Interpolated sequences and critical L -values of modular forms - PowerPoint PPT Presentation

Interpolated sequences and critical L -values of modular forms Special Session on Partition Theory and Related Topics AMS Fall Southeastern Sectional Meeting, Gainesville Armin Straub November 3, 2019 University of South Alabama n � 2 � n + k � 2 � n f ( τ ) = η (2 τ ) 4 η (4 τ ) 4 = � � α n q n A ( n ) = k k k =0 n � 1 1 , 5 , 73 , 1445 , 33001 , 819005 , 21460825 , . . . A ( p − 1 (mod p 2 ) 2 ) ≡ α p Joint work with: A ( − 1 2 ) = 16 π 2 L ( f, 2) Robert Osburn (University College Dublin) Interpolated sequences and critical L -values of modular forms Armin Straub 1 / 12

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 . 1 THM � n 3 is irrational. ζ (3) = Ap´ ery ’78 n � 1 Interpolated sequences and critical L -values of modular forms Armin Straub 2 / 12

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 . 1 THM � n 3 is irrational. ζ (3) = Ap´ ery ’78 n � 1 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 k =0 j =1 m =1 Then, B ( n ) A ( n ) → ζ (3) . But too fast for ζ (3) to be rational. Interpolated sequences and critical L -values of modular forms Armin Straub 2 / 12

Zagier’s search and Ap´ ery-like numbers n � 2 � n + k � n � � • The Ap´ ery numbers B ( n ) = for ζ (2) satisfy k k k =0 ( n + 1) 2 u n +1 = ( an 2 + an + b ) u n − cn 2 u n − 1 , ( a, b, c ) = (11 , 3 , − 1) . Q Are there other tuples ( a, b, c ) for which the solution defined by Beukers u − 1 = 0 , u 0 = 1 is integral? Interpolated sequences and critical L -values of modular forms Armin Straub 3 / 12

Zagier’s search and Ap´ ery-like numbers n � 2 � n + k � n � � • The Ap´ ery numbers B ( n ) = for ζ (2) satisfy k k k =0 ( n + 1) 2 u n +1 = ( an 2 + an + b ) u n − cn 2 u n − 1 , ( a, b, c ) = (11 , 3 , − 1) . Q Are there other tuples ( a, b, c ) for which the solution defined by Beukers u − 1 = 0 , u 0 = 1 is integral? • Apart from degenerate cases, Zagier found 6 sporadic integer solutions: * C ∗ ( n ) * C ∗ ( n ) n n � 3 � 2 � n + k � n � n � A � D � k k n k =0 k =0 � n ⌊ n/ 3 ⌋ n � (3 k )! � n �� 2 k �� 2( n − k ) � B � ( − 1) k 3 n − 3 k E � k ! 3 3 k k k n − k k =0 k =0 n n � 2 � 2 k � n � � n � C � F � ( − 1) k 8 n − k C A ( k ) k k k k =0 k =0 Interpolated sequences and critical L -values of modular forms Armin Straub 3 / 12

Modularity of Ap´ ery-like numbers • The Ap´ ery numbers 1 , 5 , 73 , 1145 , . . . n � 2 � n + k � 2 � n � A ( n ) = k k satisfy k =0 � η 12 ( τ ) η 12 (6 τ ) � n η 7 (2 τ ) η 7 (3 τ ) � = A ( n ) . η 5 ( τ ) η 5 (6 τ ) η 12 (2 τ ) η 12 (3 τ ) n � 0 modular form modular function 1 + 5 q + 13 q 2 + 23 q 3 + O ( q 4 ) q − 12 q 2 + 66 q 3 + O ( q 4 ) Interpolated sequences and critical L -values of modular forms Armin Straub 4 / 12

Modularity of Ap´ ery-like numbers • The Ap´ ery numbers 1 , 5 , 73 , 1145 , . . . n � 2 � n + k � 2 � n � A ( n ) = k k satisfy k =0 � η 12 ( τ ) η 12 (6 τ ) � n η 7 (2 τ ) η 7 (3 τ ) � = A ( n ) . η 5 ( τ ) η 5 (6 τ ) η 12 (2 τ ) η 12 (3 τ ) n � 0 modular form modular function 1 + 5 q + 13 q 2 + 23 q 3 + O ( q 4 ) q − 12 q 2 + 66 q 3 + O ( q 4 ) Not at all evidently, such a modular parametrization exists for FACT all known Ap´ ery-like numbers! • Context: f ( τ ) modular form of weight k x ( τ ) modular function y ( x ) such that y ( x ( τ )) = f ( τ ) Then y ( x ) satisfies a linear differential equation of order k + 1 . Interpolated sequences and critical L -values of modular forms Armin Straub 4 / 12

L -value interpolations THM For primes p > 2 , the Ap´ ery numbers for ζ (3) satisfy Ahlgren– Ono A ( p − 1 (mod p 2 ) , 2000 2 ) ≡ a f ( p ) with f ( τ ) = η (2 τ ) 4 η (4 τ ) 4 = a f ( n ) q n ∈ S 4 (Γ 0 (8)) . � n � 1 conjectured (and proved modulo p ) by Beukers ’87 Interpolated sequences and critical L -values of modular forms Armin Straub 5 / 12

L -value interpolations THM For primes p > 2 , the Ap´ ery numbers for ζ (3) satisfy Ahlgren– Ono A ( p − 1 (mod p 2 ) , 2000 2 ) ≡ a f ( p ) with f ( τ ) = η (2 τ ) 4 η (4 τ ) 4 = a f ( n ) q n ∈ S 4 (Γ 0 (8)) . � n � 1 conjectured (and proved modulo p ) by Beukers ’87 THM A ( − 1 2 ) = 16 π 2 L ( f, 2) Zagier 2016 ∞ � 2 � x + k � 2 � x • Here, A ( x ) = � is absolutely convergent for x ∈ C . k k k =0 • Predicted by Golyshev based on motivic considerations, the connection of the Ap´ ery numbers with the double covering of a family of K3 surfaces, and the Tate conjecture. D. Zagier Arithmetic and topology of differential equations Proceedings of the 2016 ECM, 2017 Interpolated sequences and critical L -values of modular forms Armin Straub 5 / 12

L -value interpolations, cont’d • Zagier found 6 sporadic integer solutions C ∗ ( n ) to: ∗ one of A - F ( n + 1) 2 u n +1 = ( an 2 + an + b ) u n − cn 2 u n − 1 u − 1 = 0 , u 0 = 1 n � 1 γ n, ∗ q n , so that There exists a weight 3 newform f ∗ ( τ ) = � THM 1985 - C ∗ ( p − 1 2019 2 ) ≡ γ p, ∗ (mod p ) . Interpolated sequences and critical L -values of modular forms Armin Straub 6 / 12

L -value interpolations, cont’d • Zagier found 6 sporadic integer solutions C ∗ ( n ) to: ∗ one of A - F ( n + 1) 2 u n +1 = ( an 2 + an + b ) u n − cn 2 u n − 1 u − 1 = 0 , u 0 = 1 n � 1 γ n, ∗ q n , so that There exists a weight 3 newform f ∗ ( τ ) = � THM 1985 - C ∗ ( p − 1 2019 2 ) ≡ γ p, ∗ (mod p ) . • C , D proved by Beukers–Stienstra (’85); A follows from their work • E proved using a result Verrill (’10); B through p -adic analysis • F conjectured by Osburn–S and proved by Kazalicki (’19) using Atkin–Swinnerton-Dyer congruences for non-congruence cusp forms Interpolated sequences and critical L -values of modular forms Armin Straub 6 / 12

L -value interpolations, cont’d • Zagier found 6 sporadic integer solutions C ∗ ( n ) to: ∗ one of A - F ( n + 1) 2 u n +1 = ( an 2 + an + b ) u n − cn 2 u n − 1 u − 1 = 0 , u 0 = 1 n � 1 γ n, ∗ q n , so that There exists a weight 3 newform f ∗ ( τ ) = � THM 1985 - C ∗ ( p − 1 2019 2 ) ≡ γ p, ∗ (mod p ) . • C , D proved by Beukers–Stienstra (’85); A follows from their work • E proved using a result Verrill (’10); B through p -adic analysis • F conjectured by Osburn–S and proved by Kazalicki (’19) using Atkin–Swinnerton-Dyer congruences for non-congruence cusp forms For ∗ one of A - F , except E , there is α ∗ ∈ Z such that THM Osburn S ’18 2 ) = α ∗ C ∗ ( − 1 π 2 L ( f ∗ , 2) . Interpolated sequences and critical L -values of modular forms Armin Straub 6 / 12

L -value interpolations, cont’d • Zagier found 6 sporadic integer solutions C ∗ ( n ) to: ∗ one of A - F ( n + 1) 2 u n +1 = ( an 2 + an + b ) u n − cn 2 u n − 1 u − 1 = 0 , u 0 = 1 n � 1 γ n, ∗ q n , so that There exists a weight 3 newform f ∗ ( τ ) = � THM 1985 - C ∗ ( p − 1 2019 2 ) ≡ γ p, ∗ (mod p ) . • C , D proved by Beukers–Stienstra (’85); A follows from their work • E proved using a result Verrill (’10); B through p -adic analysis • F conjectured by Osburn–S and proved by Kazalicki (’19) using Atkin–Swinnerton-Dyer congruences for non-congruence cusp forms For ∗ one of A - F , except E , there is α ∗ ∈ Z such that THM Osburn S ’18 2 ) = α ∗ C ∗ ( − 1 π 2 L ( f ∗ , 2) . x = − 1 / 2 C E ( x ) = 6 For sequence E , res π 2 L ( f E , 1) . Interpolated sequences and critical L -values of modular forms Armin Straub 6 / 12

L -value interpolations, cont’d 2 ) = α ∗ C ∗ ( − 1 π 2 L ( f ∗ , 2) * C ∗ ( n ) f ∗ ( τ ) N ∗ CM α ∗ n � 3 η (4 τ ) 5 η (8 τ ) 5 √ � n A � 32 8 Q ( − 2) η (2 τ ) 2 η (16 τ ) 2 k k =0 � n ⌊ n/ 3 ⌋ √ � (3 k )! η (4 τ ) 6 B � ( − 1) k 3 n − 3 k 16 8 Q ( − 1) 3 k k ! 3 k =0 n � 2 � 2 k √ � n � C � η (2 τ ) 3 η (6 τ ) 3 12 12 Q ( − 3) k k k =0 n � 2 � n + k √ � n � η (4 τ ) 6 D � 16 16 Q ( − 1) k n k =0 n √ � n �� 2 k �� 2( n − k ) � E � η ( τ ) 2 η (2 τ ) η (4 τ ) η (8 τ ) 2 8 6 Q ( − 2) k k n − k k =0 n √ � n � q − 2 q 2 + 3 q 3 + . . . F � 24 6 ( − 1) k 8 n − k C A ( k ) Q ( − 6) k k =0 Interpolated sequences and critical L -values of modular forms Armin Straub 7 / 12

Interpolating sequences What is the proper way of defining C ( − 1 Q 2 ) ? � ∞ EG t x e − t d t . a ( n ) = n ! is interpolated by a ( x ) = Γ( x + 1) = 0 Interpolated sequences and critical L -values of modular forms Armin Straub 8 / 12