On the behavior of pro-isomorphic zeta functions under base extension Michael M. Schein Bar-Ilan University Zeta functions and motivic integration D¨ usseldorf, July 2016 Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 1 / 21
Subgroup growth This talk will discuss joint work with Mark Berman. Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 2 / 21
Subgroup growth This talk will discuss joint work with Mark Berman. Let G be a finitely generated group. For any n ≥ 1 it has finitely many subgroups of index n . Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 2 / 21
Subgroup growth This talk will discuss joint work with Mark Berman. Let G be a finitely generated group. For any n ≥ 1 it has finitely many subgroups of index n . Let a ≤ n = |{ H ≤ G : [ G : H ] = n }| . Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 2 / 21
Subgroup growth This talk will discuss joint work with Mark Berman. Let G be a finitely generated group. For any n ≥ 1 it has finitely many subgroups of index n . Let a ≤ n = |{ H ≤ G : [ G : H ] = n }| . Can consider variations of this sequence: a ⊳ = |{ H � G : [ G : H ] = n }| n |{ � H ≃ � a ∧ = G : [ G : H ] = n }| , n where � G is the profinite completion of G . Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 2 / 21
Dirichlet series Theorem (Lubotzky-Mann-Segal) Let G be a finitely generated residually finite group. Then there exists C n ≤ n C for all n if and only if G is virtually solvable of finite such that a ≤ rank. Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 3 / 21
Dirichlet series Theorem (Lubotzky-Mann-Segal) Let G be a finitely generated residually finite group. Then there exists C n ≤ n C for all n if and only if G is virtually solvable of finite such that a ≤ rank. To study the sequences a ∗ n ( ∗ ∈ {≤ , ⊳ , ∧} ), make a Dirichlet series: � ∞ ζ ∗ a ∗ n n − s . G ( s ) = n =1 Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 3 / 21
Dirichlet series Theorem (Lubotzky-Mann-Segal) Let G be a finitely generated residually finite group. Then there exists C n ≤ n C for all n if and only if G is virtually solvable of finite such that a ≤ rank. To study the sequences a ∗ n ( ∗ ∈ {≤ , ⊳ , ∧} ), make a Dirichlet series: � ∞ ζ ∗ a ∗ n n − s . G ( s ) = n =1 Example Let G = Z . Then ∞ � � 1 1 ζ ≤ G ( s ) = ζ ⊳ G ( s ) = ζ ∧ G ( s ) = n s = 1 − p − s n =1 p is the Riemann zeta function. Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 3 / 21
Linearization If G is a torsion-free finitely generated group, there is a Lie ring L (a finite-rank free Z -module with Lie bracket) with an index-preserving correspondence: Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 4 / 21
Linearization If G is a torsion-free finitely generated group, there is a Lie ring L (a finite-rank free Z -module with Lie bracket) with an index-preserving correspondence: ← → subgroups subrings normal subgroups ← → ideals H ≤ G : � H ≃ � ← → M ≤ L : M ≃ L G Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 4 / 21
Linearization If G is a torsion-free finitely generated group, there is a Lie ring L (a finite-rank free Z -module with Lie bracket) with an index-preserving correspondence: ← → subgroups subrings normal subgroups ← → ideals H ≤ G : � H ≃ � ← → M ≤ L : M ≃ L G In this talk we concentrate on pro-isomorphic zeta functions. Note that the condition M ≃ L does not correspond to closure under the action of some subalgebra of End Z ( L ), so pro-isomorphic zeta functions do not in general fit into Roßmann’s framework of subalgebra zeta functions. Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 4 / 21
Euler decomposition Theorem (Grunewald-Segal-Smith, 1988) Let G be a finitely generated torsion-free nilpotent group. Then � ζ ∗ ζ ∗ G ( s ) = G , p ( s ) , p for any ∗ ∈ {≤ , ⊳ , ∧} , where ∞ � ζ ∗ a ∗ p k p − ks . G , p ( s ) = k =0 Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 5 / 21
Euler decomposition Theorem (Grunewald-Segal-Smith, 1988) Let G be a finitely generated torsion-free nilpotent group. Then � ζ ∗ ζ ∗ G ( s ) = G , p ( s ) , p for any ∗ ∈ {≤ , ⊳ , ∧} , where ∞ � ζ ∗ a ∗ p k p − ks . G , p ( s ) = k =0 L ( s ) = � Similarly in the linear setting, ζ ∗ p ζ ∗ L ⊗ Z Z p ( s ). Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 5 / 21
Euler decomposition Theorem (Grunewald-Segal-Smith, 1988) Let G be a finitely generated torsion-free nilpotent group. Then � ζ ∗ ζ ∗ G ( s ) = G , p ( s ) , p for any ∗ ∈ {≤ , ⊳ , ∧} , where ∞ � ζ ∗ a ∗ p k p − ks . G , p ( s ) = k =0 L ( s ) = � Similarly in the linear setting, ζ ∗ p ζ ∗ L ⊗ Z Z p ( s ). We investigate the behavior of ζ ∧ L ( s ) under base extension. Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 5 / 21
Base extension Our main question Let Γ be a Z -group scheme such that Γ( Z ) is finitely generated torsion-free nilpotent. How does ζ ∗ G ( O K ) ( s ) behave as K varies over number fields? Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 6 / 21
Base extension Our main question Let Γ be a Z -group scheme such that Γ( Z ) is finitely generated torsion-free nilpotent. How does ζ ∗ G ( O K ) ( s ) behave as K varies over number fields? Analogously, if L is a nilpotent Z -Lie ring, how does ζ ∧ L ⊗ Z O K ( s ) behave? Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 6 / 21
Base extension Our main question Let Γ be a Z -group scheme such that Γ( Z ) is finitely generated torsion-free nilpotent. How does ζ ∗ G ( O K ) ( s ) behave as K varies over number fields? Analogously, if L is a nilpotent Z -Lie ring, how does ζ ∧ L ⊗ Z O K ( s ) behave? The simplest example is not encouraging. Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 6 / 21
Base extension Our main question Let Γ be a Z -group scheme such that Γ( Z ) is finitely generated torsion-free nilpotent. How does ζ ∗ G ( O K ) ( s ) behave as K varies over number fields? Analogously, if L is a nilpotent Z -Lie ring, how does ζ ∧ L ⊗ Z O K ( s ) behave? The simplest example is not encouraging. Let A be an abelian Z -Lie ring of rank m . If [ K : Q ] = d , then A ⊗ Z O K is simply an abelian Z -Lie ring of rank md . Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 6 / 21
Base extension Our main question Let Γ be a Z -group scheme such that Γ( Z ) is finitely generated torsion-free nilpotent. How does ζ ∗ G ( O K ) ( s ) behave as K varies over number fields? Analogously, if L is a nilpotent Z -Lie ring, how does ζ ∧ L ⊗ Z O K ( s ) behave? The simplest example is not encouraging. Let A be an abelian Z -Lie ring of rank m . If [ K : Q ] = d , then A ⊗ Z O K is simply an abelian Z -Lie ring of rank md . Exercise If A is an abelian Z -Lie ring of rank m , then 1 ζ ≤ A , p ( s ) = ζ ⊳ A , p ( s ) = ζ ∧ A , p ( s ) = (1 − p − s )(1 − p 1 − s ) · · · (1 − p m − 1 − s ) . Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 6 / 21
Base extension Our main question Let Γ be a Z -group scheme such that Γ( Z ) is finitely generated torsion-free nilpotent. How does ζ ∗ G ( O K ) ( s ) behave as K varies over number fields? Analogously, if L is a nilpotent Z -Lie ring, how does ζ ∧ L ⊗ Z O K ( s ) behave? The simplest example is not encouraging. Let A be an abelian Z -Lie ring of rank m . If [ K : Q ] = d , then A ⊗ Z O K is simply an abelian Z -Lie ring of rank md . Exercise If A is an abelian Z -Lie ring of rank m , then 1 ζ ≤ A , p ( s ) = ζ ⊳ A , p ( s ) = ζ ∧ A , p ( s ) = (1 − p − s )(1 − p 1 − s ) · · · (1 − p m − 1 − s ) . Five proofs of this in Lubotzky-Segal, e.g. count Smith normal forms. Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 6 / 21
What we want To understand why we are unhappy with this very clean result, compare it with the following. Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 7 / 21
What we want To understand why we are unhappy with this very clean result, compare it with the following. Let H = � x , y , z | [ x , y ] = z � be the Heisenberg Lie ring: the simplest non-abelian Lie ring. Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 7 / 21
What we want To understand why we are unhappy with this very clean result, compare it with the following. Let H = � x , y , z | [ x , y ] = z � be the Heisenberg Lie ring: the simplest non-abelian Lie ring. Theorem (Grunewald-Segal-Smith) Let K be a number field and let [ K : Q ] = d . Then ζ ∧ H ( s ) = ζ (2 s − 2) ζ (2 s − 3) Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 7 / 21
What we want To understand why we are unhappy with this very clean result, compare it with the following. Let H = � x , y , z | [ x , y ] = z � be the Heisenberg Lie ring: the simplest non-abelian Lie ring. Theorem (Grunewald-Segal-Smith) Let K be a number field and let [ K : Q ] = d . Then ζ ∧ H ( s ) = ζ (2 s − 2) ζ (2 s − 3) � 1 ζ ∧ H ⊗O K ( s ) = (1 − ( N p ) 2 d − 2 s )(1 − ( N p ) 2 d +1 − 2 s ) p Michael M. Schein (Bar-Ilan) Pro-isomorphic zeta functions 7 / 21
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