Hypergeometric Functions over Finite Fields and Modular Forms: A Survey and New Conjectures

On some hypergeometric supercongruence conjectures of Long

“…The j = 1 case of Proposition 6.2 was conjectured by Dawsey and McCarthy in [16] and we provide an alternative proof for the j = 2, 4 cases, which were established by McCarthy-Papanikolas [38] and Ahlgren [1], respectively. A similar argument is used to establish the following result for the G2 family.…”

“…• H p In our method we use supercongruences like (1.2) and obtain formulae of L(f, 1) and L(f, 2) like (1.3) above. These two H p -function identities above resolve conjectures made by Dawsey and McCarthy in [16], which originated in studying Paley graphs [15]. The main focus of this paper is to introduce the method while explaining why it works and demonstrating how to use it.…”

“…These authors later carried out a more extensive search for more connections between values of H p (HD; 1) and coefficients of weight three newforms for primes on certain arithmetic progressions. In [16] Dawsey and McCarthy provide fifteen conjectures between H p (HD; 1) and the coefficients of weight three newforms, including the two cases that appeared in [15]. We establish the conjectures of Dawsey and McCarthy for the four cases labeled as 3-6 in Table 2 of [16] using Theorem 2.1.…”

“…In [16] Dawsey and McCarthy provide fifteen conjectures between H p (HD; 1) and the coefficients of weight three newforms, including the two cases that appeared in [15]. We establish the conjectures of Dawsey and McCarthy for the four cases labeled as 3-6 in Table 2 of [16] using Theorem 2.1.…”

“…This result leads to the analogous K 3 function. The K 3 function is involved in our proof for cases 4-6 of the conjectures made by Dawsey and McCarthy [16]. Further properties of the K 3 function will appear in the forthcoming paper of Grove.…”