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

Introduction to the show: ident: Mathematics. The day in mathematics: new theorems, new proofs, and what they open.

Ada: We're looking at the paper: "Landau-Siegel zeros over a base field and effective Brauer-Siegel".

Emmy: Here's the gist of the paper. Let k be a number field.

Ada: We prove that for every quadratic extension F/k, every real zero β of L(s, χF/k) satisfies (1 − β) log dF ⩾ 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 hKRK ⩾ c(k) d1/2K /((2π)

K:Q: log dK The Core Theorem The central achievement of this paper is proving that the Brauer–Siegel theorem holds effectively and uniformly over all Galois extensions K of a fixed base field k, providing a lower bound for the class number.

Emmy: Specifically, Theorem 1.3 asserts that for every Galois extension K/k with K!= Q, there is an effectively computable constant c(k) > 0 such that hKRK ⩾ c(k) d1/2K /((2π)

K:Q: log dK Effective Bounds on Quadratic Extensions The paper establishes effective bounds for real zeros of Dirichlet L-functions associated with quadratic extensions.

Ada: Theorem 1.1 provides a uniform upper bound for Chebotarev sums, stating that for every quadratic extension F/k, there is a constant C0 such that XNp⩽Xχ(p)=−1 log NpNp⩽ 1/2 + ε log X + Ok,ε,C(1) The zero estimate is derived from Masser’s theorem, where Proposition 4.2 asserts that µ(Γ) ⩾ nK/s This leads to Corollary 4.3, which states that no nonzero polynomial of total degree at most D vanishes at every thetaSn Deriving the Brauer–Siegel Bound Theorem 1.4 and Theorem 1.5 are derived from the zero estimate via choices of parameters Proposition 6.2 provides a bound for XNp, which is then used to show that the inequality (5.8) fails, leading to Theorem 1.4 This theorem gives a uniform upper bound for Chebotarev sums over all Galois extensions K of k Final Effective Result The effective Brauer–Siegel theorem is established by combining the results from Stark’s reduction and the bounds derived from Theorem 1.3 The proof demonstrates that the constant c3(k) is effectively computable, ensuring that all constants in Theorem 1.3 are explicit Remark 7.4 confirms that everything is effective Pruning redundant reasoning tokens The paper details the construction of a nonzero polynomial using the pigeonhole principle, which requires primes with Frobenius in S to carry enough logarithmic mass (Lemma 5.5) The mechanism involves the congruence θNp ≡ σ(θ) modulo primes above p with Frobenius σ, which makes the value at every point of a polynomial divisible by (zNp1 − zσ)b How the training signal is built The structure of the proof is organized into several sections, beginning with Theorem 1.5 and proceeding through the lattice, zero estimate, Frobenius congruence, and pigeonhole steps The key ingredients include Lemma 5.2 which describes a basis for the module of polynomials PolyD using weighted degree to ensure that substitutions do not leave PolyD Where the gains come from The essential gain comes from overcoming the ineffectivity of Siegel’s theorem, which Stark located by showing that for almost normal fields without quadratic subfields, the theorem is effective The uniform nature of Theorem 1.3 over all Galois extensions K of a fixed base field k is achieved because the obstruction to uniformity, nk log λK, is absorbed when X is large enough Remark 7.2 notes that nothing about Q(ξ) beyond its being a Galois extension of Q of degree l − 1 in which ξ is an integer of house one was used Pruning redundant reasoning tokens The comparison between the lattice method and the determinant method is discussed in Remark 7.1 The determinant method imposes vanishing to high order in one variable at the pairs (α, σα), while our method supplies the multiplicity through a box of NnK points The identity plays no part in our argument, whereas it is the cost in the determinant method How the training signal is built The final result for Theorem 1.4 shows that for every quadratic extension F/k, there is an effectively computable constant c(k) > 0 such that for every Galois extension K/k with K!= Q the following hold (i) Every real zero β ∈ (0, 1) of ζK(s) satisfies (1 − β) log dK ⩾ c(k), and (ii) hKRK ⩾ c(k) d1/2K /((2π)

K:Q: log dK The proof is given in §3 Where the gains come from The effectiveness of the constants depends on Stark’s proof of Lemma 3.3, which makes explicit the constant c2 and provides a lower bound for the residue in terms of real zeros The dependence on k enters through one analytic input only, Mertens’ theorem for k, and we comment on it in §3.2 Remark 7.4 confirms that everything is effective Pruning redundant reasoning tokens The paper concludes by showing that the bound on the coefficient of log X is at most s/G + ε, which simplifies to sG+ε2+ε4·3/2 This confirms the uniform nature of Theorem 1.4 over all Galois extensions K of k The remaining fields K have bounded discriminant, so there are finitely many of them by Hermite’s theorem, and for each of them the bound holds with the constant s/G and an OK(1) by (1.2) The comparison leads to a result that is sharper than the one obtained by Kominers’ determinant method The comparison in Remark 7.1 shows that the root of unity dilutes the constant to 1/2 + 1/

K: k: Where the gains come from The paper is organized as follows, with §6 being the proof of Theorems 1.4 and 1.5 The main advantage over Stark’s original effective cases is that this result removes the requirement for a field to have no quadratic subfield for Galois extensions of k at the cost of a constant depending only on k Remark 3.4 states that if k is almost normal over Q, every Galois extension K of k is as well, and Theorem 1.3 holds with an absolute constant in place of c(k) Pruning redundant reasoning tokens The paper acknowledges the use of a large language model for preparation but takes full responsibility for the content The references provided cover foundational work in algebraic number theory, Chebotarev density, and explicit bounds on L-functions The author thanks the Max Planck Institute for Mathematics in Bonn for its support and excellent working conditions during the preparation of this paper How the training signal is built The paper is organized as follows, with §2 Theorem 1.1 from Theorem 1.5, §3 Corollary 1.2 and Theorem 1.3, §4 the lattice of points and the zero estimate, §5 the Frobenius congruence and the pigeonhole step, and §6 Theorems 1.4 and 1.5 This structure guides the reader through the complex argument The paper is organized as follows, with §2 Theorem 1.1 from Theorem 1.5, §3 Corollary 1.2 and Theorem 1.3, §4 the lattice of points and the zero estimate, §5 the Frobenius congruence and the pigeonhole step, and §6 Theorems 1.4 and 1.

Paper summary: Ada: So, this paper, "Landau-Siegel zeros over a base field and effective Brauer-Siegel," it's tackling something really important about class numbers. It focuses on how to make the Brauer–Siegel theorem work effectively for any Galois extension K of a base field k.

Emmy: Basically, the main claim is that they prove this theorem holds uniformly over all Galois extensions K of a fixed base field k, giving us an effective lower bound for the class number. The paper says that there's an effectively computable constant c(k) such that for every quadratic extension F over k, every real zero beta of the L-function associated with chi F/k satisfies (one - beta) d F is greater than or equal to this constant <ref:2610.31402#pg1,that for every quadratic extension $F>.

Kolya: That sounds like they're dealing with something related to the analytic part of the problem, but how does that connect directly to getting a bound on the class number h K R K ? I need to understand what they're measuring there.

Ada: Exactly, Kolya. And they show that by reducing Stark’s earlier effective cases of the Brauer–Siegel theorem—which only worked for fields without quadratic subfields—they get this new result over any base field k. This new version gives us the bound on h K R K as h K R K is greater than or equal to c(k) times the square root of the discriminant divided by two pi times the degree of the extension multiplied by log discriminant.

Emmy: The reason this matters is that Stark's original work couldn't handle fields with quadratic subfields because those subfields introduced ineffectivity into Siegel's theorem. This paper removes that obstruction, making it uniform for any base field k.

Kolya: So you’re saying the uniformity comes from a uniform bound on Chebotarev sums? That seems like the mechanism they use to get that final result. What are the actual numbers governing those sums?

Ada: Yes, Kolya. They establish a uniform bound for Chebotarev sums. For any Galois extension K with group G, and any subset S of G containing the identity, they show that for any prime p of the base field, the proportion delta p of primes in K above p whose Frobenius lies in S, we have a bound like this: sum over all primes p up to some value X, of delta p times log N p over N p, is less than or equal to S / G plus some epsilon times log X plus an error term.

Emmy: That bound holds uniformly in K as long as X is large enough, specifically when X is greater than or equal to the discriminant of K raised to some power. This uniform bound is what lets them connect the analytic zero estimates to the arithmetic class number bounds.

Conclusion: Ada: So, looking at "Landau-Siegel zeros over a base field and effective Brauer-Siegel," what's the big picture here in simple terms? It's about getting a reliable lower bound for how large the class number of an algebraic number field can be.

Emmy: The core achievement is proving that this effective Brauer-Siegel theorem holds for any Galois extension K of a base field k. This means we have a concrete, calculable constant that tells us the minimum size of the product of the class number and its regulator relative to the discriminant.

Kolya: What I see is that they are moving past cases where things were only true for specific types of fields. They're showing that this effective result applies generally over any base field k, which is a big step because it removes a restriction on the structure of the base field itself.

Ada: Exactly, Kolya. And the dependence on k is very controlled; it only enters through one analytic input, Mertens' theorem for k. This makes the constant effectively computable based only on that input and our choice of epsilon.

Emmy: It's about making a deep result in class number theory applicable in a much wider, uniform way across all Galois extensions of a base field. The paper shows that the tools used to bound zeros of L-functions can be uniformly applied here.

Kolya: So the implication for someone just listening is that we have a reliable tool to estimate class numbers for these kinds of fields, and we know exactly how good our estimate is, even if it's not always the best possible one.

Ada: Right. It’s about making sure that when you calculate a class number bound, you know precisely what the constant c(k) represents and brief description of the chapter ```json

"chapter title": "Introduction and Problem Statement", "description": "Introduces the research paper's topic, its significance in current research, and clearly states the gap in knowledge that the paper aims to address." , "chapter title": "Literature Review", "description": "Summarizes existing relevant studies to provide context for the proposed work, highlighting how this new research builds upon or contrasts with previous findings." "chapter title": "Methodology", "description": "Details the specific research design, data collection methods, experimental setup, and analytical techniques used in the study." "chapter title": "Results and Discussion", "description": "Presents the findings of the study, interprets them in relation to existing literature, discusses their implications for theory and practice." "chapter title": "Conclusion and Future Work", "description": "Summarizes the main contributions of the paper and suggests directions for future research." : ```