## Departmental PhD Thesis Exam – Ren Zhu

Wednesday, May 27, 2020
2:00 p.m.

PhD Candidate:  Ren Zhu
Supervisor:   Kumar Murty
Thesis title:  The least prime whose Frobenius is an $n$-cycle

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Let $L/K$ be a Galois extension of number fields.  We consider the problem of bounding the least prime ideal of $K$ whose Frobenius lies in a fixed conjugacy class $C$. Under the assumption of Artin’s conjecture we work with Artin $L$-functions directly to obtain an upper bound in terms of irreducible characters which are nonvanishing at $C$.   As a consequence we obtain stronger upper bounds for the least prime in $C$ when many irreducible characters vanish at $C$.  We also prove a Deuring-Heilbronn phenomenon for Artin $L$-functions with nonnegative Dirichlet series coefficients as a key step.
We apply our results to the case when $\Gal(L/K)$ is the symmetric group $S_n$.  Using classical results on the representation theory of $S_n$ we give an upper bound for the least prime whose Frobenius is an $n$-cycle which is stronger than known bounds when the characters which are nonvanishing at $n$-cycles are unramified, as well a similar result for $(n-1)$-cycles.
We also give stronger bounds in the case of $S_n$-extensions over $\mathbb{Q}$ which are unramified over a quadratic field.  We also consider other groups and conjugacy classes where unconditional improvements are obtained.

A copy of the thesis can be found here: Ren Zhu PhD Thesis