Meaning of "gehören" with "an"

Question

I see the following advertisement tag line:

Auf und Abs gehören an die Börse. Nicht in den Kopf.

and I have two questions.

1. gehören meaning. Does it mean here as part of with the preposition zu? I know the other meaning of gehören is belong to without zu. What is the meaning here gehören an + AKK? Is the following sentence grammatically valid

Auf und Abs gehören zum Leben (As I translate from English Up and downs are part of life)?

2. Why there is an s after Ab as Abs? Does that imply plural?

Answer 1

1. In

   Auf und Abs gehören an die Börse. Nicht in den Kopf.

   the an belongs to an die Börse, not to gehören. You can see it in the second part of the statement:

   [Auf und Abs gehören] nicht in den Kopf.

   So I think translating gehören + Akk. here with to belong to/in/at fits quite well.

   Das Geschirr gehört in die Küche.
   Oscar gehört in die Tonne.
   Die Brille gehört auf die Nase.

   Is part of is a good translation of gehören zu (Dat.) imo.

   Klappern gehört zum Handwerk.
   Auf und Abs gehören zum Leben.

2. You're right. The s in Auf und Abs marks plural.

Answer 2

I would like to elaborate @Olafants answer a bit.

Question 1: gehören + direction (not necessarily accusative)

The construction in question is not necessarily gehören + accusative, but gehören + direction. In your sentence, the accusative is governed by the preposition an, not by gehören. The direction could as well be given by an adverb instead:

Auf und Abs gehören dorthin, nicht hierher.

Gehören + direction is an idiomatic phrase with several meanings and no complete equivalent in English. This insight is crucial here because in your example, two slightly different meanings are made use of:

1. Auf und Abs gehören an die Börse states that ups and downs should happen on (and be limited to) the stock exchange. The DWDS gives this meaning as subitem of § 3 only by an example:

⟨jmd. gehört ins Bett⟩ jmd. müsste eigentlich ins Bett.

2. [Auf und Abs gehören] nicht in den Kopf, however, matches the negation of the parent item in the DWDS, 'to be in the right place, to fit'. Thus it translates to 'to be out of place, to misfit'. The negation is actually so common that the Duden manages to give only negated examples for this meaning (§ 3) besides the variant mentioned above:

• das Fahrrad gehört nicht in die Wohnung
• das gehört nicht hierher

On the whole, I would translate your example sentence as:

Ups and downs are supposed to happen on the stock exchange, not in the head.

Question 2: Die Auf und Abs (shouldn't it be Auf-und-Abs?)

First of all: yes, it's plural, and seeing that it is a fixed phrase, the plural -s must also modify Auf, otherwise it would read *das Auf und die Abs. This aspect seems quite strange to me.

I found very little resources on the orthographical and grammatical properties of pluralized phrases that consist out of multiple nominalized adverbs. The Duden even introduces its online article on that topic with Doch müssen bei solch einem Hin und Her nicht Bindestriche her? without ever picking up that question again. Apparently, the Duden-Grammatik doesn't concern itself with the plural of multi-part nominalized phrases either. While das Auf und Ab is not hyphenated for some reason, to me it seems necessary to do so if the composition is pluralized as a whole:

die Auf-und-Abs

vs.

?die Aufs und Abs

The latter parallels ups and downs in English, but virtually never occurs in German.