Learning Path — Modular Forms & Arithmetic Geometry

You can solve representative differentiation, integration and convergence exercises and explain the hypotheses used.

You can explain the quantifiers in a statement and write these forms of proof without relying on a worked solution.

You can work with epsilon arguments and prove standard compactness and convergence results, not just calculate limits.

You can apply Cauchy’s theorem and residue methods with hypotheses; several-variable material may require further preparation.

You can prove rank–nullity, change bases and distinguish diagonalizable maps from general linear maps.

You can construct quotient groups and rings correctly and state the assumptions of the isomorphism theorems.

You can work with holomorphic functions and group actions; check the arithmetic and representation prerequisites of advanced texts.

You can prove elementary arithmetic statements and solve congruences, not just apply memorized formulas.

You can localize a ring or module, work with prime ideals and check exactness in basic module examples.

You can use Dedekind-domain and field-extension arguments; class field theory and cohomology may require more background.