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I need a good reference about the connection between modular forms and line bundles. I found only Milne's note that treats briefly this argument. I've already checked, but without finding anything, many books such as:

  • "A first course in modular forms - F. Diamond, J. Shurman"
  • "Elliptic curves and modular forms - N.Koblitz"
  • "Modular Forms - Miyake"
  • "A course in arithmetic - J.P.Serre" (the last chapter)
  • "Complex analysis - E.Freitag, R.Busam"
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Just to make sure, what exactly do you want to know that is not covered in Milne's notes? – M Turgeon Sep 24 '12 at 20:50
The problem is the following: I took notes of the course, and the relation between line bundles and modular forms is a large enough subject (about 10 pages). Unfortunately these notes are not very clear, so I need some reference. Milne's note treats the subject in a single page. – Dubious Sep 24 '12 at 20:57
To be more precise the argumets I need concerning: automorphy factors, classification of line bundles on $\mathcal H/\Gamma$ and modular forms... – Dubious Sep 24 '12 at 21:21
There is not much substance to "line bundles versus modular forms"... mostly just a way of speaking, so don't look for any big mystery to be uncovered. In fact, the non-content is why many sources don't bother to speak in such terms. – paul garrett Sep 25 '12 at 0:17
... more likely, a "non-modular" reading that treats "line bundles" might provide some illumination of that general idea. In fact, you should read about "vector bundles", to better understand the simplest case, "line bundles". Sources on algebraic geometry, even "just" analytic geometry, such as Griffiths-Harris, discuss vector bundles (and line bundles) at great length. Looking at such sources you will see that, in the basic complex-analytic context, saying "line bundle" adds little to the discussion of modular forms. – paul garrett Sep 25 '12 at 0:20

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