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On the ubiquity of modular forms and Ap´ ery-like numbers Algorithmic Combinatorics Seminar RISC (Johannes Kepler University, Linz, Austria) Armin Straub October 9, 2013 University of Illinois & Max-Planck-Institut at Urbana–Champaign f¨ ur Mathematik, Bonn Based on joint work with: Bruce Berndt Jon Borwein James Wan Wadim Zudilin Mathew Rogers University of Newcastle, Australia University of Montreal University of Illinois at Urbana–Champaign On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 1 / 47

PART I Encounters with Ap´ ery numbers and modular forms Short random walks Binomial congruences Positivity of rational functions Series for 1 /π On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 2 / 47

Ap´ ery numbers and the irrationality of ζ (3) • The Ap´ ery numbers 1 , 5 , 73 , 1445 , . . . n � 2 � n + k � 2 � n � A ( n ) = k k satisfy k =0 ( n + 1) 3 u n +1 − (2 n + 1)(17 n 2 + 17 n + 5) u n + n 3 u n − 1 = 0 . On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 3 / 47

Ap´ ery numbers and the irrationality of ζ (3) • The Ap´ ery numbers 1 , 5 , 73 , 1445 , . . . n � 2 � n + k � 2 � n � A ( n ) = k k satisfy k =0 ( n + 1) 3 u n +1 − (2 n + 1)(17 n 2 + 17 n + 5) u n + n 3 u n − 1 = 0 . ζ (3) = � ∞ 1 THM n 3 is irrational. n =1 Ap´ ery ’78 The same recurrence is satisfied by the “near”-integers proof   n n k � 2 � n + k � 2 ( − 1) m − 1 � n 1 � � � B ( n ) = j 3 +   2 m 3 � n �� n + m � k k m m j =1 m =1 k =0 Then, B ( n ) A ( n ) → ζ (3) . But too fast for ζ (3) to be rational. On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 3 / 47

Ap´ ery-like numbers • Recurrence for the Ap´ ery numbers is the case ( a, b, c ) = (17 , 5 , 1) of ( n + 1) 3 u n +1 − (2 n + 1)( an 2 + an + b ) u n + cn 3 u n − 1 = 0 . Are there other triples for which the solution defined by u − 1 = 0 , Q u 0 = 1 is integral? On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 4 / 47

Ap´ ery-like numbers • Recurrence for the Ap´ ery numbers is the case ( a, b, c ) = (17 , 5 , 1) of ( n + 1) 3 u n +1 − (2 n + 1)( an 2 + an + b ) u n + cn 3 u n − 1 = 0 . Are there other triples for which the solution defined by u − 1 = 0 , Q u 0 = 1 is integral? • Almkvist and Zudilin find 14 triplets ( a, b, c ) . The simpler case of ( n + 1) 2 u n +1 − ( an 2 + an + b ) u n + cn 2 u n − 1 = 0 was similarly investigated by Beukers and Zagier. • 4 hypergeometric, 4 Legendrian and 6 sporadic solutions On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 4 / 47

Ap´ ery-like numbers • Hypergeometric and Legendrian solutions have generating functions � 1 � 2 � � 2 , α, 1 − α � 1 � α, 1 − α − C α z � � 3 F 2 � 4 C α z , 1 − C α z 2 F 1 , � � 1 , 1 1 1 − C α z � 6 and C α = 2 4 , 3 3 , 2 6 , 2 4 · 3 3 . with α = 1 2 , 1 3 , 1 4 , 1 • The six sporadic solutions are: ( a, b, c ) A ( n ) � (3 k )! k ( − 1) k 3 n − 3 k � n �� n + k (7 , 3 , 81) � 3 k n k ! 3 � 3 �� 4 n − 5 k − 1 �� k ( − 1) k � n � 4 n − 5 k � � (11 , 5 , 125) + k 3 n 3 n � 2 � 2 k � n �� 2( n − k ) � (10 , 4 , 64) � k k k n − k � 2 � 2 � 2 k � n � (12 , 4 , 16) k k n � 2 � n �� k + l � n �� k � � (9 , 3 , − 27) k,l k l l n � 2 � 2 � n + k � n (17 , 5 , 1) � k k n On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 5 / 47

Modular forms “ Modular forms are functions on the complex plane that are in- ordinately symmetric. They satisfy so many internal symmetries that their mere existence seem like accidents. But they do exist. ” Barry Mazur (BBC Interview, “The Proof”, 1997) � a b � DEF Actions of γ = ∈ SL 2 ( Z ) : c d γ · τ = aτ + b • on τ ∈ H by cτ + d , ( f | k γ )( τ ) = ( cτ + d ) − k f ( γ · τ ) . • on f : H → C by � 0 − 1 SL 2 ( Z ) is generated by T = ( 1 1 � EG 0 1 ) and S = . 1 0 S · τ = − 1 T · τ = τ + 1 , τ On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 6 / 47

Modular forms “ There’s a saying attributed to Eichler that there are five funda- mental operations of arithmetic: addition, subtraction, multipli- cation, division, and modular forms. ” Andrew Wiles (BBC Interview, “The Proof”, 1997) A function f : H → C is a modular form of weight k if DEF • f | k γ = f for all γ ∈ SL 2 ( Z ) , • f is holomorphic (including at the cusp i ∞ ). EG τ − k f ( − 1 /τ ) = f ( τ ) . f ( τ + 1) = f ( τ ) , • Similarly, MFs w.r.t. finite-index Γ � SL 2 ( Z ) • Spaces of MFs finite dimensional, Hecke operators, . . . On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 7 / 47

Modular forms: a prototypical example • The Dedekind eta function ( q = e 2 πiτ ) η ( τ ) = q 1 / 24 � (1 − q n ) n � 1 transforms as √ η ( τ + 1) = e πi/ 12 η ( τ ) , η ( − 1 /τ ) = − iτη ( τ ) . ∆( τ ) = (2 π ) 12 η ( τ ) 24 is a modular form of weight 12 . EG On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 8 / 47

Modular forms: Eisenstein series and L -functions • For k > 1 , the Eisenstein series G 2 k ( τ ) is modular of weight 2 k . 1 � ′ G 2 k ( τ )= ( mτ + n ) 2 k d | n d k σ k ( n ) = � m,n ∈ Z ∞ = 2 ζ (2 k ) + 2(2 πi ) 2 k � σ 2 k − 1 ( n ) q n Γ(2 k ) n =1 On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 9 / 47

Modular forms: Eisenstein series and L -functions • For k > 1 , the Eisenstein series G 2 k ( τ ) is modular of weight 2 k . 1 � ′ G 2 k ( τ )= ( mτ + n ) 2 k d | n d k σ k ( n ) = � m,n ∈ Z ∞ = 2 ζ (2 k ) + 2(2 πi ) 2 k � σ 2 k − 1 ( n ) q n Γ(2 k ) n =1 • Any modular form for SL 2 ( Z ) is a polynomial in G 4 and G 6 . EG ∆ = (60 G 4 ) 3 − 27(140 G 6 ) 2 On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 9 / 47

Modular forms: Eisenstein series and L -functions • For k > 1 , the Eisenstein series G 2 k ( τ ) is modular of weight 2 k . 1 � ′ G 2 k ( τ )= ( mτ + n ) 2 k d | n d k σ k ( n ) = � m,n ∈ Z ∞ = 2 ζ (2 k ) + 2(2 πi ) 2 k � σ 2 k − 1 ( n ) q n Γ(2 k ) n =1 • Any modular form for SL 2 ( Z ) is a polynomial in G 4 and G 6 . EG ∆ = (60 G 4 ) 3 − 27(140 G 6 ) 2 n =0 b ( n ) q n is • The L -function of f ( τ ) = � ∞ � ∞ ∞ L ( f, s ) = (2 π ) s b ( n ) [ f ( iτ ) − f ( i ∞ )] τ s − 1 d τ = � n s . Γ( s ) 0 n =1 EG L ( G 2 k , s ) = 2 (2 πi ) 2 k Γ(2 k ) ζ ( s ) ζ ( s − 2 k + 1) On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 9 / 47

Modularity of Ap´ ery-like numbers • The Ap´ ery numbers 1 , 5 , 73 , 1145 , . . . n � 2 � n + k � 2 � n � A ( n ) = k k k =0 satisfy � η ( τ ) η (6 τ ) � 12 n η 7 (2 τ ) η 7 (3 τ ) � = A ( n ) . η 5 ( τ ) η 5 (6 τ ) η (2 τ ) η (3 τ ) n � 0 modular form modular function On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 10 / 47

Modularity of Ap´ ery-like numbers • The Ap´ ery numbers 1 , 5 , 73 , 1145 , . . . n � 2 � n + k � 2 � n � A ( n ) = k k k =0 satisfy � η ( τ ) η (6 τ ) � 12 n η 7 (2 τ ) η 7 (3 τ ) � = A ( n ) . η 5 ( τ ) η 5 (6 τ ) η (2 τ ) η (3 τ ) n � 0 modular form modular function Not at all evidently, such a modular parametrization exists for FACT all known Ap´ ery-like numbers! On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 10 / 47

Personal encounter in the wild I: Random walks • n steps in the plane (length 1 , random direction) On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 11 / 47

Personal encounter in the wild I: Random walks • n steps in the plane (length 1 , random direction) On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 11 / 47

Personal encounter in the wild I: Random walks • n steps in the plane (length 1 , random direction) On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 11 / 47

Personal encounter in the wild I: Random walks • n steps in the plane (length 1 , random direction) On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 11 / 47

Personal encounter in the wild I: Random walks • n steps in the plane (length 1 , random direction) On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 11 / 47

Personal encounter in the wild I: Random walks • n steps in the plane (length 1 , random direction) On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 11 / 47

Personal encounter in the wild I: Random walks • n steps in the plane (length 1 , random direction) d On the ubiquity of modular forms and Ap´ ery-like numbers Armin Straub 11 / 47