Ramanujan conjecture for non-CAP Siegel cusp forms of degree 2

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Let F∈Sk(Γ0(2)(N))F\in S_k(\Gamma^{(2)}_0(N)) be a Hecke eigenform at good primes that is non-CAP, meaning that it is of general type or Yoshida type. For each prime pp with gcd⁡(p,N)=1\gcd(p,N)=1, let αp,1,…,αp,4\alpha_{p,1},\ldots,\alpha_{p,4} be the Satake parameters of FF at pp. Ramanujan conjecture. For every such pp and 1≤i≤41\leq i\leq4, one has

∣αp,i∣=1.|\alpha_{p,i}|=1.

This is the temperedness prediction for non-CAP forms. The source notes that it is currently known for k≥3k\geq3 but remains open for k=2k=2; CAP forms are excluded because they are non-tempered.

References

Primary source

Biplab Paul and Abhishek Saha, “On Fourier coefficients and Hecke eigenvalues of Siegel cusp forms of degree 2”, arXiv:2207.06198 (2022).

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