Irreducibility of Hecke polynomials
Let \(f(z)= \sum_{n=1}^\infty a_ne^{2\pi inz}\) and \(g(z)= \sum_{n=1}^\infty b_ne^{2\pi inz}\) be two normalised Hecke eigenforms of weights \(k\) and \(k'\) respectively, with \(k'> k\geq 2\), for the full modular group \(\text{SL}_2(\mathbb Z)\). Suppose that all the coefficients \(\{a_n,b_n\}\) lie in a number field \(E\) and there are a rational prime \(p\), a positive integer \(r\), and a prime divisor \({\mathfrak p}\) in \(E\), lying above \(p\), such that the coefficients \(\{a_n,b_n\}\) are \({\mathfrak p}\)-adically integral and satisfy the congruences \(a_n=b_n\pmod {p^r}\) for every \(n\); one writes then \(f\equiv g\pmod {p^r}\). Let \[ D(s;f,g):= \zeta(2s+2-k'-k) \sum_{n=1}^\infty a_nb_nn^{-s} \] be the tensor product \(L\)-function attached to the modular forms \(f\) and \(g\). The author writes as follows: ``Let \(M_f(j)\) be the \(j\)th Tate's twist of the motive attached to \(f\) by \textit{A. J. Scholl} [Invent. Math 100, No. 2, 419--430 (1990; Zbl 0760.14002)]. Let \(M_{f,g}\) be the tensor product of the motives attached to \(f\) and \(g\). In this paper, we observe that the congruence \(f=g\pmod {p^r}\) leads (at least if \(f\) and \(g\) lie inside a Hida \(p\)-adic family and have irreducible mod \(p\) Galois representations) to the existence of global \(p^r\)-torsion for the motive \(M_{f,g}(k'-1)\). Using recent work \textit{F. Diamond, M. Flach} and \textit{L. Guo} [Math. Res. Lett. 8, No. 4, 437--442 (2003; Zbl 1022.11023); Adjoint motives of modular forms and the Tamagawa number conjecture, Preprint], we show (under mild conditions) that \(p\) does not divide the order of a certain Shafarevich-Tate group, assuming a lack of congruences mod \(p\) between \(g\) and \(f\).
scientific article; zbMATH DE number 2064726