The Euler-Kronecker invariant and the geometry of ideal lattices
We will prove a distributional formula for the Euler-Kronecker invariant γK of a number field K of any degree d, involving an arbitrary test function f on Rd, 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 γK, 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 ˉI=(covol(I)1/d)⋅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 γK 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 Q[X]/(Xd+∑dj=1cjXd−j), where c1,…,cd∈{1,…,100}, have no non-zero vector shorter than 1−o(1).