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I have a fair enough knowledge on Real Analysis, Complex Analysis and I am presently pursuing a course in Elementary Abstract Algebra. I actually want to understand the concepts of Elliptic curves and Modular forms.So what are the prerequisites to understand Algebraic and Differential Geometry?

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    $\begingroup$ Algebraic geometry and differential geometry are not really related to each other very strongly. Over the real or complex numbers they study some of the same things, but in drastically different ways. $\endgroup$ – Matt Samuel Feb 13 '16 at 4:03
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You can understand quite a bit of classical differential geometry using nothing more than multivariate calculus and perhaps (optionally) a little bit of linear algebra. Take a look at the book by Pressley, for example.

As far as I know (and I'm not an expert in these areas), algebraic geometry is not closely related to differential geometry. I don't know any good modern books on this topic. The ones I have seen are very abstract -- too much algebra and too little geometry, in my view. Your tastes may differ, of course.

There are lots of AG pointers here, from people who know much more about this area than I do.

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Many Ring's Theory are worth to have, which includes Module's Theory. These two form the land of Commutative Algebra.

For Differential Geometry, lots of Linear and Multilinear Algebra.

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