You can write summae up to half your ability. Round up or round down? I would assume the latter but I have found several instances of rounding up in the system.

Does this means that the maximum level of an ability summae is 5?

I may have given my players two 6 level ability summae without thinking it through...

I personaly dislike the fact that familiar would do anything in their free time which would benefit directly or not their magus.
Do the magus sometimes spend a season to clean the nest of his magical cuckoo?

IMO, and that's how I play it, the familiar help in lab and improves Magic theory, but other abilities are not improved : the familiar use leniency time, improving "enjoy life" ability.

Now, in principle, as OneShot says, you need not do rounding in this particular case: 6 is more than 11/2=5.5 unrounded, so with an Ability score of 11 you would not be able to write a summa level 6. This is the equivalent to rounding down.

However, Ars Magica seems to always round fractions, and when it does it almost always rounds them up (I am aware of 3 exceptions, one in RoP:I and two in Covenants). So, one could easily say that "half of 11" should be interpreted as 6, in the absence of more information. While it proves nothing, it might be worth noting that a character can write a number of Tractatus up to half his score -- and that gets rounded up explicitly. And it might be worth noting, too, that an easy, unequivocal way to say "round down" would be to say: "you can only right a Summa of a certain Level if your score in that Art or Ability at least equals twice that Level".

So in our games, and in most (but definitely not all) games I've seen, troupes tend to round up.

I agree with OneShot on this subject, and have in fact made the exact same point before, both at my local table and here.

But if you read what OneShot wrote, the whole point is that no rounding is necessary.

Score is 7.
Half of 7 is 3.5
If you cannot write a Summa with a vaue higher than half of the score of the book (the bit quoted by OneShot above), then that value cannot exceed 3.5
Since values must be integers (implicit), that means it cannot have a value higher than 3, simply because 4>3.5

If you read anything after my first line, you'd see that was the very first point I addressed In a nutshell, that assumes you want to deal with non-integers in the game mechanics, which the game never does explicitly (as far as I can tell, I may be wrong). If you rule out the existence of non-integers, then half of 11 is not 5.5, because 5.5 does not exist; you must decide whether half of 11 is 5 or 6.

Thank you for the answers. I thought (I don't know why) that 10 was the maximum level of an ability, but as addressed above, that is not true. So it seems that I am good with those 2 6 level ability summae books. I just will be more stingy from now on with the levels of the Summae.

depends.
If you have an affinity in an ability it really makes development go faster- after that it all comes down to available SQ for study. In a game I run here the covenant has fostered a certain level of competitiveness for reputation in excellence, as well as a good educational system for grogs (something to keep teachers busy when they aren't instructing apprentices or occasionally magi, at least at the beginning) and so most craftsmen have written multiple tractatus to show off their high levels of ability, and read tractate from other grogs in the same field to improve the same. Some of those who have craft affinities have reached level 14 or 15 starting an apprenticeship at 5 before they reach age 30 (of course a more skilled master also means a more productive apprenticeship), so it is certainly possible to reach those numbers, even for a (well placed and talented) grog. A magi with over a hundred years to invest in study could certainly do far better, especially if working as a group.

Integer division is also well defined in mathematics. 11 divided by 2 is 5 (with a remainder of 1). 11 divided by 2 is never 6. The only way to get six is to do fractional division AND round up. Reading the text as a mathematician, this is really quite unambiguous. As far as I have seen, rounding is always explicit when it is necessary to disambiguate.