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I am recording a song which is largely written in C major. About halfway through the song, the emphasis changes to A minor, with chord progressions resolving to the Am chord specifically.
Is this something we refer to as modulation? Or does modulation require a change in the number of flats/sharps (key signature) as well?
Yes, this is a modulation to the relative minor, which shares the same key signature. In classical music, although sometimes in popular music too, accidentals would be used to create strong cadences within the minor key. For instance, the leading note of A minor would be raised to G#, particularly when used as part of chord V, to create a major V chord, and so a strong V-I cadence. So, although the key signature doesn't change, often the introduction of raised leading notes (and sometimes sixths, too), can be a clue that you have modulated to the relative minor key (when reading a score, for instance).
When first reading your question title, I thought you might be referring to a move from major to minor while retaining the same tonic note, eg. C Major to C Minor. This is quite different, and is a modulation to the tonic minor. This would require a change of key signature, to three flats in this case, along with (possibly) the characteristic raised leading notes.
@BobBroadley is right, but there's also a point of view in which this particular phenomenon is described by the term "mutation." I don't think the "mutation" people would disagree that it's a modulation, but they find "mutation" more specific and therefore more helpful. (I learned "mutation" myself, but I haven't studied this stuff for a long time so I don't want to speak as an expert.)
Like Bob,from the title, I thought you meant C to Cm, the tonic or parallel minor. But, yes, it's a modulation, if a 'minor' one. Forgive the pun.The key signature won't change, although when reading the dots, there is usually the clue of the G# (in Am), as that note is not diatonic to C, but often is used in Am.
You could say that the C maj. is Ionian, compared with the Aeolian mode that Am brings. All part and parcel of the same family, but with a different centre.
A modulation is any change of tonic note or root pitch, so yes you can modulate without actually changing the key signature.
Interestingly, changing from C major to C minor is not by this definition a modulation in the strictest sense. But it is common that people will also call this dramatic change a "modulation" as well. I don't believe this is strictly correct, but if there is another term for this change in common usage then I don't know of it. Wikipedia refers to this as a parallel key modulation, but that's a bit of an oxymoron if you believe modulation refers to tonal center. But the term modulation itself comes from a time when it was not as common to write music following parallel modes while maintaining the same tonal center. So there you have it.
