The low-lying zeros of L-functions associated to non-Galois cubic fields
Typ
Examensarbete för masterexamen
Master's Thesis
Master's Thesis
Program
Övrigt, MSc
Publicerad
2023
Författare
Ahlquist, Victor
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
We study the low-lying zeros of Artin L-functions associated to non-Galois cubic number
fields through their one- and two-level densities. In particular, we find new precise estimates
for the two-level density with a power-saving error term. We apply the L-functions Ratios
Conjecture to study these densities for a larger class of test functions than unconditional
computations allow. By reviewing a known Ratios Conjecture prediction, due to Cho,
Fiorilli, Lee, and Södergren, for the one-level density, we isolate a phase transition in the
lower-order terms, which reveals a striking symmetry. Our computations show that the
same symmetry exists in the one-level density of several other families, that have previously
been studied in the literature, and this motivates us to formulate a conjecture extending one
part of the Katz–Sarnak prediction for families of symplectic symmetry type. Moreover, we
isolate several phase transitions in the lower-order terms of the two-level density. To the
best of our knowledge, this is the first time such phase transitions have been observed in
any n-level density with n ≥ 2.
Beskrivning
Ämne/nyckelord
Mathematics, number theory, cubic fields, L-functions, low-lying zeros, onelevel density, two-level density, phase transition