The third conditional for "if I could"

What is the third conditional for "if I could"? For example, we say:

If I had studied hard, I would have passed the exam. How about this:

If I could study, I would have passed the exam.[Is there a past perfect for could?]

Although could is the past tense of can, past tense in modals often doesn't mean past time. In the if I could, the past tense could has a counterfactual meaning rather than a past time meaning. Of course, the verb following the modal can't be past tense, so to give it a past time meaning, use the perfect aspect (i.e., could + have + past participle). So your example about studying could be changed to

If I could have studied, I would have passed.

Another option, as Irene has pointed out, is to avoid could and use be able, leading to

If I had been able to study, I would have passed.

The idea of zero, first, second, and third conditional has been much criticszed. See, for example, here.

When you want to use could in the Past Perfect, you use the expression be able to in the correct form. So your sentence will be:

If I had been able to study, I would have passed the exam.

The same rule applies when you want to use the Present Perfect form:

I haven't been able to talk to him recently.

Or the Future form:

I will be able to travel around the world some day.

There are two moods of speech at play here: the subjunctive and optative moods. Sometimes, you have to decide which of those two you are expressing.

Frequently, non-native speakers of English ignore the concepts of subjunctive and optative moods.

It is usual for people to say,

If you take your shoes off before coming in, I will be a very happy host.

However, you could also say,

If you took your shoes off before coming in, I would be a very happy host.

The subjunctive mood is found in most Romance and Germanic languages as well as ancient Sanskrit.

In English, the subjunctive/optative moods are designated by using the past tense.

Subjunctive mood
The mood implies the notion of impossibility or improbability. The famous example found in many books introducing the subjunctive is

If I were a bird, I would sing all day.

You should not say

If I am a bird, I will sing all day

because it is impossible for you to be a bird. And since the English language does not have a different set of tenses for the subjunctive/optative moods, we fall into using the past tense - in this case were and would.

Politeness and the optative mood
Frequently, non-native speakers of English do not realise that the following is essentially rude or assertive:

Can you please take off your shoes before coming in?

You should say

Could you please take your shoes off before coming in?

It is spoken in a combination of the optative and subjunctive moods. It bears an underlying message that the speaker is saying

• you have a choice, it is not imperative that you take your shoes off. However, it would be the joy of my heart if you refrain from muddying up the floor of my living room.
• I realise it is impossible for me to compel you to take your shoes off. However, if you decide to do the impossible for me, you will be my hero.

The present, past and future tenses

Future assertive mood:

When you bake a cake, I will certainly eat it.

After you have baked me a cake, I will certainly eat it.

Future optative:

If you baked me a cake, I would certainly eat it.

If you could bake me a cake, I would certainly eat it.

You are saying to your sister - I know it would trouble you a lot, but please, please, pretty please bake me a cake.

Future subjunctive:

Even if you were to bake me a cake, I would certainly not eat it.

Even if you could bake me a cake, I could certainly not eat it.

Were you to bake me a cake, would I even eat it?

You are telling your sister - It is impossible for you to bake a cake because you lack the skills to bake it, but even if you could, it would certainly be so bad that I would not eat it.

Present assertive:

You are baking a cake and I am expecting to eat it.

Present optative:

You could be baking a cake now, and I could be standing here watching and hoping to eat it.

Present subjunctive:

If you were baking a cake, I would be standing here watching and hoping to eat it. (But you are not because you are too occupied with FB or SO).

Past assertive:

You should have baked me the cake and I would have eaten it.

Past optative denoted by the past perfect:

If you had baked me a cake, I would have eaten it.

Had you baked me the cake, I would have eaten it.

If you had baked me a cake, I would have eaten it.

Past subjunctive:

Had you baked me that silly cake, would I have even eaten it?

Had you studied harder, would you even have passed the exam?

We could even extend an extra leg to conjure a past perfect subjunctive/optative by cascading subjunctives/optatives:

Had we been in the situation of you having baked me a cake, I would have been feeling the effects of having eaten your cake.

We could have an infinite recursion of subjunctives/optatives:

Had we been in the status where we could have been in the status of being in the status of being in ..... of having you baked me a cake, I would have been in the status where I could have been in the status of being in the status of being in .... of having eaten that cake.