A modular supercongruence for 6 F 5 : An Ap ery-like story - PowerPoint PPT Presentation

A modular supercongruence for 6 F 5 : An Ap´ ery-like story Palmetto Number Theory Series (PANTS XXVIII) University of Tennessee Armin Straub September 17, 2017 University of South Alabama ˆ 1 ˇ ˙ ˇ 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 ˇ ” b p p q p mod p 3 q 2 6 F 5 ˇ 1 1 , 1 , 1 , 1 , 1 p ´ 1 Joint work with: Robert Osburn Wadim Zudilin (University College Dublin) (University of Newcastle/ Radboud Universiteit) A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 1 / 15

Ap´ ery numbers and the irrationality of ζ p 3 q • The Ap´ ery numbers 1 , 5 , 73 , 1445 , . . . ˆ n ˙ 2 ˆ n ` k ˙ 2 n ÿ A p n q “ k k satisfy k “ 0 p n ` 1 q 3 A p n ` 1 q “ p 2 n ` 1 qp 17 n 2 ` 17 n ` 5 q A p n q ´ n 3 A p n ´ 1 q . A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 2 / 15

Ap´ ery numbers and the irrationality of ζ p 3 q • The Ap´ ery numbers 1 , 5 , 73 , 1445 , . . . ˆ n ˙ 2 ˆ n ` k ˙ 2 n ÿ A p n q “ k k satisfy k “ 0 p n ` 1 q 3 A p n ` 1 q “ p 2 n ` 1 qp 17 n 2 ` 17 n ` 5 q A p n q ´ n 3 A p n ´ 1 q . ζ p 3 q “ ř 8 1 THM n 3 is irrational. n “ 1 Ap´ ery ’78 The same recurrence is satisfied by the “near”-integers proof ˙ 2 ˜ n ¸ ˆ n ˙ 2 ˆ n ` k n k ÿ ÿ ÿ p´ 1 q m ´ 1 1 2 m 3 ` n ˘` n ` m ˘ B p n q “ j 3 ` . k k m “ 1 k “ 0 j “ 1 m m Then, B p n q A p n q Ñ ζ p 3 q . But too fast for ζ p 3 q to be rational. A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 2 / 15

Hypergeometric series Trivially, the Ap´ ery numbers have the representation EG ˆ n ˙ 2 ˆ n ` k ˙ 2 ÿ n A p n q “ k k k “ 0 ˆ ´ n, ´ n, n ` 1 , n ` 1 ˇ ˙ ˇ ˇ “ 4 F 3 ˇ 1 . 1 , 1 , 1 • Here, 4 F 3 is a hypergeometric series: ˆ a 1 , . . . , a p ˇ ˙ 8 ÿ ˇ z n p a 1 q k ¨ ¨ ¨ p a p q k ˇ p F q ˇ z “ n ! . p b 1 q k ¨ ¨ ¨ p b q q k b 1 , . . . , b q k “ 0 A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 3 / 15

Hypergeometric series Trivially, the Ap´ ery numbers have the representation EG ˆ n ˙ 2 ˆ n ` k ˙ 2 ÿ n A p n q “ k k k “ 0 ˆ ´ n, ´ n, n ` 1 , n ` 1 ˇ ˙ ˇ ˇ “ 4 F 3 ˇ 1 . 1 , 1 , 1 • Here, 4 F 3 is a hypergeometric series: ˆ a 1 , . . . , a p ˇ ˙ 8 ÿ ˇ z n p a 1 q k ¨ ¨ ¨ p a p q k ˇ p F q ˇ z “ n ! . p b 1 q k ¨ ¨ ¨ p b q q k b 1 , . . . , b q k “ 0 • Similary, we have the truncated hypergeometric series ˆ a 1 , . . . , a p ˇ ˙ ÿ M ˇ z n p a 1 q k ¨ ¨ ¨ p a p q k ˇ p F q ˇ z “ n ! . p b 1 q k ¨ ¨ ¨ p b q q k b 1 , . . . , b q M k “ 0 A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 3 / 15

A first connection to modular forms • The Ap´ ery numbers A p n q satisfy 1 , 5 , 73 , 1145 , . . . ˆ η 12 p τ q η 12 p 6 τ q ˙ n ÿ η 7 p 2 τ q η 7 p 3 τ q “ A p n q . η 5 p τ q η 5 p 6 τ q η 12 p 2 τ q η 12 p 3 τ q n ě 0 modular form modular function 1 ` 5 q ` 13 q 2 ` 23 q 3 ` O p q 4 q q ´ 12 q 2 ` 66 q 3 ` O p q 4 q q “ e 2 πiτ A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 4 / 15

A first connection to modular forms • The Ap´ ery numbers A p n q satisfy 1 , 5 , 73 , 1145 , . . . ˆ η 12 p τ q η 12 p 6 τ q ˙ n ÿ η 7 p 2 τ q η 7 p 3 τ q “ A p n q . η 5 p τ q η 5 p 6 τ q η 12 p 2 τ q η 12 p 3 τ q n ě 0 modular form modular function 1 ` 5 q ` 13 q 2 ` 23 q 3 ` O p q 4 q q ´ 12 q 2 ` 66 q 3 ` O p q 4 q q “ e 2 πiτ ? EG As a consequence, with z “ 1 ´ 34 x ` x 2 , ˆ 1 ˇ ˙ ÿ 2 , 1 2 , 1 ˇ 17 ´ x ´ z 1024 x A p n q x n “ ˇ ? 2 2 p 1 ` x ` z q 3 { 2 3 F 2 ˇ ´ . p 1 ´ x ` z q 4 1 , 1 4 n ě 0 For contrast, the Ap´ ery numbers are the diagonal coefficients of EG S 2014 1 . p 1 ´ x 1 ´ x 2 qp 1 ´ x 3 ´ x 4 q ´ x 1 x 2 x 3 x 4 A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 4 / 15

A second connection to modular forms For primes p ą 2 , the Ap´ ery numbers satisfy THM Ahlgren– ˆ p ´ 1 ˙ Ono ’00 p mod p 2 q A ” a p p q 2 where a p n q are the Fourier coefficients of the Hecke eigenform 8 ÿ η p 2 τ q 4 η p 4 τ q 4 “ a p n q q n n “ 1 of weight 4 for the modular group Γ 0 p 8 q . • conjectured by Beukers ’87, and proved modulo p • similar congruences modulo p for other Ap´ ery-like numbers • by the Deligne–Weil bounds, | a p p q| ă 2 p 3 { 2 , the congruence determines the modular form A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 5 / 15

The “super” in these congruences Fourier coefficients a p p q Ap´ ery sequence A p n q A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 6 / 15

The “super” in these congruences Fourier coefficients a p p q Œ point counts on modular curves modulo p Œ character sums Œ Gaussian hypergeometric series Œ harmonic sums Œ truncated hypergeometric series Œ Ap´ ery sequence A p n q A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 6 / 15

The “super” in these congruences Fourier coefficients a p p q Œ point counts on modular curves modulo p Œ equalities character sums Œ Gaussian hypergeometric series Œ harmonic sums “easy” mod p Œ truncated hypergeometric series Œ Ap´ ery sequence A p n q A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 6 / 15

Kilbourn’s extension of the Ahlgren–Ono supercongruence ˆ 1 ˇ ˙ THM 2 , 1 2 , 1 2 , 1 ˇ ˇ p mod p 3 q , Kilbourn 2 ” a p p q 4 F 3 ˇ 1 2006 1 , 1 , 1 p ´ 1 for primes p ą 2 . Again, a p n q are the Fourier coefficients of 8 ÿ η p 2 τ q 4 η p 4 τ q 4 “ a p n q q n . n “ 1 A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 7 / 15

Kilbourn’s extension of the Ahlgren–Ono supercongruence ˆ 1 ˇ ˙ THM 2 , 1 2 , 1 2 , 1 ˇ ˇ p mod p 3 q , Kilbourn 2 ” a p p q 4 F 3 ˇ 1 2006 1 , 1 , 1 p ´ 1 for primes p ą 2 . Again, a p n q are the Fourier coefficients of 8 ÿ η p 2 τ q 4 η p 4 τ q 4 “ a p n q q n . n “ 1 • This result proved the first of 14 related supercongruences conjectured by Rodriguez-Villegas (2001) between • truncated hypergeometric series 4 F 3 and • Fourier coefficients of modular forms of weight 4 . • Despite considerable progress, 11 of these remain open. McCarthy (2010), Fuselier–McCarthy (2016) prove one each; McCarthy (2010) proves “half” of each of the 14. 2017/5/4 : Preprint by Long–Tu–Yui–Zudilin proving all 14 congruences. • The 14 supercongruence conjectures were complemented with 4 ` 4 conjectures for 2 F 1 and 3 F 2 . A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 7 / 15

A supercongruence for 6 F 5 ˆ 1 ˇ ˙ THM 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 ˇ ˇ p mod p 3 q , 2 OSZ 6 F 5 ˇ 1 ” b p p q 2017 1 , 1 , 1 , 1 , 1 p ´ 1 for primes p ą 2 . Here, b p n q are the Fourier coefficients of ÿ 8 η p τ q 8 η p 4 τ q 4 ` 8 η p 4 τ q 12 “ b p n q q n , n “ 1 the unique newform in S 6 p Γ 0 p 8 qq . • Conjectured by Mortenson based on numerical evidence, which further suggests it holds modulo p 5 . A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 8 / 15

A supercongruence for 6 F 5 ˆ 1 ˇ ˙ THM 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 ˇ ˇ p mod p 3 q , 2 OSZ 6 F 5 ˇ 1 ” b p p q 2017 1 , 1 , 1 , 1 , 1 p ´ 1 for primes p ą 2 . Here, b p n q are the Fourier coefficients of ÿ 8 η p τ q 8 η p 4 τ q 4 ` 8 η p 4 τ q 12 “ b p n q q n , n “ 1 the unique newform in S 6 p Γ 0 p 8 qq . • Conjectured by Mortenson based on numerical evidence, which further suggests it holds modulo p 5 . • A result of Frechette, Ono and Papanikolas expresses the b p p q in terms of Gaussian hypergeometric functions. • Osburn and Schneider determined the resulting Gaussian hypergeometric functions modulo p 3 in terms of sums involving harmonic sums. A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 8 / 15

A brief impression of the available ingredients THM In terms of Gaussian hypergeometric series, b p p q “ ´ p 56 F 5 p 1 q ` p 44 F 3 p 1 q ` p 32 F 1 p 1 q ` p 2 . • Conjectured by Koike; proven by Frechette, Ono and Papanikolas (2004). • Here, φ p is the quadratic character mod p , ǫ p the trivial character, and ˇ ˆ ˙ ˇ φ p , φ p , . . . , φ p ˇ n ` 1 F n p x q “ n ` 1 F n ˇ x , ǫ p , . . . , ǫ p p the finite field version of ˆ 1 ˇ ˙ ˇ 2 , 1 2 , . . . , 1 ˇ n ` 1 F n 2 ˇ x . 1 , . . . , 1 A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 9 / 15

A brief impression of the available ingredients THM In terms of Gaussian hypergeometric series, b p p q “ ´ p 56 F 5 p 1 q ` p 44 F 3 p 1 q ` p 32 F 1 p 1 q ` p 2 . • Conjectured by Koike; proven by Frechette, Ono and Papanikolas (2004). • Here, φ p is the quadratic character mod p , ǫ p the trivial character, and ˇ ˆ ˙ ˇ φ p , φ p , . . . , φ p ˇ n ` 1 F n p x q “ n ` 1 F n ˇ x , ǫ p , . . . , ǫ p p the finite field version of ˆ 1 ˇ ˙ ˇ 2 , 1 2 , . . . , 1 ˇ n ` 1 F n 2 ˇ x . 1 , . . . , 1 • Since p nn ` 1 F n p x q P Z , it follows easily that ˆ 1 ˇ ˙ ˇ 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 ˇ b p p q ” ´ p 5 6 F 5 p 1 q ” 6 F 5 2 p mod p q . ˇ 1 1 , 1 , 1 , 1 , 1 p ´ 1 A modular supercongruence for 6 F 5 : An Ap´ ery-like story Armin Straub 9 / 15