# First formulation of the Dedekind and Hasse-Weil conjectures

I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated:

• The Dedekind conjecture (that the quotient $\zeta_K(s)/\zeta(s)$ is entire for any number field $K$)

The only information I've been able to find is that Dedekind himself proved the conjecture in the case of pure cubic fields in 1873. This definitely gives an upper bound. I'm also curious to know for which extensions $K$ over $\mathbb{Q}$ he expected the conjecture to be true.

Edit (1) I've found the paper in question:

Richard Dedekind, Über die Anzahl von Idealklassen in reinen kubischen Zahlkörpern (1900)

Here is a link to his complete works. I can't read german, so the question is still the same, but I thought the link might help. Discussion about the zeta function of number fields seems to start in page 166 (section 6).

Edit (2) Another clue, mentioned by Vesselin Dimitrov in the comments, and suggesting that the conjecture might be due to Landau:

"On page 34 of Bombieri's article The classical theory of zeta and L-functions in the Milan Journal of Mathematics, the conjecture is attributed to Landau. So it probably appears in Landau's Handbuch, much earlier than in Artin"

• The Hasse-Weil conjecture (the zeta function of an algebraic variety has a meromorphic continuation to the complex plane and a functional equation)

(note: this has been nicely answered by Francois Ziegler in the comments)

The problem with this one seems to be that it applies to several different kinds of functions, and both definitions and result came in several stages. I think that the first and most basic definitions is Artin's zeta function of a curve over a finite field. This was generalized to arbitrary algebraic varieties, and also to schemes of finite type over $\mathbb{Z}$. Yet the most common definition is over number fields. There's also the local/global and L/zeta distinction.

In this case I'm just trying to get a feeling of when/where the conjecture was first formulated in some generality.

• Do you require much earlier than Weil Number-theory and algebraic geometry, Proc. Int. Cong. Math. 1950, vol. 2, pp. 90-100, which ends: "I should like to conclude with a brief discussion of a very interesting conjecture, due, I believe, to Hasse (...) we are thus led to consider the product of these zeta-functions for all $\mathfrak p$, which is precisely the function previously defined by Hasse, of which he conjectured that it can be continued analytically over the whole plane, that it is meromorphic, and that it satisfies a functional equation."? – Francois Ziegler Oct 17 '15 at 22:50
• @FrancoisZiegler Thanks, that's close! I guess what's I'm looking for is somwhere where Hasse might have written this down. – Myshkin Oct 18 '15 at 8:34
• That seems to have been oral. Lang (1995, p. 1302) writes: "In 1950, as far as I know, Hasse had not published his conjecture, but he did publish it in 1954; see his comments on the first page of [Ha 54]." There Hasse writes that he had given this problem to his late student Humbert "towards the end of the 1930s". – Francois Ziegler Oct 18 '15 at 11:12
• @FrancoisZiegler That's exactly what I was looking for! If you want to put that as an answer, I can accept it and take the part regarding the Dedekind conjecture to another question. – Myshkin Oct 18 '15 at 11:56
• Happy to help, I think it's fine if I leave it as a comment. Never ask two questions in one :-) – Francois Ziegler Oct 18 '15 at 12:10

Regarding the first of these conjectures, I believe it was first explicitly stated (in the more general setting of a relative extension $K/k$) in Artin's 1923 paper [Über die Zetafunktionen gewisser algebraischer Zahlkörper, Math. Ann. 89, pp. 147-156]. This of course is a very special case of the general Artin holomorphy conjecture, and it is easy in the case of a normal extension. Dedekind did the case of pure cubic fields, but doesn't this result of his date to much later than 1873? Artin quotes the following paper from 1900, and attributes to it the result on pure cubic fields: [Dedekind, Über die Anzahl von Idealklassen in reinen kubischen Zahlkörpern, the Crelle journal, vol. 121, pp. 40-123].

• I'm not sure that's it. The first time he published the proof was the in the 1900, but the result seems to be from 1873. See for example here, where van der Waall mentions the ealier date, but references the 1900 paper. I would be surprised if Dedekind didn't made the conjecture first, at least in a modest generality in the 60-70s, but I guess it's not entirely impossible that it was Artin 50 years later. – Myshkin Oct 18 '15 at 8:39
• I didn't know about this; thanks for the reference! Yes, probably Dedekind thought of the general case, but it could be that this never appeared in print before Artin. Have you looked into Dedekind's Gesammelte mathematische Werke? – Vesselin Dimitrov Oct 18 '15 at 15:37
• @Myshkin: On page 34 of Bombieri's article The classical theory of zeta and $L$-functions in the Milan Journal of Mathematics, the conjecture is attributed to Landau. So it probably appears in Landau's Handbuch, much earlier than in Artin. – Vesselin Dimitrov Nov 13 '15 at 18:01
• Since it is easy for normal extensions, can you sketch the proof :) ? @GHfromMO – reuns Sep 19 '16 at 3:27
• @user1952009: The above response was written by Vesselin Dimitrov, but let me give you some pointers. It is clear that $f(s):=\zeta_K(s)/\zeta_k(s)$ is meromorphic, while Aramata and Brauer independently proved that some positive integral power of $f(s)$ is holomorphic (as a product of certain Hecke $L$-functions associated with certain subfields of $K$). So $f(s)$ is meromorphic without a pole, hence it is holomorphic. For more details see the papers of Aramata (Proceedings of the Imperial Academy of Japan 9 (1933), 31-34) and Brauer (American Journal of Mathematics 69 (1947), 243-250). – GH from MO Sep 19 '16 at 5:35

From Milne's article: The Riemann hypothesis... "According to Weil's recollections (OEuvres, II, p.529),* Hasse defined the Hasse-Weil zeta function for an elliptic curve over $\mathbb{Q}$, and set the Hasse-Weil conjecture in this case as a thesis problem! Initially, Weil was sceptical of the conjecture, but he proved it for curves of the form $Y^{m}=aX^{n}+b$ over number fields by expressing their zeta functions in terms of Hecke $L$-functions.** In particular, Weil showed that the zeta functions of the elliptic curves $Y^{2}=aX^{3}+b$ and $Y^{2}=aX^{4}+b$ can be expressed in terms of Hecke $L$-functions, and he suggested that the same should be true for all elliptic curves with complex multiplication. This was proved by Deuring in a "beautiful series" of papers... "

*"Peu avant la guerre, si mes souvenirs sont exacts, G. de Rham me raconta qu'un de ses étudiants de Genève, Pierre Humbert, était allé à Göttingen avec l'intention d'y travailler sous la direction de Hasse, et que celui-ci lui avait proposé un problème sur lequel de Rham désirait mon avis. Une courbe elliptique $C$ étant donnée sur le corps des rationnels, il s'agissait principalement, il me semble, d'étudier le produit infini des fonctions zêta des courbes $C_{p}$ obtenues en réduisant $C$ modulo $p$ pour tout nombre premier $p$ pour lequel $C_{p}$ est de genre $1$; plus précisément, il fallait rechercher si ce produit possède un prolongement analytique et une equation fonctionnelle. J'ignore si Pierre Humbert, ou bien Hasse, avaient examiné aucun cas particulier. En tout cas, d'après de Rham, Pierre Humbert se sentait découragé et craignait de perdre son temps et sa peine."

**Weil also saw that the analogous conjecture over global function fields can sometimes be deduced from the Weil conjectures.

• The quoted recollections of Weil are available in full here. – Francois Ziegler Oct 18 '15 at 12:31

Artin (1923) wrote that Dedekind proved the case of this conjecture concerning pure cubic number fields. Dedekind published his article in 1900, but writes that it is a reworking of a draft he had written in 1871/72. The conjecture was named after Dedekind by van der Waall in 1974.

Despite Artin's claim, the integrality of the quotient $\zeta_K(s)/\zeta(s)$ is not addressed at all by Dedekind (Hecke proved that $\zeta_K(s)$ extends to a meromorphic function on the complex plane in 1917). Dedekind proved that $\zeta_K(s)/\zeta(s)$ for pure cubic extensions may be written as a linear combination of Epstein zeta functions. Epstein proved in 1903 that these are meromorphic with simple poles in $s = 1$, and Artin realized in 1923 that this implies that the quotient is entire - I guess this explains his remark.

In any case Dedekind had nothing to do with this conjecture.

Edit. Landau [Über die Wurzeln der Zetafunktion eines algebraischen Zahlkörpers, Math. Ann. 79 (1919), 388-401] writes on p. 390: "I have to be careful with its formulation since it is not known whether $\zeta_k(s)/\zeta(s)$ is always an entire function".

• For completeness, the paper of van der Waall is "On a conjecture of Dedekind on zeta-functions" Indagationes Math. 78 (1975), 83-86. He writes near the start that since Dedekind showed the ratio is holomorphic for $K$ a pure cubic field, "we will equip the following conjecture with Dedekind's name". – KConrad Jun 23 '18 at 22:54