Computing zeta functions of groups and rings Tobias Rossmann (joint - PowerPoint PPT Presentation

Computing zeta functions of groups and rings Tobias Rossmann (joint with Christopher Voll) Universität Bielefeld Braunschweig, May 2013

Some counting problems • Given a finitely generated nilpotent group G , let � � a n ( G ) = # H � G : | G : H | = n . • Given a matrix algebra A � M d ( Z ) , let � � Λ : Λ is a submodule of Z d & | Z d : Λ | = n a n ( A ) = # . • Given an additively finitely generated ring L , let � � a n ( L ) = # Λ : Λ is a subring of L & | L : Λ | = n . • Many variations: normal subgroups of G , ideals of L , . . . Let Γ be one of the above and a n = a n ( Γ ) . Goal: compute ( a 1 , a 2 , . . . )

Zeta functions a nm = a n a m for gcd ( n , m ) = 1 � Dirichlet generating functions Definition The (subgroup/submodule/subring/. . .) zeta function of Γ is ∞ � a n n − s . ζ Γ ( s ) = n = 1 Fact m � a n = O ( m α ) . ζ Γ ( s ) converges for Re ( s ) > α ⇐ ⇒ n = 1 Example ∞ � n − s , the Riemann zeta function. ζ Z ( s ) = n = 1

Local zeta functions Definition The local zeta function of Γ at the prime p is ∞ � a p k p − ks . ζ Γ , p ( s ) = k = 0 Theorem (Grunewald, Segal & Smith 1988) � 1 ζ Γ ( s ) = ζ Γ , p ( s ) (“Euler product”) p prime 2 ζ Γ , p ( s ) ∈ Q ( p − s )

Local zeta functions Theorem (Grunewald & du Sautoy 2000) Given Γ , there are Q -varieties V 1 , . . . , V m and W 1 , . . . , W m ∈ Q ( X , Y ) s.t. for almost all primes p, m � # V i ( F p ) · W i ( p , p − s ) . ζ Γ , p ( s ) = i = 1 Remark Key steps: 1 Express ζ Γ , p ( s ) as a p -adic integral. 2 Evaluate the integral using a resolution of singularities. Usually infeasible! Goal Develop practical methods for computing such V i and W i under non-degeneracy assumptions on Γ .

This project Key ingredients 1 a new concept of non-degeneracy for a class of p -adic integrals 2 an effective method for evaluating non-degenerate integrals 3 a method that modifies the integrals in order to remove degeneracies (WIP) Inspiration 1 Khovanskii et al. (1970s): explicit resolution of singularities under non-degeneracy assumptions w.r.t. certain Newton polyhedra 2 Denef et al. (1980–): Igusa’s local zeta function enumerating solns of f ( x ) ≡ 0 mod p n 3 Gr¨ obner bases machinery, toric geometry

Cone integrals Theorem (Grunewald & du Sautoy 2000) Let Γ have Hirsch length/dimension/additive rank d. Then there are polynomials f 1 . . . , f r over Q s.t. for almost all primes p, � ζ Γ , p ( s ) = ( 1 − p − 1 ) − d | x 11 | s − 1 · · · | x dd | s − d d µ ( x ) , p p V p where    x 11 · · · · · · x 1 d       .  ... �  .   x 22 .  � V p = x =  ∈ Tr d ( Z p ) � x 11 · · · x dd | f 1 ( x ) , . . . , f r ( x ) .   � . ...   � .   .        x dd

Non-degenerate cone integrals Definition The Newton polytope New ( f ) of f = � a e X e : convex hull of { e : a e � = 0 } . Fact Faces τ ⊆ New ( f 1 · · · f r ) define canonical sub-polynomials f i , τ of the f i . � � Write f = ( f 1 , . . . , f r ) . For J ⊆ { 1, . . . , r } , write f J , τ = f j , τ j ∈ J . Definition f is non-degenerate (w.r.t. New ( f 1 · · · f r ) ) if ⇒ rk ( f ′ f J , τ ( x ) = 0 = J , τ ( x )) = # J for faces τ ⊆ New ( f 1 · · · f r ) , subsets J ⊆ { 1, . . . , r } and x ∈ ( C × ) n .

Evaluating non-degenerate cone integrals Recall: given Γ , we obtain f = ( f 1 , . . . , f r ) s.t. ζ Γ , p is a cone integral involving f . Theorem (R. & Voll) Suppose f is non-degenerate. Then there are explicit W τ , J ∈ Q ( X , Y ) indexed by faces τ ⊆ New ( f 1 · · · f r ) and subsets J ⊆ { 1, . . . , r } s.t. � c τ , J ( p ) W τ , J ( p , p − s ) ζ Γ , p ( s ) = τ , J � � p ) n � for almost all p, where c τ , J ( p ) = # u ∈ ( F × � f j , τ ( u ) = 0 ⇐ ⇒ j ∈ J . � Heuristic observation Typical forms of degeneracy can be fixed using a “toric reduction process” (WIP) inspired by Gr¨ obner bases machinery.

Examples We have � | a | s − 1 | x | s − 2 | z | s − 3 d µ ( a , . . . , z ) , ζ Z [ X ] / X 3 , p ( s ) = ( 1 − p − 1 ) − 3 V p where � � a � � � b c � � xz | aby − b 2 x − acx , abz , x 3 , bx 2 V p = ∈ Tr 3 ( Z p ) . . x y � � . . z Newton polytope = △ , 7 cases. Non-degenerate: � Result: ( 1 + p 1 − 2 s )( 1 + p − s + p 1 − 2 s + ( p 2 − p ) p − 3 s + ( p 3 − p 2 ) p − 4 s − p 3 − 5 s − p 4 − 6 s − p 4 − 7 s ) . ( 1 − p − s )( 1 − p 2 − 3 s ) 2 ( 1 − p 4 − 5 s ) Very similar: ζ sl 2 ( Z ) , p ( s ) du Sautoy & Taylor (2002): manual resn of singularities; 8 pages

Examples Submodule ζ -functions for semisimple repns: L. Solomon et al. (1970s) • U 3 ( Z ) � Z 3 ≡ n 3 ( Z ) � Z 3 : ζ p ( s ) ζ p ( 2 s − 1 ) ζ p ( 3 s − 1 ) ζ p ( 4 s − 2 ) ζ p ( 4 s − 1 ) • Nilradical of the Borel subalgebra of sp 4 ( Z ) acting on Z 4 : ζ p ( s ) ζ p ( 2 s − 1 ) ζ p ( 3 s − 1 ) ζ p ( 4 s − 2 ) 2 ζ p ( 6 s − 3 ) ζ p ( 4 s − 1 ) ζ p ( 6 s − 2 ) • U 4 ( Z ) � Z 4 ≡ n 4 ( Z ) � Z 4 : 1 − p 1 − 4 s + · · · 35 terms · · · − p 10 − 30 s ( 1 − p − s )( 1 − p 1 − 2 s )( 1 − p 1 − 3 s )( 1 − p 1 − 4 s )( 1 − p 2 − 4 s )( 1 − p 2 − 5 s )( 1 − p 2 − 6 s )( 1 − p 3 − 7 s )( 1 − p 4 − 8 s ) • Can do: gl 2 ( Z ) (subrings), U 5 ( Z ) � Z 5 , Z [ X ] / X 4 , “commutative Heisenberg rings” of rank � 5, . . . many known examples