Jason Martin
Jason Martin

Ph.D. (2006) Cornell University

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Dissertation
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Abstract: Given a number field, K, let r_{1} be the number of real embeddings of K, and let r_{2} be the number of conjugate pairs of complex embeddings. We put t = r_{1} / (r_{1} + 2r_{2}), and call t the infinity type or signature type of K. Let n be the degree of K over Q, and let D be the discriminant, then we define the root discriminant, rd_{K}, as
rd_{K} = ^{n}√ D.
Fix values of t and n then let R_{n,t} denote the minimum root discriminant taken over all number fields of degree n and infinity type t. Finally, put
α(t) = liminf_{n→∞}R_{n,t}.
The constants, α(t), were first defined by J. Martinet, so we refer to them as Martinet's Constants. They describe the smallest discriminant growth one can expect in any infinite family of number fields. An open question in number theory is to find the values of these constants. Serre, Poitou, Stark, and Odlyzko worked on the problem of finding lower bounds with Odlyzko's work providing the best known lower bounds of
α(t) ≥ A^{1 – t}B^{t},
where, if one assumes the Generalized Riemann Hypothesis, A ≈ 44.7 and B ≈ 215.3$. In 1978, Martinet gave upper bounds of α(0) ≤ 92.3 and α(1) ≤ 1058.6, and his bounds were the best known until recently when Hajir and Maire showed that α(0) ≤ 82.2 and α(1) ≤ 954.3. In my work, I have constructed explicit examples for signature types of 1/4, 1/3, 1/2, 3/5, 2/3, 5/7, and 1. This is significant because although there exists (even without GRH) a curve giving the lower bound of α(t), no such curve is even conjectured to exist on the upper bounds. Yet, these values are all within 10% of the curve one would interpolate by using Hajir and Maire's values, and the explicit examples which show that α(3/5) < 342.42 and α(1) < 913.50 are below the HajirMaire curve.