For GL_3 base change to a real quadratic field, new middle-degree automorphic periods satisfy a Hida-type congruence formula and yield a p-adic divisibility of periods, conditional on the Calegari-Geraghty hypotheses and up to an uncomputed constant.
A proof of Kirillov 's conjecture
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Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations
For GL_3 base change to a real quadratic field, new middle-degree automorphic periods satisfy a Hida-type congruence formula and yield a p-adic divisibility of periods, conditional on the Calegari-Geraghty hypotheses and up to an uncomputed constant.