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A Brief Survey of Arithmetic Equivalence

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A Brief Survey of Arithmetic Equivalence A brief survey of Arithmetic Equivalence Santiago Arango Pi~neros Universidad de Los Andes, Colombia. PIMS Workshop on Arithmetic Topology, June 2019 Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 1 / 13 The Dedekind zeta function Let K be a number field, and let OK be its ring of integers. The Dedekind zeta function of K is defined by the Dirichlet series X −s Y −s −1 ζK (s) := N(I ) = 1 − N(p) ; Re(s) > 1: I ⊆OK p where the sum ranges over nonzero ideals in OK , the product ranges over nonzero prime ideals in OK and N(I ) := #(OK =I ) is the absolute norm. ζK (s) admits an analytic continuation to C − f1g and satisfies a functional equation relating the argument s to 1 − s. Example ζ (s) = P N(I )−s = P n−s = ζ(s) is the Riemann zeta function. Q I ⊆Z n≥1 Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 2 / 13 The Dedekind zeta function Let K be a number field, and let OK be its ring of integers. The Dedekind zeta function of K is defined by the Dirichlet series X −s Y −s −1 ζK (s) := N(I ) = 1 − N(p) ; Re(s) > 1: I ⊆OK p where the sum ranges over nonzero ideals in OK , the product ranges over nonzero prime ideals in OK and N(I ) := #(OK =I ) is the absolute norm. ζK (s) admits an analytic continuation to C − f1g and satisfies a functional equation relating the argument s to 1 − s. Example ζ (s) = P N(I )−s = P n−s = ζ(s) is the Riemann zeta function. Q I ⊆Z n≥1 Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 2 / 13 The Dedekind zeta function Let K be a number field, and let OK be its ring of integers. The Dedekind zeta function of K is defined by the Dirichlet series X −s Y −s −1 ζK (s) := N(I ) = 1 − N(p) ; Re(s) > 1: I ⊆OK p where the sum ranges over nonzero ideals in OK , the product ranges over nonzero prime ideals in OK and N(I ) := #(OK =I ) is the absolute norm. ζK (s) admits an analytic continuation to C − f1g and satisfies a functional equation relating the argument s to 1 − s. Example ζ (s) = P N(I )−s = P n−s = ζ(s) is the Riemann zeta function. Q I ⊆Z n≥1 Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 2 / 13 Arithmetic Equivalence Definition Two number fields K1 and K2 are said to be arithmetically equivalent if ζK1 (s) = ζK2 (s). We denote this by K1 ≈ K2. ζK (s) governs the arithmetic of K to the extent that each rational prime has the same decomposition type in two arithmetically equivalent fields. Theorem (Perlis, 1977) Arithmetically equivalent number fields have the same degree, discriminant, signature, roots of unity, normal closure, normal core and product of the class number with the regulator. Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 3 / 13 Arithmetic Equivalence Definition Two number fields K1 and K2 are said to be arithmetically equivalent if ζK1 (s) = ζK2 (s). We denote this by K1 ≈ K2. ζK (s) governs the arithmetic of K to the extent that each rational prime has the same decomposition type in two arithmetically equivalent fields. Theorem (Perlis, 1977) Arithmetically equivalent number fields have the same degree, discriminant, signature, roots of unity, normal closure, normal core and product of the class number with the regulator. Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 3 / 13 Arithmetic Equivalence Definition Two number fields K1 and K2 are said to be arithmetically equivalent if ζK1 (s) = ζK2 (s). We denote this by K1 ≈ K2. ζK (s) governs the arithmetic of K to the extent that each rational prime has the same decomposition type in two arithmetically equivalent fields. Theorem (Perlis, 1977) Arithmetically equivalent number fields have the same degree, discriminant, signature, roots of unity, normal closure, normal core and product of the class number with the regulator. Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 3 / 13 Almost Conjugate Subgroups Definition Let H1; H2 be subgroups of a finite group G. We say that H1 and H2 are almost conjugate if for every G-conjugacy class C, #(H1 \C) = #(H2 \C): We denote this by H1 ≈ H2. Example (Gassman) Let G = S6 and consider the subgroups H1 = fe; (12)(34); (13)(24); (14)(23)g; H2 = fe; (12)(56); (34)(56); (14)(23)g: Both H1 and H2 are isomorphic to the Klein four group, but they are not conjugate within G. Nevertheless, H1 ≈ H2. Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 4 / 13 Almost Conjugate Subgroups Definition Let H1; H2 be subgroups of a finite group G. We say that H1 and H2 are almost conjugate if for every G-conjugacy class C, #(H1 \C) = #(H2 \C): We denote this by H1 ≈ H2. Example (Gassman) Let G = S6 and consider the subgroups H1 = fe; (12)(34); (13)(24); (14)(23)g; H2 = fe; (12)(56); (34)(56); (14)(23)g: Both H1 and H2 are isomorphic to the Klein four group, but they are not conjugate within G. Nevertheless, H1 ≈ H2. Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 4 / 13 Gassman's Theorem Let K1 and K2 be number fields, and fix L=Q any Galois number field containing K1K2, G := Gal(L=Q), H1 := Gal(L=K1), H2 := Gal(L=K2). Theorem (Gassman, 1926) ζK1 (s) = ζK2 (s) if and only if H1 ≈ H2. L f1g ⊂ ⊂ () K1 ≈ K2 H1 ≈ H2 ⊂ ⊂ Q G Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 5 / 13 Gassman's Theorem Let K1 and K2 be number fields, and fix L=Q any Galois number field containing K1K2, G := Gal(L=Q), H1 := Gal(L=K1), H2 := Gal(L=K2). Theorem (Gassman, 1926) ζK1 (s) = ζK2 (s) if and only if H1 ≈ H2. L f1g ⊂ ⊂ () K1 ≈ K2 H1 ≈ H2 ⊂ ⊂ Q G Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 5 / 13 Isospectral Riemmanian Manifolds Let (M; g) be a compact and connected Riemmanian manifold. Let ∆M be the Laplace-Beltrami operator, 1 1 ∆M : C (M) ! C (M); f 7! −div grad f : Theorem For (M; g) as above, the eigenspaces of ∆M are finite dimensional, and the corresponding eigenvalues form a countable discrete sequence of non-negative real numbers 0 = λ0 < λ1 ≤ λ2 ≤ · · · . The ordered sequence of nonzero eigenvalues of ∆M (listed with multiplicity) is the eigenvalue spectrum of M, denoted by λ(M). Riemannian manifolds with the same spectrum are called isospectral. Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 6 / 13 Isospectral Riemmanian Manifolds Let (M; g) be a compact and connected Riemmanian manifold. Let ∆M be the Laplace-Beltrami operator, 1 1 ∆M : C (M) ! C (M); f 7! −div grad f : Theorem For (M; g) as above, the eigenspaces of ∆M are finite dimensional, and the corresponding eigenvalues form a countable discrete sequence of non-negative real numbers 0 = λ0 < λ1 ≤ λ2 ≤ · · · . The ordered sequence of nonzero eigenvalues of ∆M (listed with multiplicity) is the eigenvalue spectrum of M, denoted by λ(M). Riemannian manifolds with the same spectrum are called isospectral. Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 6 / 13 Isospectral Riemmanian Manifolds Let (M; g) be a compact and connected Riemmanian manifold. Let ∆M be the Laplace-Beltrami operator, 1 1 ∆M : C (M) ! C (M); f 7! −div grad f : Theorem For (M; g) as above, the eigenspaces of ∆M are finite dimensional, and the corresponding eigenvalues form a countable discrete sequence of non-negative real numbers 0 = λ0 < λ1 ≤ λ2 ≤ · · · . The ordered sequence of nonzero eigenvalues of ∆M (listed with multiplicity) is the eigenvalue spectrum of M, denoted by λ(M). Riemannian manifolds with the same spectrum are called isospectral. Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 6 / 13 Isospectral Riemmanian Manifolds Let (M; g) be a compact and connected Riemmanian manifold. Let ∆M be the Laplace-Beltrami operator, 1 1 ∆M : C (M) ! C (M); f 7! −div grad f : Theorem For (M; g) as above, the eigenspaces of ∆M are finite dimensional, and the corresponding eigenvalues form a countable discrete sequence of non-negative real numbers 0 = λ0 < λ1 ≤ λ2 ≤ · · · . The ordered sequence of nonzero eigenvalues of ∆M (listed with multiplicity) is the eigenvalue spectrum of M, denoted by λ(M). Riemannian manifolds with the same spectrum are called isospectral. Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 6 / 13 The zeta function in Riemannian Geometry (M; g) compact connected Riemmanian manifold. The zeta function of M is defined by X −s ζM (s) := λi ; for Re(s) 0; i≥1 where f0 < λ1 ≤ λ2 ≤ · · · g = λ(M). ζM (s) has a meromorphic continuation to the whole plane with at worst finitely many simple poles. Also, ζM1 (s) = ζM2 (s) if and only if λ(M1) = λ(M2). Example 1 2 2 Consider S with the usual metric. The Laplacian is ∆ = −d =dθ , and 1 λ(S ) = f0; 1; 1; 4; 4; 9; 9; 16; 16;::: g. Therefore, X −2s ζS1 (s) = 2n = 2ζ(2s): n≥1 Santiago Arango Pi~neros (Los Andes) Arithmetic Equivalence PIMS 2019 7 / 13 The zeta function in Riemannian Geometry (M; g) compact connected Riemmanian manifold. The zeta function of M is defined by X −s ζM (s) := λi ; for Re(s) 0; i≥1 where f0 < λ1 ≤ λ2 ≤ · · · g = λ(M).
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