For G = SL_ℓ with ℓ prime, and for G = Sp4 or Sp6, the sum over semisimple conjugacy classes of modified L-functions L_S(M_{Gγ}) is a Lefschetz-type function of the base change degree m; assuming Gross's trace formula conjectures, this governs sums of multiplicities of cuspidal representations.
Lan, Arithmetic compactifications of PEL -type S himura varieties , Princeton University Press, 2013
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The number of cuspidal representations over a function field and its behavior under base changes
For G = SL_ℓ with ℓ prime, and for G = Sp4 or Sp6, the sum over semisimple conjugacy classes of modified L-functions L_S(M_{Gγ}) is a Lefschetz-type function of the base change degree m; assuming Gross's trace formula conjectures, this governs sums of multiplicities of cuspidal representations.