Congruences connecting modular forms and truncated hypergeometric series Minisymposium on Symbolic Combinatorics 2017 SIAM Conference on Applied Algebraic Geometry Armin Straub July 31, 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) Congruences connecting modular forms and truncated hypergeometric series Armin Straub 1 / 19
The wonderful world of A “ B ˆ n ˙ 2 ˆ n ` k ˙ˆ 2 k ˙ ˆ 3 n ` 1 ˙ˆ n ` k ˙ 3 EG ÿ n ÿ n p´ 1 q n ` k “ k k n n ´ k k k “ 0 k “ 0 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 2 / 19
The wonderful world of A “ B ˆ n ˙ 2 ˆ n ` k ˙ˆ 2 k ˙ ˆ 3 n ` 1 ˙ˆ n ` k ˙ 3 EG ÿ n ÿ n p´ 1 q n ` k “ k k n n ´ k k k “ 0 k “ 0 ˆ n ˙ 2 ˆ n ` k ˙ 2 EG ÿ n Ap´ ery ’78 u n “ k k k “ 0 satisfies the difference equation p n ` 1 q 3 u n ` 1 “ p 2 n ` 1 qp 17 n 2 ` 17 n ` 5 q u n ´ n 3 u n ´ 1 . Congruences connecting modular forms and truncated hypergeometric series Armin Straub 2 / 19
The wonderful world of A “ B ˆ n ˙ 2 ˆ n ` k ˙ˆ 2 k ˙ ˆ 3 n ` 1 ˙ˆ n ` k ˙ 3 EG ÿ n ÿ n p´ 1 q n ` k “ k k n n ´ k k k “ 0 k “ 0 ˙ 2 ˜ n ¸ ˆ n ˙ 2 ˆ n ` k EG ÿ n ÿ ÿ k p´ 1 q m ´ 1 1 Ap´ ery ’78 u n “ j 3 ` 2 m 3 ` n ˘` n ` m ˘ k k k “ 0 j “ 1 m “ 1 m m satisfies the difference equation p n ` 1 q 3 u n ` 1 “ p 2 n ` 1 qp 17 n 2 ` 17 n ` 5 q u n ´ n 3 u n ´ 1 . Congruences connecting modular forms and truncated hypergeometric series Armin Straub 2 / 19
The wonderful world of A “ B ˆ n ˙ 2 ˆ n ` k ˙ˆ 2 k ˙ ˆ 3 n ` 1 ˙ˆ n ` k ˙ 3 EG ÿ n ÿ n p´ 1 q n ` k “ k k n n ´ k k k “ 0 k “ 0 ˙ 2 ˜ n ¸ ˆ n ˙ 2 ˆ n ` k EG ÿ n ÿ ÿ k p´ 1 q m ´ 1 1 Ap´ ery ’78 u n “ j 3 ` 2 m 3 ` n ˘` n ` m ˘ k k k “ 0 j “ 1 m “ 1 m m satisfies the difference equation p n ` 1 q 3 u n ` 1 “ p 2 n ` 1 qp 17 n 2 ` 17 n ` 5 q u n ´ n 3 u n ´ 1 . ˆ n ˙ 2 ˆ n ` k ˙ 2 ` EG n ÿ ˘ 1 ´ 2 k p 2 H k ´ H n ` k ´ H n ´ k q “ 1 k k k “ 0 Scott Ahlgren, Shalosh B. Ekhad, Ken Ono, Doron Zeilberger A binomial coefficient identity associated to a conjecture of Beukers Electronic Journal of Combinatorics, Vol. 5, 1998, #R10 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 2 / 19
The wonderful world of A ” B ˆ n ˙ 2 ˆ n ` k ˙ 2 ` EG ÿ n ˘ again 1 ´ 2 k p 2 H k ´ H n ` k ´ H n ´ k q “ 1 k k k “ 0 • Below, p ą 2 is a prime and n “ p p ´ 1 q{ 2 . ˆ n ˙ 3 ˆ n ` k ˙ 3 ` EG ÿ n ˘ p´ 1 q k OSZ 1 ´ 3 k p 2 H k ´ H n ` k ´ H n ´ k q 2017 k k k “ 0 ˆ n ˙ 2 ˆ n ` k ˙ 2 n ÿ p mod p 2 q ” k k k “ 0 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 3 / 19
The wonderful world of A ” B ˆ n ˙ 2 ˆ n ` k ˙ 2 ` EG ÿ n ˘ again 1 ´ 2 k p 2 H k ´ H n ` k ´ H n ´ k q “ 1 k k k “ 0 • Below, p ą 2 is a prime and n “ p p ´ 1 q{ 2 . ˆ n ˙ 3 ˆ n ` k ˙ 3 ` EG ÿ n ˘ p´ 1 q k OSZ 1 ´ 3 k p 2 H k ´ H n ` k ´ H n ´ k q 2017 k k k “ 0 ˆ n ˙ 2 ˆ n ` k ˙ 2 n ÿ p mod p 2 q ” k k k “ 0 ˆ n ˙ 2 ˆ n ` k ˙ 2 ˆ n ˙ 2 ˆ n ` k ˙ˆ 2 k ˙ EG ÿ n ÿ n ” p´ 1 q n p mod p 2 q OSZ 2017 k k k k n k “ 0 k “ 0 • We have no general algorithmic approach to such congruences. • Instead, we had to find suitable intermediate identities . Congruences connecting modular forms and truncated hypergeometric series Armin Straub 3 / 19
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 . Congruences connecting modular forms and truncated hypergeometric series Armin Straub 4 / 19
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. Congruences connecting modular forms and truncated hypergeometric series Armin Straub 4 / 19
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 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 5 / 19
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 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 5 / 19
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 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 6 / 19
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 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 6 / 19
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 p p q A 2 where a p n q are the Fourier coefficients of the Hecke eigenform ÿ 8 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 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 7 / 19
The “super” in these congruences Fourier coefficients a p p q Ap´ ery sequence A p n q Congruences connecting modular forms and truncated hypergeometric series Armin Straub 8 / 19
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 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 8 / 19
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 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 8 / 19
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 Congruences connecting modular forms and truncated hypergeometric series Armin Straub 9 / 19
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