"frequency" of fields for which the p-adic regulator vanishes (mod p)

Question

There is a very nice question which arises in the study of the Discrete Logarithm Problem which I wish to present here. The question, in a general setting, is to specify an empirical expression for the "frequency" of fields for which the $p$-adic regulator vanishes $(\bmod p)$. More specifically: If $k$ is a number field and $p$ an odd prime, we let

$$R_{p}(k) \in \operatorname{Rems} :=\{0, 1, ..., p-1\}$$

be the remainder of the p-adic regulator modulo $p$.

One would expect that the values of $R_{p}(k)$ are uniformly distributed within $\operatorname{Rems}$.

I would not expect the proof of such a statement to be within reach with presently known techniques. However, there are some questions concerning the way to even state the uniform distribution.

I think of two possible variants for this:

1. Arrange the number fields according to some "measure" that may depend on the degree and the discriminant, or only one of these parameters.

Therefore $\|k\| = \operatorname{disc}(k)$ and one takes averages either over all fields or over all fields of fixed degree. Let then

$$N_B = \{ \text{number fields}\ k : \| k \| < B\ \text{and}\ R_{p}(k) = 0 \},$$

while

$$F_B = \{ \text{number fields}\ k : \| k \| < B \}.$$

$N_B(d)$ and $F_B(d)$ are defined similarly, restricting only to fields of degree $d$.

The first conjectures: Fix $p$. Then $$ \text{Conjecture 1:}\quad \lim_{B\rightarrow +\infty} N_B/F_B = 1/p. $$ $$ \text{Conjecture 2:}\quad \lim_{B\rightarrow +\infty} N_B(d)/F_B(d) = 1/p. $$

2. The second approach fixes $k$ and lets $p$ vary. Let $P \subset N$ be the set of all natural primes, then let $$\chi(p) = \begin{cases} 0&\text{if}\ R_{p}(k) > 0,\\ 1&\text{if}\ R_{p}(k) = 0.\\ \end{cases} $$

Then the conjectures state: $$ \text{Conjecture 3:}\quad \lim_{B\rightarrow +\infty} \frac{\displaystyle \sum_{ p \in P, p < B} \chi(p) p }{B/\log(B)} = 1. $$ $$ \text{Conjecture 4:}\quad \lim_{B\rightarrow +\infty} \frac{\displaystyle \sum_{ p \in P, p < B} \chi(p) }{\log \log B} = 1 $$

QUESTIONS:

a) Are analogous statements about uniform distribution of, say, the class number modulo $p$ or some other field invariants already discussed in the literature? And if yes, how does one express the uniform distributions?

b) Are any secondary phenomena, that may give raise to some constants in the above conjectures, especially in Conjectures 3 and 4? I have in mind things like the correction factors in the Hardy-Littlewood Conjecture or other similar ones.

c) Do there exist some alternative ways for stating uniform distribution?

Answer 1

Some "equidistributions $\mod{p}$" are discussed in Washington's "Introduction to Cyclotomic Fields". I adress here only your first question, and have nothing to say about questions b) and c).

The first equidistribution is about irregular primes, and is discussed after Theorem 5.17 (I refer, here and henceforth, to the second edition), on page 62-63. The idea is to assume that the classes of Bernoulli numbers are random $\mod{p}$, in the sense that the probability that $p\mid B_j$ for some $j\in[2,4,6,\dots,p-3]$ is $1/p$. In this case one coops with a Poisson distribution and the deduced heuristic is that approximately $60\%$ of primes are regular, which agrees with computations.

A second distribution argument is briefly touched upon after Theorem 5.37 and concerns the possibility that the Bernoulli number $B_{(p-1)/2}$ be random $\mod{p}$. This would have consequences on the residue $\mod{p}$ of the class number of $\mathbb{Q}(\sqrt{p})$: as it is explained in Exercice 5.9, though, one can show that $B_{(p+1)/2}$ is not random $\mod{p}$, as a consequence of Brauer-Siegel (at least for $p\equiv 3\pmod{4})$.

A third one concerns Vandiver's conjecture and is discussed in the Remark on page 158 ($\S$ 8.4). It shows that the naive approach of assuming that the $i$-th cyclotomic unit be a $p$-th power with probability $1/p$ fails dramatically, while there is a more refined example that combines the above assumption with the needed hypothesis that $p$ be irregular; the very first heuristics I mentioned, about equidistribution of the $j$-th Bernoulli number $B_j\pmod{p}$, implies that possible exceptions to Vandiver's up to $p\leq x$ go like $1/2\log\log(x)$, and is therefore of no significance that we haven't found any counterexample so far. The very same section of Washington's book has a final remark on Vandiver based on the equidistribution assumption that $p\mid h^+$ with probability $1/p$ (where $h^+$ is the class number of the totally real field $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$). In this case again, Vandiver could be false for a prime $p\leq x$ with probability $$ \log\log(x) $$ which is bigger than before but roughly analogous. In both cases, since computations (at the time of Washingtons' writing, namely 1996) can go up to $p\sim 4.000.000$ and $\log\log(4.000.000)=2.72...$, that no exception has been found is "not too strange".

I should perhaps add that it seems that P. Mihailescu has some argument for turning Washington's heuristic "in favor" of Vandiver here.