Now I want for $student{$student_id} to be a reference to an array of references to the key/value pair stored at $catalog{$course_number}.

I don't see the need for that. Why not just store the key in the array and then use that to look up the value from %catalog whenever you need it?

push @{$students{$student_id}}, $course_id;

# ... then to retrieve later ...

for my $cid (@{$students{$student_id}}) {
    print "Course ID $cid, Tutor: $catalog{$cid}{tutor}\n"; # or whate
+ver
}
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