Moments of elliptic integrals and critical $$L$$ L -values

Abstract

We compute the critical L-values of some weight 3, 4, or 5 modular forms, by transforming them into integrals of the complete elliptic integral K . In doing so, we prove closed-form formulas for some moments of K 3 . Many of our L-values can be expressed in terms of Gamma functions, and we also obtain new lattice sum evaluations.

On a q-deformation of modular forms

“…(see [28,Theorem 5]), where L(f 1 , s) denotes the Dirichlet L-function of (2.1). This relationship between the hypergeometric series and modular form is somewhat deeper because of the chain of related congruences…”

Interpolated Sequences and Critical L-Values of Modular Forms

“…is the unique normalized Hecke eigenform in S 4 (Γ 0 (8)) and η(τ ) is the Dedekind eta function. We note that, expressing the left-hand side as a hypergeometric series, the identity (2) was previously established by Rogers, Wan and Zucker [30]. The evaluation (2) can be seen as a continuous counterpart to the congruence…”

“…upon taking τ = i, in which case λ(i) = 1 2 and θ 3 (i) 2 = Γ 2 (1/4) 2π 3/2 . On the other hand, it is shown by Rogers, Wan and Zucker [30] that…”

Feynman integrals and critical modular $L$-values

“…The proofs of Theorem 1 and Theorem 2, given in Section 4, depend on the modular realization of the integral I(t) given in [8] and recent results on critical L-values due to Rogers, Wan, and Zucker [29]. Other supplementary results which are required for our proofs will be proved in Section 2 and Section 3.…”

“…When t = 64, 8, −8, the K3 surface X t is singular and the order of complex multiplication corresponding to t has discriminant D = −15, −8, −24, respectively. In theory, for each t given above, one can evaluate I(t) at the corresponding CM point τ using (29) and identify the term ̟ 2 (−1/6τ ) with a critical L-value of some weight 3 cusp form using [29,Thm. 5].…”