Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic lattices with fixed invariant trace field Jiming Ma Fudan University The Second China-Russia Workshop on Knot Theory and Related Topics Novosibirsk, August 21-25, 2015
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Outline Distribution of primes 1 Quaternion algebra 2 Number fields Quaternion algebra Orders in quaternion algebra Arithmetic hyperbolic 3-manifolds 3 Arithmetic hyperbolic 3-manifolds Examples Maximal arithmetic hyperbolic 3-manifolds 4 Higher dim Volume formula and finiteness of Arith hyper 3-mfd Distribution of Max lattices with a fixed trace field K The proof of the distribution
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Let π ( x ) = ♯ { p ∈ Z + , p is a prime number , p ≤ x } , people want to know the asymptotic behavior of π ( x ) as x → ∞ .
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Let π ( x ) = ♯ { p ∈ Z + , p is a prime number , p ≤ x } , people want to know the asymptotic behavior of π ( x ) as x → ∞ . The Riemann Hypothesis: Riemann 1859 ζ ( s ) = � ∞ 1 1 n s = � 1 − p − s , all zeros of ζ ( s ) have real part n = 1 p prime 1 2 except trivial ones.
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Let π ( x ) = ♯ { p ∈ Z + , p is a prime number , p ≤ x } , people want to know the asymptotic behavior of π ( x ) as x → ∞ . The Riemann Hypothesis: Riemann 1859 ζ ( s ) = � ∞ 1 1 n s = � 1 − p − s , all zeros of ζ ( s ) have real part n = 1 p prime 1 2 except trivial ones. The prime number theorem: J. Hadamard, independently, C. Vallée-Poussin, 1896 x x π ( x ) = log x + o ( log x )
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Ingham’ Theorem, 1932. the supremum of real parts of the zeros of ζ ( s ) is the infimum of numbers β such that the error is O ( x β ) in the prime number theorem.
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Ingham’ Theorem, 1932. the supremum of real parts of the zeros of ζ ( s ) is the infimum of numbers β such that the error is O ( x β ) in the prime number theorem. So the Riemann Hypothesis is: The Riemann Hypothesis: equivalent form 1 x 2 + ǫ ) , for any ǫ > 0. π ( x ) = log x + o ( x
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds There are many results after distribution of primes:
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds There are many results after distribution of primes: Margulis: closed primitive geodesics of bounded length in a negative curved manifold;
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds There are many results after distribution of primes: Margulis: closed primitive geodesics of bounded length in a negative curved manifold; Mirzkhani: simple closed geodesics of bounded length in a hyperbolic surface;
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds There are many results after distribution of primes: Margulis: closed primitive geodesics of bounded length in a negative curved manifold; Mirzkhani: simple closed geodesics of bounded length in a hyperbolic surface; Eskin-Mirzkhani: pseudo-Anosov maps of bounded dilatation in the mapping class group of a surface (i.e., geodesics of bounded length in moduli space with the Teichmüller metric);
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds There are many results after distribution of primes: Margulis: closed primitive geodesics of bounded length in a negative curved manifold; Mirzkhani: simple closed geodesics of bounded length in a hyperbolic surface; Eskin-Mirzkhani: pseudo-Anosov maps of bounded dilatation in the mapping class group of a surface (i.e., geodesics of bounded length in moduli space with the Teichmüller metric); Kahn-Markovic: surface subgroups in π 1 of a hyperbolic 3-manifold (Hamenstädt; the existence of surface subgroups of a discrete subgroup of some Lie groups).
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Number fields Q [ x ] Let K be a number field, i.e., K = ( f ( x )) , f ( x ) is a monic irreducible polynomial in Q [ x ] . We assume f ( x ) has n roots, a 1 , a 2 = ¯ a 1 ∈ C − R , a i ∈ R for 3 ≤ i ≤ n .
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Number fields Q [ x ] Let K be a number field, i.e., K = ( f ( x )) , f ( x ) is a monic irreducible polynomial in Q [ x ] . We assume f ( x ) has n roots, a 1 , a 2 = ¯ a 1 ∈ C − R , a i ∈ R for 3 ≤ i ≤ n . Now the field K i = Q ( a i ) is isomorphic to K , for each i ,
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Number fields Q [ x ] Let K be a number field, i.e., K = ( f ( x )) , f ( x ) is a monic irreducible polynomial in Q [ x ] . We assume f ( x ) has n roots, a 1 , a 2 = ¯ a 1 ∈ C − R , a i ∈ R for 3 ≤ i ≤ n . Now the field K i = Q ( a i ) is isomorphic to K , for each i , K → K 1 , 2 ֒ → C is a pair of complex places of K
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Number fields Q [ x ] Let K be a number field, i.e., K = ( f ( x )) , f ( x ) is a monic irreducible polynomial in Q [ x ] . We assume f ( x ) has n roots, a 1 , a 2 = ¯ a 1 ∈ C − R , a i ∈ R for 3 ≤ i ≤ n . Now the field K i = Q ( a i ) is isomorphic to K , for each i , K → K 1 , 2 ֒ → C is a pair of complex places of K K → K i ֒ → R , 3 ≤ i ≤ n , are n − 2 real places of K .
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Quaternion algebra R K be the set of algebraic intergers in K , as an abelian group, R K = Z | K : Q | .
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Quaternion algebra R K be the set of algebraic intergers in K , as an abelian group, R K = Z | K : Q | . R K is a ring, P K be the set of prime ideals in R K .
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Quaternion algebra R K be the set of algebraic intergers in K , as an abelian group, R K = Z | K : Q | . R K is a ring, P K be the set of prime ideals in R K . Definition A quaternion algebra A = A K over a field K is:
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Quaternion algebra R K be the set of algebraic intergers in K , as an abelian group, R K = Z | K : Q | . R K is a ring, P K be the set of prime ideals in R K . Definition A quaternion algebra A = A K over a field K is: A is a 4-dim vector space over K with basis 1, i , j , ij ;
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Quaternion algebra R K be the set of algebraic intergers in K , as an abelian group, R K = Z | K : Q | . R K is a ring, P K be the set of prime ideals in R K . Definition A quaternion algebra A = A K over a field K is: A is a 4-dim vector space over K with basis 1, i , j , ij ; multiplication i 2 = a · 1 ∈ K − { 0 } , j 2 = b · 1 ∈ K − { 0 } , and ji = − ij ;
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Quaternion algebra R K be the set of algebraic intergers in K , as an abelian group, R K = Z | K : Q | . R K is a ring, P K be the set of prime ideals in R K . Definition A quaternion algebra A = A K over a field K is: A is a 4-dim vector space over K with basis 1, i , j , ij ; multiplication i 2 = a · 1 ∈ K − { 0 } , j 2 = b · 1 ∈ K − { 0 } , and ji = − ij ; extending the multiplication linearly, A is an associative algebra with K ⊂ centre ( A ) .
Distribution of primes Quaternion algebra Arithmetic hyperbolic 3-manifolds Maximal arithmetic hyperbolic 3-manifolds Quaternion algebra R K be the set of algebraic intergers in K , as an abelian group, R K = Z | K : Q | . R K is a ring, P K be the set of prime ideals in R K . Definition A quaternion algebra A = A K over a field K is: A is a 4-dim vector space over K with basis 1, i , j , ij ; multiplication i 2 = a · 1 ∈ K − { 0 } , j 2 = b · 1 ∈ K − { 0 } , and ji = − ij ; extending the multiplication linearly, A is an associative algebra with K ⊂ centre ( A ) . A = ( a , b ) K , the Hilbert symbol.
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