Под фото есть подпись.

Because есть here does not just mean the missing present-tense is. It has an existential meaning: there is / there exists.

So:

Под фото есть подпись. = There is a caption under the photo.

Compare:

• Это подпись. = This is a caption.
  Here Russian does not use есть.
• Под фото есть подпись. = There is a caption under the photo.
  Here есть is natural because you are stating that something exists in that place.

In everyday Russian, есть is often omitted in some contexts, but in this sentence it sounds normal and clear.

Because под can take different cases depending on the meaning:

• location: under / beneath → usually instrumental
• direction/movement: to under → usually accusative

Here the meaning is location: the caption is already positioned under the photo. So Russian uses the instrumental pattern.

With a normal declinable noun, you would clearly see that:

• под столом = under the table
• под фотографией = under the photograph

But фото is an indeclinable noun, so its form stays фото in all cases. Even though the case required is instrumental, the word itself does not change.

So:

• под фото = under the photo
• под фотографией = under the photograph

Both are possible; фото is just shorter.

Подпись is in the nominative singular.

That is because it is the thing that exists in the sentence:

• есть подпись = there is a caption

In existential sentences with есть, the thing being introduced is usually in the nominative:

• На столе есть книга. = There is a book on the table.
• Под фото есть подпись. = There is a caption under the photo.

A useful comparison is the negative version:

• Под фото нет подписи. = There is no caption under the photo.

After нет, Russian uses the genitive: подписи.

Yes. Фото is a common shorter word for photograph / photo, while фотография is the full form.

So both are possible:

• Под фото есть подпись.
• Под фотографией есть подпись.

The main differences are:

• фото: shorter, very common in everyday speech
• фотография: fuller, a bit more formal or neutral

There is also a grammar difference:

• фото is indeclinable
• фотография declines normally, so after под it becomes под фотографией

Подпись can mean different things depending on context:

1. signature
   as in writing your name on a document

2. caption
   as in text under a photo or image

In Под фото есть подпись, it means caption.

This is a common source of confusion because English usually separates these meanings more clearly.

Also compare надпись:

• подпись = caption, or signature
• надпись = inscription, written text, lettering

So under a photo, подпись is the right word for a caption.

Russian word order is flexible, but different orders sound slightly different in focus.

The original:

Под фото есть подпись.

This is very natural if you want to start with the location under the photo and then say what is there.

Also possible:

Есть подпись под фото.

This can work too, though it may sound a little less neutral depending on context.

Less natural in most ordinary situations:

Подпись есть под фото.

This sounds more contrastive or emphatic, as if you are insisting: There is a caption under the photo.

So the original sentence is a very good neutral version.

Because фото is one of those Russian nouns that are indeclinable.

That means the word keeps the same form in different cases:

• фото
• нет фото
• к фото
• под фото

The grammar still works normally: the preposition still requires a certain case, but the noun itself does not show a visible ending change.

This is common with some borrowed or shortened nouns in Russian.

You would say:

Под фото нет подписи.

Notice the change:

• positive: есть подпись
• negative: нет подписи

After нет, Russian uses the genitive, so подпись becomes подписи.

This is a very useful pattern:

• На столе есть книга. = There is a book on the table.

• На столе нет книги. = There is no book on the table.

• Под фото есть подпись.
• Под фото нет подписи.

The stress is:

Под фо́то есть по́дпись.

Word by word:

• под — usually unstressed
• фо́то — stress on the first syllable
• есть — usually stressed lightly if said clearly
• по́дпись — stress on the first syllable

A natural pronunciation would sound roughly like:

под ФО-то есть ПОД-пись

The unstressed vowels may sound weaker in normal speech, but those are the main stress positions.