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Session S21 - Galois representations and automorphic forms

Tuesday, July 13, 15:20 ~ 16:00 UTC-3

Density questions on arithmetic equivalence

Guillermo Mantilla Soler

Universidad Konrad Lorenz and Aalto University, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

It is a classic result that two number fields have equal Dedekind zeta functions if and only if the arithmetic type of a prime p is the same in both fields for almost all prime p. Here, almost all means with the possible exception of a set of Dirichlet density zero. In this talk we'll show that the condition density zero can be improved to a specific positive density that depends solely on the degree of the fields. More specifically, for every positive n we exhibit a positive constant c_{n} such that any two degree n number fields K and L are arithmetically equivalent if and only if the set of primes p such that the arithmetic type of p in K and L is not the same has Dirichlet density at most c_n. We also show that to check whether or not two number fields are arithmetically equivalent it is enough to check equality between finitely many coefficients of their zeta functions, and we give an upper bound for such a number.

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