Yoo, Hwajong Non-optimal levels of a reducible mod \(\ell \) modular representation. (English) Zbl 1444.11080 Trans. Am. Math. Soc. 371, No. 6, 3805-3830 (2019). Summary: Let \( \ell \geq 5\) be a prime and let \( N\) be a square-free integer prime to \( \ell \). For each prime \( p\) dividing \( N\), let \( a_p\) be either \( 1\) or \( -1\). We give sufficient criteria for the existence of a newform \( f\) of weight 2 for \( \Gamma _0(N)\) such that the mod \( \ell \) Galois representation attached to \( f\) is reducible and \( U_p f = a_p f\) for primes \( p\) dividing \( N\). The main techniques used are level raising methods based on an exact sequence due to Ribet. Cited in 9 Documents MSC: 11F33 Congruences for modular and \(p\)-adic modular forms 11F80 Galois representations 11G18 Arithmetic aspects of modular and Shimura varieties Keywords:Eisenstein ideals; non-optimal levels; reducible Galois representation PDFBibTeX XMLCite \textit{H. Yoo}, Trans. Am. Math. 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