Ling Long

In this paper we study the compatible family of degree- 4 4 Scholl representations ρ ℓ \rho _{\ell } associated with a space S S of weight κ > 2 \kappa > 2 noncongruence cusp forms satisfying Quaternion Multiplication over a biquadratic extension of Q \mathbb {Q} . It is shown that ρ ℓ \rho _\ell is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for G L 4 \mathrm {GL}_4 over Q \mathbb {Q} . Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in S S and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.

We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of algebraic varieties and exhibit Atkin and Swinnerton-Dyer type congruences. In particular, when r=3, they are related to K3 surfaces. For special values of \lambda, with s=1 and r=3, our congruences are stronger than what can be predicted by the theory of formal groups because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Rodriguez-Villegas for the \lambda=1 case and confirm some other supercongruence conjectures at special values of \lambda.

: Let p be a prime congruent to 1 or 3 modulo 8 so that the equation is solvable in integers. In this paper, we obtain closed-form expressions for a and b in terms of Jacobsthal sums. This is analogous to a classical identity of Jacobsthal.

This paper is concerned with a compatible family of 4-dimensional ℓ \ell -adic representations ρ ℓ \rho _{\ell } of G Q := G a l ( Q ¯ / Q ) G_{\mathbb {Q}}:=\mathrm {Gal}(\overline {\mathbb {Q}}/\mathbb {Q}) attached to the space of weight-3 cuspforms S 3 ( Γ ) S_3(\Gamma ) on a noncongruence subgroup Γ ⊂ S L 2 ( Z ) \Gamma \subset \mathrm {SL}_2(\mathbb Z) . For this representation we prove that: [1.] It is automorphic: the L L -function L ( s , ρ ℓ ∨ ) L(s, \rho _{\ell }^{\vee }) agrees with the L L -function for an automorphic form for GL 4 ( A Q ) \text {GL}_4(\mathbb A_{{\mathbb Q}}) , where ρ ℓ ∨ \rho _{\ell }^{\vee } is the dual of ρ ℓ \rho _{\ell } . [2.] For each prime p ≥ 5 p \ge 5 there is a basis h p = { h p + , h p − } h_p = \{ h_p ^+, h_p ^- \} of S 3 ( Γ ) S_3(\Gamma ) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the q q -expansion coefficients of a newform f f of level 432. The structure of this basis depends on the class of p p modulo 12. The key point is that the representation ρ ℓ \rho _{\ell } admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long.

In this article, we provide an application of hypergeometric evaluation identities, including a strange valuation of Gosper, to prove several supercongruences related to special valuations of truncated hypergeometric series. In particular, we prove a conjecture of van Hamme.