Interpolated sequences and critical L -values of modular forms Minisymposium on Computer Algebra and Special Functions OPSFA-15, Hagenberg, Austria Armin Straub July 25, 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 / 11
A victory for the French peasant. . . ∗ • 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 . ζ (3) = � ∞ 1 THM n 3 is irrational. n =1 Ap´ ery ’78 ∗ Someone’s “sour comment” after Henri Cohen’s report on Ap´ ery’s proof at the ’78 ICM in Helsinki. Interpolated sequences and critical L -values of modular forms Armin Straub 2 / 11
A victory for the French peasant. . . ∗ • 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 . 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 k =0 j =1 m =1 Then, B ( n ) A ( n ) → ζ (3) . But too fast for ζ (3) to be rational. ∗ Someone’s “sour comment” after Henri Cohen’s report on Ap´ ery’s proof at the ’78 ICM in Helsinki. Interpolated sequences and critical L -values of modular forms Armin Straub 2 / 11
A victory for the French peasant. . . ∗ • 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 . ζ (3) = � ∞ 1 THM n 3 is irrational. n =1 Ap´ ery ’78 “ After a few days of fruitless effort the specific problem was Alfred van der Poorten — A proof that Euler missed. . . (1979) ” mentioned to Don Zagier (Bonn), and with irritating speed he showed that indeed the sequence satisfies the recurrence. ∗ Someone’s “sour comment” after Henri Cohen’s report on Ap´ ery’s proof at the ’78 ICM in Helsinki. Interpolated sequences and critical L -values of modular forms Armin Straub 2 / 11
A victory for the French peasant. . . ∗ • 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 . ζ (3) = � ∞ 1 THM n 3 is irrational. n =1 Ap´ ery ’78 “ After a few days of fruitless effort the specific problem was Alfred van der Poorten — A proof that Euler missed. . . (1979) ” mentioned to Don Zagier (Bonn), and with irritating speed he showed that indeed the sequence satisfies the recurrence. Nowadays, there are excellent implementations of this creative telescoping , including: • HolonomicFunctions by Koutschan (Mathematica) • Sigma by Schneider (Mathematica) • ore algebra by Kauers, Jaroschek, Johansson, Mezzarobba (Sage) (These are just the ones I use on a regular basis. . . ) ∗ Someone’s “sour comment” after Henri Cohen’s report on Ap´ ery’s proof at the ’78 ICM in Helsinki. Interpolated sequences and critical L -values of modular forms Armin Straub 2 / 11
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 / 11
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 / 11
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 4 / 11
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 4 / 11
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 5 / 11
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 5 / 11
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 5 / 11
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 5 / 11
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