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Math 259: Introduction to Analytic Number Theory How small can |disc(K)| be for a number field K of degree n = r 1 + 2r 2?
Let K be a number field of degree n = r 1 + 2r 2 , where as usual r 1 and r 2 are respectively the numbers of real embeddings and conjugate complex embeddings of K. Let OK be the ring of algebraic integers of K, and DK = disc(K/Q) the discriminant. Minkowski proved that every ideal class of K contains some ideal J ⊆ OK of norm at most (n!/nn)(4/π)r^2 |DK |^1 /^2. (See for instance [Marcus 1977].) In particular, the principal ideal class contains such a J (which might as well be taken to be OK itself), and since the norm of J is at least 1 we recover the Minkowski bound
|DK | ≥
( (^) π 4
) 2 r 2 ( (^) nn n!
In particular, it readily follows that |DK | > 1 once n > 1 (that is, except for K = Q); that is, Q has no nontrivial unramified extension. This is a key ingredient of the Kronecker-Weber theorem, which asserts that any finite extension of Q with abelian Galois group is contained in a cyclotomic extension Q(e^2 πi/n).
Asymptotically as n→∞, Minkowski’s bound is
log |DK | ≥ (2 − o(1))n − 2 log(π/4)r 2. (2)
That is, we have the lower bound (π/4)^2 r^2 /ne^2 −o(1) on the “root-discriminant” |DK |^1 /n. (Note for future reference the numerical values: (π/4)^2 r^2 /ne^2 is ap-
proximately (7.389)r^1 /n(5.803)^2 r^2 /n.) It is known that the root-discriminant is invariant under unramified extensions; for instance (1) also implies that some other number fields — such as the quadratic fields Q(e^2 πi/^3 ), Q(i), Q(
- whose discriminants − 3 , − 4 , 5 have the smallest absolute values — have no nontriv- ial unramified extension. Subsequent work extended Minkowski’s “geometry of numbers” to show log |DK | is bounded below by larger linear combinations of r 1 , r 2.
In the other direction, Golod and Safareviˇˇ c proved that quadratic number fields K 0 whose discriminants have many prime factors have an infinite “class field tower”, and thus unramified extensions K with [K : K 0 ]→∞. Such K all have root-discriminant |DK 0 |^1 /^2. There is thus an upper limit to improvements on the constants in (2). One survey of such constructions and the resulting upper limits is [Schoof 1986].
Much less is known here than for the analogous question on curves C of high genus with many points over a fixed finite field k. (See the Remarks below.) The best lower bounds for all but the smallest few n are now obtained by a method independent of Minkowski’s approach, and similar to the techniques that yield upper bounds on #C(k). The method, attributed to Stark [1974, 1975] by Odlyzko [1991], uses the Euler and Hadamard products for the zeta function ζK to transform the functional equation for ζK into a formula for log |DK | in
terms of r 1 , r 2 , and the nontrivial zeros of ζK. In a series of papers starting from [Odlyzko 1975], the bounds were progressively improved until reaching their present form:
Theorem. Let K be a number field of degree n = r 1 + 2r 2. Then
log |DK | > (log 4π + γ − o(1))n + r 1 (3)
as n→∞, where γ = −Γ′(1) =. 577... is Euler’s constant. If moreover ζK satisfies the Generalized Riemann Hypothesis then
log |DK | > (log 8π + γ − o(1))n + (π/2)r 1 (4)
as n → ∞.
Numerically, the root-discriminant of K is asymptotically bounded below by (60.8)r^1 /n(22.38)^2 r^2 /n, and by (215.3)r^1 /n(44.7)^2 r^2 /n^ under the GRH. For many applications one needs also explicit estimates on the o(1) terms for specific values of (r 1 , r 2 ). Odlyzko carried out extensive numerical computations to obtain good lower bounds for many (r 1 , r 2 ). See [Odlyzko 1991] for a survey of the methods used and some of the applications, which include the theorem that each of the nine imaginary quadratic fields of class number 1 has no nontrivial unramified extensions. (NB the last of these fields has root-discriminant
We present only a simple proof of the asymptotic estimate under GRH, making no attempt to optimize the o(1) error. The same approach yields the uncondi- tional bound (3); see the Exercises.
We begin by obtaining Artin’s formula for |DK |:
Proposition. For all real s > 1 we have
log |DK | = r 1
log π −
(s/2)
log 2π −
(s)
s − 1
s
ζ K′ ζK (s) + 2
ρ
Re
s − ρ
where ρ runs over the nontrivial zeros of ζK (s) counted with multiplicity.
Proof : Recall that the functional equation for ζK may be written in the form
ξK (s) := Γ(s/2)r^1 Γ(s)r^2 (4−r^2 π−n|DK |)s/^2 ζK (s) = ξK (1 − s), (6)
and that (s^2 − s)ξK (s) is an entire function of s of order 1. Translation by 1 /2 yields the entire function (s^2 − 14 )ξK (s + 12 ) symmetric under the involution s 7 → −s. The logarithmic derivative of the Hadamard product for this function yields the partial-fraction decomposition
ξ K′ (s) ξK (s)
= B −
s
1 − s
m s − (^12)
ρ
s − ρ
ρ − (^12)
Here m is the multiplicity of the zero, if any, of ζK (s) at s = 1/2; and ρ runs over the nontrivial zeros of ζK (s) counted with multiplicity, excluding 1/2. Since (7)
Condition (i) holds for large enough M because (Γ′/Γ)(s) and (Γ′/Γ)(s/2) are both analytic functions of s in a circle of radius 1 > 1 /2 about s 0 ). To verify that (ii) also holds as M →∞, let^1 ρ = 1/2+it, and note that Re(1/(s− 12 −ρ)) = Re(1/(s − it)) = /(^2 + Im(ρ)^2 ), and the value of the M -th partial sum of the Taylor expansion differs from this by
Re
[1 + 2( − it)]M^ ( + it)
(1 + ^2 + t^2 )−M/^2.
The positive /(^2 + t^2 ) clearly dominates the error (1 + ^2 + t^2 )−M/^2 uniformly in t once M is sufficiently large.
Now divide (8) by 2mm!, sum from m = 0 to M − 1, and set s = s 0 to obtain
log |DK | > r 1
log π−
(s 0 / 2 − 1 /4)−
+2r 2
log 2π−
(s 0 − 1 /2)−
+O(1);
since was arbitrarily small and s 0 arbitrarily close to 1, we are done.
Remarks
Besides the problem of evaluating limits such as lim infn→∞ log |DK |/n, many other natural questions remain wide open in this context where analogous ques- tions for high-genus curves with many rational points over a finite field have been settled for some time. We list several of these open questions:
- It is not known how to construct class field towers explicitly. Can one construct an explicit infinite sequence of number fields K with bounded root-discriminant?
- When a class field tower over K 0 can be proved infinite, the resulting unramified extensions K have [K : K 0 ] limited to a very sparse set of positive integers, namely those whose prime factors are contained in a given finite set S. Does there exist θ > 0 an infinite sequence of number fields K with bounded root-discriminant whose degrees cover at least xθ of the integers n < x as x→∞?
- More ambitiously: Can there be such a sequence that covers every n? Equivalently, is lim supn→∞ log |DK |/n finite?
- In another direction: in a class field tower over a fixed number field, the ratios r 1 /n are limited to a small subset of [0, 1] ∩ Q. Does there exist a finite R such that the number fields K with |DK | < Rn^ have ratios r 1 /n that form a dense subset of [0, 1], or even of an interval of positive length in [0, 1]?
- The ratio r 1 /n can be regarded as a measure of the behavior of the “archimedean place” of Q in K. Similar questions can be posed concern- ing the splitting or ramification of a given set of “nonarchmedean places” (rational primes) in K. (^1) The customary ρ = 1/2 + iγ may lead to confusion in the presence of Euler’s constant γ.
Another notable application of the method of Odlyzko et al. is Mestre’s lower bound on the conductor of an elliptic curve E/Q of given rank, assuming GRH as well as the conjecture of Birch and Swinnerton-Dyer for the L-function L(E, s) of the curve. Similar bounds have been obtained for even more complicated L-functions.
Exercises
- Fill in the missing steps in our proof of (4) by checking the derivation of the formula (6) or log |DK | and proving the formulas (9) for the logarithmic derivative of Γ(s) at s = 1/2 and s = 1/4.
- Show that the Odlyzko bound (4) still holds under the weakened hypothesis that all zeros of ζK (s) are either real or on the critical line σ = 1/2. (This hypothesis allows also for nontrivial zeros on (0, 1).) Can you find a yet weaker hypothesis on the zeros under which (4) remains true?
- Use the same methods to prove the unconditional lower bound (3).
- Suppose that the rational prime 2 splits completely in K (whence the Eu- ler product for ζK (s) contains the factor (1 − 2 −s)−n). Obtain lowers bounds on |DK |, both unconditionally and under GRH, that improve on (3,4). Gener- alize.
References
[Marcus 1977] Marcus, D.A.: Number Fields. New York: Springer, 1977.
[Odlyzko 1975] Odlyzko, A.M.: Some analytic estimates of class numbers and discriminants, Invent. Math. 29 (1975) #3, 275–286.
[Odlyzko 1991] Odlyzko, A.M.: Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, S´em. Th. des Nombres Bordeaux (2) 2 (1990) #1, 119–141.
[Schoof 1986] Schoof, R.: Infinite class field towers of quadratic fields, J. reine angew. Math. 372 (1986), 209–220.
[Stark 1974] Stark, H.M.: Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152.
[Stark 1975] Stark, H.M.: The analytic theory of algebraic numbers, Bull. Amer. Math. Soc. 81) (1975), 961–972. Invent. Math. 23 (1974), 135–152.
