}: = \frac{{ - B + i\sqrt {\left| D \right|} }}{{2A}}\). Set w _Q : = 6 if Q is equivalent (i.e. related by a modular transformation) to A′x ² + A′xy + A′y ² for some A′, set w _Q : = 4 if Q is equivalent to A′x ² + A′y ², and set w _Q : = 2 otherwise. Then for D < 0 we have³⁵

where the sum is over equivalence classes of quadratic forms of discriminant D. If D is fundamental then h(D) is the number of summands in (11). For example, \({\rm{dim}}\,{W_{ - 7}} = \frac{1}{2}{J_{O'N}}\left( {\frac{{ - 1 + i\sqrt 7 }}{2}} \right) = 8288256\) because every quadratic form with discriminant −7 is equivalent to x ² + xy + 2y ². So the identity (11) hints at a connection to class numbers. Say that D is not a square modulo p if there is no integer n such that D is congruent to n ² modulo p. By establishing analogues of (11) for a _g (D), for g a two-fold, three-fold, five-fold or seven-fold symmetry in the O’Nan group, we obtain the following results.

Theorem 2

Let D < 0 be a fundamental discriminant. If D < −8 is even then −24h(D) is congruent to a(D) modulo 16. If D is not a square modulo 3 then −24h(D) is congruent to a(D) modulo 9. If p is 5 or 7, and if D is not a square modulo p then −24h(D) is congruent to a(D) modulo p.

For D < 0 a fundamental discriminant define elliptic curves E ₁₄(D) and E ₁₅(D) as follows.

By proving analogues of (11) for higher order symmetries in the O’Nan group we obtain new relationships between class numbers and the rational solutions to these equations.

Theorem 3

Let D  < 0 be a fundamental discriminant. Suppose that D is congruent to 1 modulo 2 and is not a square modulo 7, and let g be a two-fold symmetry in the O’Nan group. Then Sel₇ (E ₁₄ (D)) is non-trivial if and only if a _g (D) is congruent to 3h(D) − 9h ⁽²⁾ (D) modulo 7. Also, if \({L_{{E_{14}}(D)}}(1) \ne\, 0\) then the Birch–Swinnerton-Dyer conjecture is true for E ₁₄ (D), and |Ш(E ₁₄ (D))| is congruent to 0 modulo 7 if and only if a _g (D) is congruent to 3h(D) − 9h ⁽²⁾ (D) modulo 7.

Theorem 4

Let D  < 0 be a fundamental discriminant. Suppose that D is congruent to 1 modulo 3 and is not a square modulo 5, and let g be a three-fold symmetry in the O’Nan group. Then Sel₅ (E ₁₅ (D)) is non-trivial if and only if a _g (D) is congruent to 2h(D) − 4h ⁽³⁾ (D) modulo 5. Also, If \({L_{{E_{15}}(D)}}(1) \ne\, 0\) then the Birch–Swinnerton-Dyer conjecture is true for E ₁₅ (D), and Ш(E ₁₅ (D))| is congruent to 0 modulo 5 if and only if a _g (D) is congruent to 2h(D) − 4h ⁽³⁾ (D) modulo 5.

Note that a _g (D) is the same for all two-fold symmetries g, and this is also true for three-fold symmetries. Similar to the a(D), the a _g (D) are computed effectively by recursion relations when g is a two-fold or three-fold symmetry. The h ^(p)(D) are analogues of the class numbers h(D) that are defined by restricting to modular transformations Q(ax + by, pcx + dy) where ad − pbc = 1.

Recent work of Skinner³⁶, following earlier work³⁷ of Skinner–Urban, shows that (8) holds modulo certain primes p, when certain conditions on the underlying elliptic curve E are satisfied. We prove Theorems 3 and 4 by verifying these conditions for p = 2 and E = E ₁₄(D), and for p = 3 and E = E ₁₅(D), when D < 0 is fundamental.

Define a further two families of elliptic curves as follows.

If we assume (8) then the analogues of (11) for eleven-fold and nineteen-fold symmetries in the O’Nan group lead to the statement that if p = 11 or p = 19, and D < 0 is a fundamental discriminant that is not a square modulo p, then Sel _p (E _p (D)) is non-trivial if and only if a(D) is congruent to −24h(D) modulo p.