Four-variable $p$-adic triple product $L$-functions and the trivial zero conjecture

Abstract

We construct the four-variable primitive p-adic L-functions associated with the triple product of Hida families and prove the explicit interpolation formulae at all critical values in the balanced range. Our construction is to carry out the p-adic interpolation of Garrett's integral representation of triple product L-functions via the p-adic Rankin-Selberg convolution method. As an application, we obtain the cyclotomic p-adic L-function for the motive associated with the triple product of elliptic curves and prove the trivial zero conjecture for this motive. Contents 1. Introduction 1 2. Computation of the local zeta integral: the p-adic case 6 3. Computation of the local zeta integral: the ramified case 14 4. Computation of the local zeta integral: the archimedean case 17 5. Classical and p-adic modular forms 25 6. A p-adic family of pull-backs of Siegel Eisenstein series 28 7. Four-variable p-adic triple product L-functions 33 8. The trivial zero for the triple product of elliptic curves 39 References 46

Algebraicity of critical values of triple product $L$-functions in the balanced case

“…Böcherer and Schulze-Pillot [BSP96] proved the algebraicity of all critical values under the assumptions that lcm(N 1 , N 2 , N 3 ) > 1 is square-free and ω 1 = ω 2 = ω 3 = 1. In their construction of p-adic triple product L-function, Hsieh and Yamana [HY19] obtain the algebraicity of all critical values as a byproduct under the assumption that f 1 , f 2 , and f 3 are simultaneously ordinary at an odd prime p. For numerical computation of critical values of triple product L-functions, there are results of Mizumoto [Miz00] and Ibukiyama-Katsurada-Poor-Yuen [IKPY14]. For explicit central value formula for the triple product L-function, we refer to the results of Gross and Kudla [GK92], [BSP96, § 5], and the author and Cheng [CC19,§ 6].…”

“…More precisely, we choose an auxiliary good place at which the local section defining the Eisenstein series is supported in the big cell. For the local computation of zeta integral, we follow the method employed in [HY19]. It turns out that when m = m 0 + 1 2 , we can find input data so that the resulting local zeta integral is non-vanishing at s = m. Whereas when m = m 0 + 1 2 and either (ω 1 ω 2 ω 3 ) 2 = 1 or f 1 , f 2 , f 3 are not simultaneously CM by an imaginary quadratic field, we need the Sato-Tate conjecture to guarantee the existence of good place such that some local zeta integral is non-vanishing at s = m. For m = m 0 + 1 2 and f 1 , f 2 , f 3 are simultaneously CM by an imaginary quadratic field, we can reduce to the previous case (ω 1 ω 2 ω 3 ) 2 = 1 by showing that the algebraicity holds for L(m, f 1 × f 2 × f 3 ) if and only if it holds for L(m, f 1 × f 2 × f 3 ⊗ χ) for any even Dirichlet character χ.…”