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\(p\)-adic measures and square roots of special values of triple product \(L\)-functions. (English) Zbl 1034.11034
Let \(f,g,h\) be cuspidal Hecke eigenforms for \(SL(2,\mathbb Z)\) with weights \(k,\ell,m\) respectively, with \(k\geq \ell\geq m\) and \(k\geq \ell+m\). Let \(L(s,f\otimes g\otimes h)\) be the triple product \(L\)-function, where the center is \(\frac{k+\ell+m-2}{2}=:\frac{w+1}{2}\). The triple \(L\)-functions have been studied by P. Garrett [Ann. Math. (2) {125}, 209–235 (1987; Zbl 0625.10020)] and I. Piatetski-Shapiro and S. Rallis [Compos. Math. {64}, 31–115 (1987; Zbl 0637.10023)]. M. Harris and S. Kudla [Ann. Math. (2) {133}, 605–672 (1991; Zbl 0731.11031)] showed that the modified central \(L\)-value \[ \frac{L\left(\frac{w+1}{2},f\otimes g\otimes h\right)} {\pi^{2k} (f,f)_k^2 C(k,\ell,m)} \] is a square in \({\mathbb Q}(f,g,h)\), where \({\mathbb Q}(f,g,h)\) is the field generated over \(\mathbb Q\) by the Fourier coefficients of \(f\), \(g\) and \(h\), \((\cdot, \cdot)_k\) is the normalized Petersson inner product, and \(C(k,\ell, m)\in \mathbb Q\) is a universal constant only depending on \(k, \ell\) and \(m\).
The present paper under review studies the \(p\)-adic interpolation of the square root of the modified central \(L\)-value above. The authors construct \(p\)-adic measures which interpolate the square root, as \(f\) varies in a Hida family \(\mathbf f\) of \(p\)-adic modular forms. As a special case of the main theorem (Theorem 2.2.8), the authors prove \[ \left(\frac{D_H(\mathbf f, g,h)(k)}{H(k)K(k)} \right )^2=\frac{L\left(\frac{w+1}{2},f_k\otimes g\otimes h\right)} {\pi^{2k} (f_k,f_k)_k^2 C(k,\ell,m)}, \] where \(f_k\) is the primitive form associated to the specialization in weight \(k\) of \(\mathbf f\), \(D_H(\mathbf f, g,h)\) is the analytic function associated to the constructed \(p\)-adic measure, and \(H(k)\) and \(K(k)\) are normalizing factors depending on \(\mathbf f\). The construction of the \(p\)-adic measure modifies Hida’s approach to \(p\)-adic interpolation of Rankin products [H. Hida, Invent. Math. 79, 159–195 (1985; Zbl 0573.10020)].

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F33 Congruences for modular and \(p\)-adic modular forms
11G18 Arithmetic aspects of modular and Shimura varieties
11F85 \(p\)-adic theory, local fields
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