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Irreducibility of Hecke polynomials. (English) Zbl 1162.11335

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 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 F. Diamond, M. Flach and 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 \(f\).
Reviewer: B. Z. Moroz (Bonn)

MSC:

11F25 Hecke-Petersson operators, differential operators (one variable)
11F33 Congruences for modular and \(p\)-adic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F80 Galois representations
11M41 Other Dirichlet series and zeta functions
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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