The main result of this important paper is a factorization formula for the \(p\)-adic \(L\)-function of the tensor square of a \(p\)-ordinary modular form as the product of the symmetric square \(p\)-adic \(L\)-function of the modular form and the Kubota-Leopoldt \(p\)-adic \(L\)-function; this result can be seen as a \(p\)-adic analogue of the well-known factorization of the corresponding complex analytic objects. Let \(f\in S_k(\Gamma_1(N_f),\chi)\) be a normalized cuspidal eigenform of level \(N_f\), character \(\chi\) and weight \(k\). Let \(\psi\) be a Dirichlet character of level \(N_\psi\). We have then an equality of complex \(L\)-series \[ L(f\otimes f\otimes\psi,s)=L(\mathrm{Sym}^2f\otimes\psi,s)\cdot L(\chi\cdot\psi,s-k+1). \] Let now \(p\) be a prime which does not divide \(N_\psi\). The \(p\)-adic object corresponding to the complex ones considered above have been defined by H. Hida [in: Automorphic forms, Shimura varieties, and \(L\)-functions. Vol. II, Proc. Conf., Ann Arbor/MI (USA) 1988, Perspect. Math. 11, 95–142 (1990; Zbl 0705.11033)] (that defined \(L_p(f\otimes f\otimes\psi,\sigma)\) for \(\sigma\in\mathcal{W}=\mathrm{Hom}_{\mathrm{cont}}(\mathbb{Z}_p^\times,\mathbb{C}_p^\times)\) interpolating the complex \(L\)-series \(L(f\otimes f\otimes\psi,s)\) for suitable choices of \(\sigma\) and \(s\)), by Schmidt-Hida (that defined \(L_p(\mathrm{Sym}^2f\otimes\psi,\sigma))\) and Kubota-Loepoldt. The main result of the paper reads as a factorization of the \(p\)-adic analytic objects which looks formally equal to the complex case: \[ L_p(f\otimes f\otimes\psi,\sigma)=L_p(\mathrm{Sym}^2f\otimes\psi,\sigma)\cdot L_p(\chi'\cdot\psi,z\cdot\sigma/\kappa) \] where \(\chi=\chi'\chi\) is the decomposition of \(\chi\) into its prime to \(p\) factor \(\chi'\) and its \(p\)-power factor \(\chi_p\), \(\kappa\in\mathcal{W}\) is defined by \(\kappa(z)=z^k\chi_p(z)\) and \(\sigma\), \(\psi\) satisfy the condition \(\sigma(-1)=-\psi(-1)\) (the last condition can be removed if \(p\nmid N_f\)). The author remarkably applies this factorization to obtain the proof of Greenberg’s exceptional zero conjecture for the adjoint \(p\)-adic \(L\)-function.
The strategy for proving such a result is quite interesting. The first step is to put \(f\) in a Hida family (for which one needs the ordinary assumption), and consider \(2\)-variable analogues of the \(1\)-variable objects considered above, in which now the variables are \(\sigma\in\mathcal{W}\) as above and the weight variable \(\kappa\in\mathcal{W}\); the factorization formula is then a consequences of a more general factorization formula involving these two variable analogues. The advantage of this second factorization formula is that there is a set of \(p\)-adically dense points which can be used to check, via specializations, the factorization formula, and then extend it to all points (including the point we are actually interested in for the \(1\)-variable factorization formula) by a density argument. The second step consists then in the use of recent work of Bertolini-Darmon-Rotger [M. Bertolini et al., J. Algebr. Geom. 24, No. 2, 355–378 (2015; Zbl 1325.14034); ibid. 24, No. 3, 569–604 (2015; Zbl 1328.11073)] and Lei-Loeffler-Zerbes [A. Lei et al., Ann. Math. (2) 180, No. 2, 653–771 (2014; Zbl 1315.11044)] which express these specializations of the \(2\)-variable analogue of \(L_p(f\otimes f\otimes\psi,\sigma)\) in terms of the \(p\)-adic logarithm \(\log_p(b_{f,\psi,\beta})\) of certain \(p\)-adic numbers \(b_{f,\psi,\beta}\in U_\eta=(\mathcal{O}_K^\times\otimes\bar{\mathbb{Q}})^{\eta^{-1}}\), called Beilinson-Flach units, coming from arithmetic intersection theory of algebraic cycles on the product of two modular curves (here, to fill the notation, \(\eta=\chi\psi\alpha\beta^{-1}\), where \(\alpha\) and \(\beta\) are auxiliary Dirichlet characters of \(p\)-power conductor having the same parity of \(\chi\) and \(\psi\) respectively, and \(K\) is the real cyclotomic field cut out by the even character \(\eta\)). With this in hand, exploiting the interpolation formula of the symmetric square \(p\)-adic \(L\)-function in two variables, the factorization formula reduces to show that the Beilinson-Flach units \(b_{f,\psi,\beta}\) compute the algebraic special values \(\frac{L(\mathrm{Sym}^2f,\beta^{-1},1)}{\text{period}}\) in the \(\bar{\mathbb{Q}}\)-vector spaces \(U_\eta\), which is the third step of the proof (here and in the following in the denominator we indicate by “\(\text{period}\)” a suitable complex number that makes the quotient algebraic). This third step is archived by exploiting the interpolation properties of the \(p\)-adic \(L\)-series considered, and the factorization formula satisfied by the correspoding complex \(L\)-series. More precisely, one first uses the factorization of complex \(L\)-series to obtain the formula \[ \frac{L'(f\otimes f\otimes\psi\beta^{-1},1)}{\text{period}} \overset{\cdot}{=} L(\mathrm{Sym}^2f\otimes\psi\beta^{-1},1)\cdot L'(\eta,0) \] where here and in the following we indicate by \(\overset{\cdot}{=}\) an equality which holds up to explicit algebraic factors. One the one hand, \(L'(\eta,0)\) is computed by the Dirichlet class number formula. Further, combining Beilinson regulator formula and a general compatibility result involving the Archimedean regulator, the cycle class map and intersection pairing, one shows the formula \[ \frac{L'(f\otimes f\otimes\psi\beta^{-1},1)}{\text{period}} \overset{\cdot}{=} \log_\infty (b_{f,\psi,\beta}) \] where \(\log_\infty\) is the usual Archimedean logarithm. Combining this (and using that \(U_\eta\) is a \(1\)-dimensional \(\bar{\mathbb{Q}}\)-vector space on which \(\log_\infty\) is injective) one obtains the sough for relation between \(b_{f,\psi,\beta}\) and \(\frac{L(\mathrm{Sym}^2f,\beta^{-1},1)}{\text{period}}\), thus concluding the proof.