9
$\begingroup$

What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known range?

As mentioned below http://arxiv.org/pdf/1502.07953.pdf provides some computation which gets a lower bound from exhaustive checking.

Is there a theoretical way to get upper bound?

A new post https://mathoverflow.net/questions/217689/some-issues-on-idoneal-numbers-in-available-research-literature has been created moving some portion of modification of post originally made here.

$\endgroup$
  • 2
    $\begingroup$ The existence of an additional idoneal number does not imply RH is false, but rather GRH is false for $L$-functions of quadratic characters associated to imaginary quadratic fields. $\endgroup$ – KConrad Sep 6 '15 at 14:28
  • $\begingroup$ @KConrad corrected. is there computational bounds? $\endgroup$ – user76479 Sep 6 '15 at 18:39
  • $\begingroup$ Perhaps it would be best to leave this one as a question for bounds (the original one), and make another question about the 66-67 issue. That way it is more likely to get an answer. $\endgroup$ – Myshkin Sep 7 '15 at 18:08
  • $\begingroup$ @Myshkin mathoverflow.net/questions/217689/… $\endgroup$ – user76479 Sep 7 '15 at 18:27
  • $\begingroup$ We're at revision 14; hopefully the question is in a stable state now. Remember that each edit bumps the question to the front page and knocks another one off (already you have been asking many questions recently, with front page coverage). $\endgroup$ – Todd Trimble Sep 8 '15 at 0:29
6
$\begingroup$

Regarding upper bounds, there are none. In more detail:

By genus theory we know that for discriminants with one class per genus, the class group satisfies $$ \mathcal C(-d)\cong \left(\mathbb Z/2\right)^{g-1}, $$ where $g$ is the number of prime divisors of $d$. Obviously $d$ is bigger than the absolute value of the smallest fundamental discriminant with $g$ prime divisors, $$ d_g\overset{\text{def.}}=3\cdot4\cdot5\cdot7\dots \cdot p_g. $$ From lower bounds on the size of $p_g$, the $g$th prime and on $\theta(x)=\sum_{p<x}\log(p)$, one can show that $$ d_g>g^g. $$ Since $ 2^{g-1} \ll \sqrt{g^g}, $ good lower bounds for the class number rule out the possibility for one class per genus for large $g$. Chowla used this idea to show that the number of classes per genus tends to infinity with $d$.

In 1973, Peter Weinberger showed that on GRH, no fundamental discriminant $-d<-5460$ has one class per genus, and unconditionally there is at most one more such $d$. Weinberger used Tatuzawa's version of Siegel's Theorem to deduce there is at most one such $d$ bigger than $d_{11}=401120980260\approx 4\times 10^{11}$, and sieving to eliminate the $d<d_{11}$.

In contrast, Oesterle explicitly observed that the lower bound due to Goldfeld-Gross-Zagier is not strong enough to finish the classification of discriminants with one class per genus: $ \log(g^g) $ is $\ll 2^{g-1} $. Iwaniec and Kowalski observed that even the full strength of the Birch Swinnerton-Dyer conjecture, 'the best effective lower bounds which current technology allows us to hope for' would not suffice, as $ \log(g^g)^r$ is $\ll 2^{g-1}$ for any $r$. In fact, the outlook is still more bleak: Watkins observes that if the discriminant $-d$ is divisible by all the primes up to $(\log\log d)^3$ (as $d_g$ certainly is), the product over primes dividing $d$ in the Goldfeld-Gross-Zagier lower bound is so small the resulting bound is worse than the trivial bound.

Regarding lower bounds, sieving can be used, as Weinberger did and as has been done with all class number problems going back to the work of D.H. Lehmer. In resolving class number problems, beginning with the work of Stark on class number $1$ all the way up to the work of Watkins resolving all class numbers $\le 100$, there is always an intermediate range beyond where sieving is feasible and below where the deep theorems (e.g. Gross-Zagier-Goldfeld) take effect. In this intermediate range one uses instead the Deuring-Heilbronn phenomenon, in which a counter-example to GRH (real a zero of a quadratic Dirichlet $L$-function close to $s=1$) will influence the location of zeros (on the critical line) of other $L$-functions. Practical application of the Deuring-Heilbronn phenomenon was initiated by Stark in his PhD thesis, who used it to show a $10$th discriminant with class number $1$ would have to have $d>\exp(2.2\cdot 10^7)$.

These ideas can be adapted to the study of discriminations with one class per genus (unpublished). One can show that there is no such discriminant in the range $10^{67}\le d\le 10^{28000}$. Extending this interval upwards requires finding new quadratic Dirichlet $L$ functions with very low-lying zeros. What one really desires is to extend the interval downward to meet up with the range eliminated by sieving. This is technically very difficult (see Watkins' thesis for the analogous problem for class number $\le 100$) and is the main reason the work is unpublished.

$\endgroup$
  • $\begingroup$ So there an unknown range between $10^{11}$ and $10^{67}$? $\endgroup$ – 1.. Aug 17 '18 at 10:04
  • $\begingroup$ @Freeman Yes, and of course for $d\ge 10^{28000}$ as well. $\endgroup$ – Stopple Aug 17 '18 at 14:19
3
$\begingroup$

The best lower bound seems to be $2^{40}$:

Mosunov-Jacobson, Unconditional Class Group Tabulation of Imaginary Quadratric Fields (2015)

Also, see Idoneal Numbers and some Generalizations for a nice overview, particularly section 2.7.

$\endgroup$
  • $\begingroup$ Corollary $23$ on page $13$ suggests there could be two more idoneal numbers. Is this true? $\endgroup$ – user76479 Sep 6 '15 at 20:14
  • $\begingroup$ @Arul I'm confused too. The 2015 Mosunov & Jacobson paper also says that 67th doesn't exists (pp. 24). Perhaps Kani is missing some result not in Weinberger article? The argument in page 14 seems solid. $\endgroup$ – Myshkin Sep 6 '15 at 20:47
  • $\begingroup$ I don't see how you get $10^{40}$ from their paper, when it only computes to $2^{40}$. If you read Weinberger's paper matwbn.icm.edu.pl/ksiazki/aa/aa22/aa2221.pdf in Theorem 1 it says there is at most one $d$ of a complex quadratic field, but you need to realize (as Kani goes over) that there could be two idoneal numbers corresponding to this, if $d$ is even. So I think Mosunov-Jacobson misinterpreted Weinberger. $\endgroup$ – kantelope Sep 7 '15 at 1:43
  • $\begingroup$ @kantelope You're obviously right, it is $2^{40}$. As for the other matter, you are most likely right too. But I would like to know for sure if Kani (and us) are not missing something, perhaps a later result. It might make for another good MO question. $\endgroup$ – Myshkin Sep 7 '15 at 2:04
  • 2
    $\begingroup$ @Myshkin: Weinberger's result applies to fundamental discriminants. By a theorem of Grosswald, any nonfundamental discriminant $-f$ with one class per genus (i.e. idoneal) and $f>315$ is of the form $-f=-4d$ where $-d$ is fundamental and also has one class per genus. Combining these two results gives at most two more, one fundamental and one not. $\endgroup$ – Stopple Sep 8 '15 at 0:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy