# The Euler-Kronecker invariant and the geometry of ideal lattices

We will prove a distributional formula for the Euler-Kronecker invariant $\gamma_K$ of a number field $K$ of any degree $d$, involving an arbitrary test function $f$ on $\mathbb{R}^d$, as well as the "Heegner locus" of ideal lattice shapes of the number field. For $d = 2$, it can be seen as an extension of the classical limit formulas of Kronecker and Hecke. We then apply the formula with a careful choice of $f$ and obtain an enhancement of the Stark-Ihara lower bound on $\gamma_K$, and from there a GRH conditional proof of the following statement about number fields in the large degrees asymptotic: if the discriminant of $K$ is smaller than the double exponential $\exp((1.001)^d)$ in the degree $d$, an asymptotic unit fraction of the ideal lattice shapes $\bar{I} = (\mathrm{covol}(I)^{1/d}) \cdot I$ of $K$ contain no non-zero vectors shorter than $0.9999$. (How about a converse to this statement? This and plenty of other related questions in this area remain unsolved.) Unconditionally, we explain how our $\gamma_K$ inequality allows to interpret recent works of Breuillard and Varju and of Bary-Soroker, Koukoulopoulos and Kozma as a statement about the random ideal lattice of a random number field: most of the ideal lattice shapes of most of the number fields $\mathbb{Q}[X] / (X^d + \sum_{j=1}^d c_j X^{d-j})$, where $c_1, \ldots, c_d \in \{1, \ldots, 100\}$, have no non-zero vector shorter than $1 - o(1)$.