## Departmental PhD Thesis Exam – Val Chiche-Lapierre

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Monday, July 15, 2019
1:10 p.m.
BA6183

PhD Candidate:  Val Chiche-Lapierre
Supervisor:   Jacob Tsimerman
Thesis title:   Length of elements in a Minkowski basis for an order in a number field
(or a ring of integers of a number field)
Exam type:    One-defense

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Abstract:

Suppose K is a number field of degree n, and R is an order in K with discriminant D. If K has r real embeddings and s pairs of complex embeddings then we can look at R as a lattice in \R^r x \C^s. We call the length of elements of R their Euclidean length in \R^r x \C^s and denote it by |.|. Let v1=1,v2,…,vn be a Minkowski basis for R. We are interested in the asymptotic lengths of these vi’s for a family or orders with arbitrarily large discriminant D. By the theory of Minkowski bases we have that 1\leq |v2| \leq … \leq |vn| and \prod |v_i| \asymp |D|^{1/2} and by \cite{J}, we also know that |v_n| << |D|^{1/n}.

We say a family of orders in number fields have Minkowski type \delta_2,…,\delta_n if the members of the family have arbitrarily large discriminant and each have a Minkowski basis of the form v1=1,v2,…,vn with |vi| \asymp |D|^{\delta_i} for each i, where D is the discriminant.

In the thesis, we are interested in possible Minkowski types. The first question is: Can we find sufficient and necessary bounds on some rational numbers \delta_2,…,\delta_n such that there is a family of orders in number fields having Minkowski type \delta_2,…,\delta_n?

We already know the following necessary conditions: \delta_2 \leq … \leq \delta_n and \delta_2+…+\delta_n=1/2 by Minkowski basis theory, and \delta_n \leq 1/n by \cite{J}. We prove that bounds of the form \delta_k << \delta_i+\delta_j for each i+j=k are sufficient bounds, and if K has no non trivial subfield, we conjecture that these bounds are actually necessary. We can prove this in some cases (of n,i,j,k). In particular, for n=3,4,5,6, we prove that all these bounds are necessary.

The second question is: For some fixed \delta_2,…\delta_n, “how many” orders in number fields have Minkowski type \delta_2,…,\delta_n. We will make sense of what we mean by “how many” using the Delone-Faddeev correspondence (n=3), and the correspondence of Bhargava (n=4,5). Using these correspondences and counting, we are also able to give a sieving argument to count those orders that are maximal (and therefore are ring of integers of number fields).

A copy of the thesis can be found in this link: val_chichelapierre_thesis