This article gives bounds towards the Ramanujan (or Ramanunan-Petersson/Selberg) Conjecture that hold for a cuspidal automorphic representation on the general linear groups \(\mathrm{GL}_2\), \(\mathrm{GL}_3\), \(\mathrm{GL}_4\) over an arbitrary number field.
Let \(K\) be a number field with ring of adeles \(\mathbb{A}_K\), and let \(\pi\) be an irreducible cuspidal automorphic representation of \(\mathrm{GL}_n(\mathbb{A}_K)\) with unitary central character. Write \(\pi\cong \otimes'_v \pi_v\) with \(\pi_v\) an irreducible unitary generic representation of \(\mathrm{GL}_n(K_v)\), where \(v\) runs over the places of \(K\). The Ramanujan conjecture asserts that each \(\pi_v\) is tempered. If not then \(\pi_v\) is parabolically induced from a standard parabolic subgroup with Levi factor \(M\cong \mathrm{GL}_{n_1}(K_v)\times \cdots \times \mathrm{GL}_{n_r}(K_v)\) and a representation \(\tau[\sigma]=\tau_1[\sigma_\pi(v,1)]\otimes\cdots\otimes \tau_r[\sigma_\pi(v,r)]\) of \(M\), where each \(\tau_i\) is a tempered irreducible representation of \(\mathrm{GL}_{n_i}(K_v)\) and \(\tau[\sigma]\) denotes the twist of \(\tau\) by \(|\det g|_v^\sigma\). Let \(m(\pi,v)\) denote \(\max_j|\sigma_{\pi}(v,j)|\) if \(\pi_v\) is not tempered and \(0\) otherwise.
Let \(H_n(\delta)\) be the statement that for any number field \(K\) and any irreducible cuspidal automorphic representation \(\pi\) of \(\mathrm{GL}_n(\mathbb{A}_K)\) with unitary central character, one has \(m(\pi,v)\leq \delta\) for all places \(v\) of \(K\). The best general bound, due to W. Luo et al. [Proc. Symp. Pure Math. 66, 301–310 (1999; Zbl 0965.11023)] is \(H_n(1/2-1/(n^2+1))\), while H. H. Kim and F. Shahidi established \(H_2(1/9)\) [Duke Math. J. 112, No. 1, 177–197 (2002; Zbl 1074.11027)]. The authors’ principal result is that the statements \(H_2(7/64)\), \(H_3(5/14)\) and \(H_4(9/22)\) hold. As a corollary, the authors observe that their result gives an improvement on the subconvexity bound for the central value of the \(\mathrm{GL}_2\) standard \(L\)-function in twisted aspect. The authors’ main result is an improvement over existing bounds for all fields other than \(\mathbb{Q}\) and imaginary quadratic fields (for example, the bound \(7/64\) was established for \(\mathbb{Q}\) by Kim and Sarnak [Appendix 2 to H. H. Kim, J. Am. Math. Soc. 16, No. 1, 139–183 (2003; Zbl 1018.11024)]).
To establish their main result, the authors show that if \(\pi\) is an irreducible cuspidal automorphic representation of \(\mathrm{GL}_n(\mathbb{A})\) with unitary central character such that \(L(s,\pi,\text{Sym}^2)\) converges absolutely for \(\mathrm{Re}(s)>1\), then \(m(\pi,v)\leq 1/2-2/n(n+1)\). The main result follows from this statement as in Kim-Sarnak [loc. cit.], which uses the approach of W. Duke and H. Iwaniec [in: Automorphic forms and analytic number theory, Proc. Conf., Montréal/Can. 1989, 43–47 (1990; Zbl 0745.11030)] for determining information about the coefficients of a Dirichlet series from its properties under twisting. The obstruction to applying prior methods for a general number field \(K\) is that when the group of units \(\mathcal{O}_K^\times\) in the ring of integers \(\mathcal{O}_K\) is infinite, for a general ideal \(\mathcal{M}\) the image of \(\mathcal{O}_K^\times\) in \((\mathcal{O_K}/\mathcal{M})^\times\) is frequently all of \((\mathcal{O_K}/\mathcal{M})^\times\). The authors overcome this by introducing a test function on ideals that takes into account not only residue classes modulo an ideal but also certain Archimedean information.