abstract:

We construct the three-variable $p$-adic triple product $L$-functions attached to Hida families of elliptic newforms and prove the explicit interpolation formulae at all critical specializations by establishing explicit Ichino's formulae for the trilinear period integrals of automorphic forms. Our formulae perfectly fit the conjectural shape of $p$-adic $L$-functions predicted by Coates and Perrin-Riou. As an application, we prove the factorization of certain unbalanced $p$-adic triple product $L$-functions into a product of anticyclotomic $p$-adic $L$-functions for modular forms. By this factorization, we obtain a construction of the square root of the anticyclotomic $p$-adic $L$-functions for elliptic curves in the definite case via the diagonal cycle Euler system \`a la Darmon and Rotger and obtain a Greenberg-Stevens style proof of anticyclotomic exceptional zero conjecture for elliptic curves due to Bertolini and Darmon.

摘要：

我们构造了附加在椭圆新形式Hida系列上的三变量$ p $ -adic三乘积$ L $-函数，并通过为自同形式的三线性周期积分建立了显式的Ichino公式，证明了在所有关键专业领域的显式插值公式。我们的公式完全符合Coates和Perrin-Riou预测的$ p $ -adic $ L $-函数的猜想形状。作为一个应用，我们证明了某些不平衡的$ p $ -adic三元积L $-函数的因式分解为模块化形式的抗环原子$ p $ -adic $ L $-函数的乘积。通过这种分解，