Institute for Advanced Study 4 Octoer 017
Disclaimer All statements are to e understood as essentially true (i.e., true once the statement is modified slightly).
Arithmetic invariant theory This talk is aout arithmetic invariant theory. Suppose: G is a linear algeraic group (e.g., GL, GL 6 ) V is a finite-dimensional rational representation of G (e.g. Sym (V ), 3 V 6 ) Definition Arithmetic invariant theory is the study of Parametrizing the orits G(Q) on V (Q) or Parametrizing the orits G(Z) on V (Z) if G, V have structures over the integers. Frequently, the orits tend to e parametrized y interesting arithmetic ojects.
Binary quadratic forms Example G = GL, V = Sym (V ) det 1. GL (Q) acts on Sym (Q ) = symmetric matrices: Make integral: ( a g c ) = det(g) 1 g ( a c ) g t. G(Z) = GL (Z) GL (Q) {( ) } a V Z = : a,, c Z, lattice inside Sym (R ) c Equivalently: V Z = {ax + xy + cy : a,, c Z} f ((x, y)) = ax +xy+cy (g f )((x, y)) = det(g) 1 f ((x, y)g).
Orits on inary quadratic forms over Q If f (x, y) = ax + xy + cy, then ( Disc(f) := a 4ac = 4 det c ) is an invariant of the orit: I.e., Disc(f ) = Disc(g f ) for g GL. Orits over Q (with Disc(s) 0): If s 1, s Sym (Q ) and Disc(s 1 ) = Disc(s ) 0, then g GL (Q) with det(g) 1 gs 1 t g = s. Orits parametrized y Disc, or quadratic étale algeras over Q, GL (Q)s Q( Disc(s)).
Orits on inary quadratic forms over Z For D Z, set V D Z = {f (x, y) : Disc(f ) = D} = {s = ( a c ) } : 4 det(s) = D. D must e a square modulo 4 for this set to e nonempty; i.e., D 0, 1 modulo 4. Orits over Z: 1 quadratic rings S D over Z, GL (Z)s S D := Z[x]/(x Dx + D D 4 ), D = Disc(s); plus elements in Cl(S D ): explicit module for S D.
Bhargava s HCL There is no a priori reason to elieve parametrizing orits G(Z) on V (Z) would yield interesting results However, Bhargava found many examples of such interesting parametrizations Example (Bhargava) 1 GL (Z) SL 3 (Z) acting on Z Sym (Z 3 ) GL (Z) SL 3 (Z) SL 3 (Z) acting on Z Z 3 Z 3 3 GL (Z) SL 6 (Z) acting on Z (Z 6 ). Orits parametrize cuic rings T over Z (e.g. Z[7 1/3 ]), plus finer data For example, in case, orits parametrize (T, I 1, I ) with I 1, I Cl(T ), I 1 I = 1.
Twisted orit prolems Work in arithmetic invariant theory over Z, i.e., parametrizations G(Z) on V (Z), has occurred when the group G is split, e.g. Split G(Z) = GL (Z) SL 3 (Z) SL 3 (Z) V (Z) = Z Z 3 Z 3 = Z M 3 (Z). Prolem What aout twisted, or non-split cases? Twisted Let S e a quadratic ring over Z, SL 3 (Z) SL 3 (Z) = SL 3 (Z Z) SL 3 (S). σ : S S, σ(a + D) = a D, a, Z, M 3 (Z) H 3 (S) = {h M 3 (S) : h = t h σ } GL (Z) SL 3 (S) acts on Z H 3 (S)
Sample theorem Theorem (P.) The orits of GL (Z) SL 3 (S) on Z H 3 (S) parametrize triples (T, I, β) up to equivalence where T is a cuic ring I is a T Z S-fractional ideal β (T Z Q) the norm of I in T Q is principal, generated y β Key idea: Relate orits O in Z H 3 (S) that are ig (technically: in the open orit for (GL GL 3 (S))(Q)) To orits Õ in H 3(S) T that are small (technically: for which the stailizer of the line spanned y an element in the orit is a paraolic sugroup of GL 3 (T S)(Q).) The lifted orits Õ are easier to understand from the point of view of AIT, which is why this helps you
Binary quadratic forms, again Example Using lifted orits Õ for inary quadratic forms Recall: If s 1, s Sym (Q ), and Disc(s 1 ) = Disc(s ) 0, then g GL (Q) with det(g) 1 gs 1 t g = s. Proof: Suppose Set Then s = ( a / / c ), D = Disc(s) = 4 det(s) = 4ac. ( D s = s + 1 1 ) ( = + D a D c det( s) = 0, so s is a rank one matrix in M (Q( D)) Eigenvector property: s ( ) 1 1 s = D s. ).
Proof, continued Proof sketch 1 Suppose s 1, s Sym (Q ) with Disc(s 1 ) = Disc(s ). Easy: Since s 1 and s are rank one, there is g GL (Q D) with g s t 1 g σ = s. D 3 g = g 1 + g, some g 1, g M (Q). 4 Set h = g 1 + g s ( ) 1 M (Q). 1 5 Immediate from Eigenvector property: h s t 1 h = s. 6 Thus hs t 1 h = s and det(h) = 1.
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From the point of view of automorphic forms... Question From the point of view of automorphic forms, why might you e interested in arithmetic invariant theory? Most asic answer: Help you understand the Fourier expansions of automorphic forms. For example, suppose f (Z) = T Sym (Z 3 ) a f (T )e πi tr(tz) is a Siegel modular form of degree three (i.e., for GSp 6 ). Then if m SL 3 (Z), a f (T ) = a f (m T ). Orits GL 3 (Z)\Sym (Z 3 ) Orders in quaternion algeras
Fourier coefficients on GSp 6 If O is a quaternion order, and T Sym (Z 3 ), T O, define a f (O) := a f (T ). Theorem (Evdokimov + ɛ) Suppose f (Z) as aove, weight k level one Hecke Eigenform. Suppose furthermore that B is a quaternion algera, ramified at infinity, and B 0 is a maximal order in B. Then n 1,O B 0 a f (Z + no) n s [B 0 : O] s k+3 = a(b L(π, Spin, s) 0) ζ(s 3k + 6)ζ D. B (s 3k + 8) Here ζ D B (s) is the Riemann zeta function, with the Euler factors at primes dividing the discriminant of B removed, and the sum is over all positive integers n and quaternion orders O contained in B 0.