Landau-Siegel zeros over a base field and effective Brauer-Siegel

Abstract

Let $k$ be a number field. We prove that for every quadratic extension $F/k$, every real zero $\beta \in (0,1)$ of $L(s,{\chi }_{F/k})$ satisfies $(1-\beta )\log d_F\ge c(k)$ with an effective constant $c(k)>0$, and we deduce (through a reduction of Stark) that the Brauer-Siegel theorem holds effectively and uniformly over all Galois extensions $K$ of $k$, in the form $h_K{R}_K\ge c(k)d_K^{1/2}/((2\pi {)}^{[K:\mathbb{Q}]}\log d_K)$. Stark's effective cases of the Brauer-Siegel theorem excluded the fields with quadratic subfields, since those subfields carried the ineffectivity of Siegel's theorem; over a base field $k$, the first result removes this obstruction. Both rely on a uniform bound for Chebotarev sums. Let $K/k$ be a Galois extension with group $G$, and suppose $S\subset G$ contains the identity. For any prime $\mathfrak{p}$ of $k$, let ${\delta }_{\mathfrak{p}}$ be the proportion of the primes of $K$ above $\mathfrak{p}$ whose Frobenius lies in $S$. Then ${\sum }_{N\mathfrak{p}\le X}{\delta }_{\mathfrak{p}}\log N\mathfrak{p}/N\mathfrak{p}\le (|S|/|G|+\epsilon )\log X+O(1)$ holds uniformly in $K$ as soon as $X\ge d_K^C$, with $C$ and the implied constant depending only on $\epsilon$, $C$, and the degree $[K:\mathbb{Q}]$. The proof is the argument of OpenAI in the form given by Kerr, simple vanishing at the points $(\iota \theta {)}_{\iota \in S}$, with $\theta$ in a box in ${\mathcal{O}}_K$, and Masser's zero estimate. The field enters only through Masser's invariant of the lattice of points, which we compute to be $[K:\mathbb{Q}]/|S|$ by averaging over the Galois group of the normal closure. Over $\mathbb{Q}$, the determinant method gives sharper Chebotarev bounds, and we explain the comparison.

arXiv commentv2: references corrected and verified; related work (Frei and Kai; Cureton and Thorner, Liu) added; attribution of the determinant method revised; mathematics unchanged. 19 pages, 6 diagrams