Dedekind Sums: A Geometric Viewpoint Matthias Beck San Francisco State University math.sfsu.edu/beck
“Ubi materia, ibi geometria.” Johannes Kepler (1571-1630)
“Ubi number theory, ibi geometria.” Variation on Johannes Kepler (1571-1630)
Ehrhart Theory Integral (convex) polytope P – convex hull of finitely many points in Z d P ∩ 1 t P ∩ Z d � t Z d � � � For t ∈ Z > 0 , let L P ( t ) := # = # Dedekind Sums: A Geometric Viewpoint Matthias Beck 3
Ehrhart Theory Integral (convex) polytope P – convex hull of finitely many points in Z d P ∩ 1 t P ∩ Z d � t Z d � � � For t ∈ Z > 0 , let L P ( t ) := # = # Theorem (Ehrhart 1962) If P is an integral polytope, then... ◮ L P ( t ) and L P ◦ ( t ) are polynomials in t of degree dim P ◮ Leading term: vol( P ) (suitably normalized) ◮ (Macdonald 1970) L P ( − t ) = ( − 1) dim P L P ◦ ( t ) Dedekind Sums: A Geometric Viewpoint Matthias Beck 3
Ehrhart Theory Integral (convex) polytope P – convex hull of finitely many points in Z d P ∩ 1 t P ∩ Z d � t Z d � � � For t ∈ Z > 0 , let L P ( t ) := # = # Theorem (Ehrhart 1962) If P is an integral polytope, then... ◮ L P ( t ) and L P ◦ ( t ) are polynomials in t of degree dim P ◮ Leading term: vol( P ) (suitably normalized) ◮ (Macdonald 1970) L P ( − t ) = ( − 1) dim P L P ◦ ( t ) Alternative description of a polytope: x ∈ R d : A x ≤ b x ∈ R d � � � � P = ≥ 0 : A x = b ⇄ Dedekind Sums: A Geometric Viewpoint Matthias Beck 3
Ehrhart Theory Rational (convex) polytope P – convex hull of finitely many points in Q d P ∩ 1 t P ∩ Z d � t Z d � � � For t ∈ Z > 0 , let L P ( t ) := # = # Theorem (Ehrhart 1962) If P is an rational polytope, then... ◮ L P ( t ) and L P ◦ ( t ) are quasi-polynomials in t of degree dim P ◮ Leading term: vol( P ) (suitably normalized) ◮ (Macdonald 1970) L P ( − t ) = ( − 1) dim P L P ◦ ( t ) Alternative description of a polytope: x ∈ R d : A x ≤ b x ∈ R d � � � � P = ≥ 0 : A x = b ⇄ Quasi-polynomial – c d ( t ) t d + c d − 1 ( t ) t d − 1 + · · · + c 0 ( t ) where c k ( t ) are periodic Dedekind Sums: A Geometric Viewpoint Matthias Beck 3
An Example in Dimension 2 ( x, y ) ∈ R 2 � � ∆ := ≥ 0 : ax + by ≤ 1 ( a = 7 , b = 4 , t = 23) Dedekind Sums: A Geometric Viewpoint Matthias Beck 4
An Example in Dimension 2 ( x, y ) ∈ R 2 � � ∆ := ≥ 0 : ax + by ≤ 1 ( a = 7 , b = 4 , t = 23) ( m, n ) ∈ Z 2 � � L ∆ ( t ) = # ≥ 0 : am + bn ≤ t Dedekind Sums: A Geometric Viewpoint Matthias Beck 4
An Example in Dimension 2 ( x, y ) ∈ R 2 � � ∆ := ≥ 0 : ax + by ≤ 1 ( a = 7 , b = 4 , t = 23) ( m, n ) ∈ Z 2 � � L ∆ ( t ) = # ≥ 0 : am + bn ≤ t ( m, n, s ) ∈ Z 3 � � = # ≥ 0 : am + bn + s = t Dedekind Sums: A Geometric Viewpoint Matthias Beck 4
An Example in Dimension 2 ( x, y ) ∈ R 2 � � ∆ := ≥ 0 : ax + by ≤ 1 ( a = 7 , b = 4 , t = 23) ( m, n ) ∈ Z 2 � � L ∆ ( t ) = # ≥ 0 : am + bn ≤ t ( m, n, s ) ∈ Z 3 � � = # ≥ 0 : am + bn + s = t 1 = const (1 − x a ) (1 − x b ) (1 − x ) x t Dedekind Sums: A Geometric Viewpoint Matthias Beck 4
An Example in Dimension 2 ( x, y ) ∈ R 2 � � ∆ := ≥ 0 : ax + by ≤ 1 ( a = 7 , b = 4 , t = 23) ( m, n ) ∈ Z 2 � � L ∆ ( t ) = # ≥ 0 : am + bn ≤ t ( m, n, s ) ∈ Z 3 � � = # ≥ 0 : am + bn + s = t 1 = const (1 − x a ) (1 − x b ) (1 − x ) x t 1 dx � = (1 − x a ) (1 − x b ) (1 − x ) x t +1 2 πi | x | = ǫ Dedekind Sums: A Geometric Viewpoint Matthias Beck 4
An Example in Dimension 2 ( x, y ) ∈ R 2 � � ∆ := ≥ 0 : ax + by ≤ 1 1 f ( x ) := (1 − x a ) (1 − x b ) (1 − x ) x t +1 1 � L ∆ ( t ) = f dx 2 πi | x | = ǫ Dedekind Sums: A Geometric Viewpoint Matthias Beck 5
An Example in Dimension 2 ( x, y ) ∈ R 2 � � ∆ := ≥ 0 : ax + by ≤ 1 gcd ( a, b ) = 1 1 ξ a := e 2 πi/a f ( x ) := (1 − x a ) (1 − x b ) (1 − x ) x t +1 1 � L ∆ ( t ) = f dx 2 πi | x | = ǫ a − 1 b − 1 � � = Res x =1 ( f ) + Res x = ξ k a ( f ) + Res x = ξ j b ( f ) k =1 j =1 Dedekind Sums: A Geometric Viewpoint Matthias Beck 5
An Example in Dimension 2 ( x, y ) ∈ R 2 � � ∆ := ≥ 0 : ax + by ≤ 1 gcd ( a, b ) = 1 1 ξ a := e 2 πi/a f ( x ) := (1 − x a ) (1 − x b ) (1 − x ) x t +1 1 � L ∆ ( t ) = f dx 2 πi | x | = ǫ a − 1 b − 1 � � = Res x =1 ( f ) + Res x = ξ k a ( f ) + Res x = ξ j b ( f ) k =1 j =1 � 1 t 2 � � 3 � 2 ab + t ab + 1 a + 1 + 1 a + 3 b + 3 + a b + b a + 1 = 2 b 12 ab a − 1 b − 1 +1 1 + 1 1 � � (1 − ξ kb a ) (1 − ξ k a ) ξ kt � � � � a b 1 − ξ ja 1 − ξ j ξ jt a j =1 k =1 b b b Dedekind Sums: A Geometric Viewpoint Matthias Beck 5
An Example in Dimension 2 (Pick’s or) Ehrhart’s Theorem implies that � 1 t 2 2 ab + t ab + 1 a + 1 � + 1 � 3 a + 3 b + 3 + a b + b a + 1 � L ∆ ( t ) = 2 b 12 ab a − 1 b − 1 +1 1 + 1 1 � � (1 − ξ kb a ) (1 − ξ k a ) ξ kt � � � � a b 1 − ξ ja 1 − ξ j ξ jt a k =1 j =1 b b b has constant term L ∆ (0) = 1 Dedekind Sums: A Geometric Viewpoint Matthias Beck 6
An Example in Dimension 2 (Pick’s or) Ehrhart’s Theorem implies that � 1 t 2 2 ab + t ab + 1 a + 1 � + 1 � 3 a + 3 b + 3 + a b + b a + 1 � L ∆ ( t ) = 2 b 12 ab a − 1 b − 1 +1 1 + 1 1 � � (1 − ξ kb a ) (1 − ξ k a ) ξ kt � � � � a b 1 − ξ ja 1 − ξ j ξ jt a k =1 j =1 b b b has constant term L ∆ (0) = 1 and hence a − 1 b − 1 1 1 a ) + 1 1 � � (1 − ξ kb a ) (1 − ξ k � � � � a b 1 − ξ ja 1 − ξ j j =1 k =1 b b = 1 − 1 � 3 a + 3 b + 3 + a b + b a + 1 � 12 ab Dedekind Sums: A Geometric Viewpoint Matthias Beck 6
An Example in Dimension 2 (Recall that ξ a := e 2 πi/a ) a − 1 b − 1 1 1 a ) + 1 1 � � (1 − ξ kb a ) (1 − ξ k � � � � a b 1 − ξ ja 1 − ξ j j =1 k =1 b b = 1 − 1 � 3 a + 3 b + 3 + a b + b a + 1 � 12 ab However... a − 1 a − 1 1 1 a ) = − 1 � πkb � � πk � + a − 1 � � cot cot (1 − ξ kb a ) (1 − ξ k a 4 a a a 4 a k =1 k =1 is essentially a Dedekind sum. Dedekind Sums: A Geometric Viewpoint Matthias Beck 7
Dedekind Sums � x − ⌊ x ⌋ − 1 if x / ∈ Z , 2 Let ( ( x ) ) := if x ∈ Z , and define the Dedekind sum as 0 b − 1 � � ka � � � � k � � � s ( a, b ) := b b k =1 b − 1 1 � πja � � πj � � = cot cot . 4 b b b j =1 Dedekind Sums: A Geometric Viewpoint Matthias Beck 8
Dedekind Sums � x − ⌊ x ⌋ − 1 if x / ∈ Z , 2 Let ( ( x ) ) := if x ∈ Z , and define the Dedekind sum as 0 b − 1 � � ka � � � � k � � � s ( a, b ) := b b k =1 b − 1 1 � πja � � πj � � = cot cot . 4 b b b j =1 Since their introduction by Dedekind in the 1880’s, these sums and their generalizations have appeared in various areas such as analytic (transformation law of η -function) and algebraic number theory (class numbers), topology (group action on manifolds), combinatorial geometry (lattice point problems), and algorithmic complexity (random number generators). Dedekind Sums: A Geometric Viewpoint Matthias Beck 8
Dedekind Sums � x − ⌊ x ⌋ − 1 if x / ∈ Z , 2 Let ( ( x ) ) := if x ∈ Z , and define the Dedekind sum as 0 b − 1 � � ka � � � � k � � � s ( a, b ) := b b k =1 b − 1 1 � πja � � πj � � = cot cot . 4 b b b j =1 The identity L ∆ (0) = 1 implies... � a � s ( a, b ) + s ( b, a ) = − 1 4 + 1 b + 1 ab + b 12 a the Reciprocity Law for Dedekind sums. Dedekind Sums: A Geometric Viewpoint Matthias Beck 8
Dedekind Sum Reciprocity b − 1 s ( a, b ) = 1 � πja � � πj � � cot cot . 4 b b b j =1 the Reciprocity Law � a � s ( a, b ) + s ( b, a ) = − 1 4 + 1 b + 1 ab + b 12 a together with the fact that s ( a, b ) = s ( a mod b, b ) implies that s ( a, b ) is polynomial-time computable (Euclidean Algorithm). Dedekind Sums: A Geometric Viewpoint Matthias Beck 9
Ehrhart Theory Revisited P ∩ 1 � t P ∩ Z d � � t Z d � For t ∈ Z > 0 , let L P ( t ) := # = # . Theorem (Ehrhart 1962) If P is an rational polytope, then... ◮ L P ( t ) and L P ◦ ( t ) are quasi-polynomials in t of degree dim P . ◮ Leading term: vol( P ) (suitably normalized) ◮ (Macdonald 1970) L P ( − t ) = ( − 1) dim P L P ◦ ( t ) In particular, if t P ◦ ∩ Z d = ∅ then L P ( − t ) = 0 . Dedekind Sums: A Geometric Viewpoint Matthias Beck 10
Rademacher Reciprocity If t P ◦ ∩ Z d = ∅ then L P ( − t ) = 0 . t ∆ ◦ = ( x, y ) ∈ R 2 � � > 0 : ax + by < t does not contain any lattice points for 1 ≤ t < a + b which gives for these t a − 1 b − 1 ξ jt ξ kt 1 a ) + 1 � � a b (1 − ξ kb a ) (1 − ξ k � � � � a b 1 − ξ ja 1 − ξ j j =1 k =1 b b � 1 = − t 2 � � 3 � 2 ab + t ab + 1 a + 1 − 1 a + 3 b + 3 + a b + b a + 1 . 2 b 12 ab Dedekind Sums: A Geometric Viewpoint Matthias Beck 11
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