Modular Forms and Elliptic Curves

Description: This is a Master level course at IMPA. The first part of the course is devoted to the theory of elliptic curves. After a quick review of the theory of algebraic curves we study the Mordell-Weil group of elliptic curves over the field of rational numbers, local fields and finite fields. In the second part of the course we develope the theory of modular forms for congruence subgrups of SL(2,Z) and Hecke operators.

References:

1.  The arithmetic of elliptic curves, Joseph Silverman.

2.  Elliptic curves, modular forms and their L-functions, Álvaro Lozano-Robledo.

3.  A first course in modular forms, Fred Diamond and Jerry Shurman.

4.  Problems in the theory of modular forms, M. Ram Murty, Michael Dewar and Hester Graves.

Notes: My lecture notes and solved exercises.

Period and time: January 4 to February 26, 2021. Monday-Tuesday-Wednesday 13h00-15h00 (google meet).

• Lecture 1: Elliptic curves appearing in nature: Congruent number problem and Fermat last theorem. Counting sums, generating functions and q-series: Fibonacci sequence, Euler partition function and q-exponential formula. Jacobi triple product identity and Euler pentagonal number theorem. Counting sums of squares, Jacobi’s 2-square theorem. Arithmetic modularity theorem.

• Lecture 2: Solving Diophantine equations. Affine algebraic varieties. Projective varieties.

• Lecture 3: Ramification theory. Divisors. Differentials. Riemann-Roch theorem and Hurwitz formula.

• Lecture 4: Weierstrass equations. Modular invariant and isomorphism classes. Legendre form.

• Lecture 5: Addition law on rational points of an elliptic curve. Isomorphism with the degree zero subgroup of the Picard group. Announcement of Mordell-Weil theorem and weak Mordell-Weil theorem. Examples of Mordell-Weil groups. Torsion part of Mordell-Weil group, Mazur’s theorem, Nagell-Lutz theorem. Siegel’s theorem and record of ranks of elliptic curves.

• Lecture 6: Review and examples of local fields, Hensel’s lemma. Minimal Weierstrass equations and reduction modulo π. Points with non-singular reduction, kernel of reduction and short exact sequence. Determining torsion points via reduction modulo π, examples. Integrality conditions on torsion points.

• Lecture 7: Formal groups, examples, formal homomorphisms, the formal multiplication by m map. The normalized invariant differential of a formal group, structure of multiplication by m map. Group associated to a formal group over a complete local ring and its torsion points. Formal group of an elliptic curve and its relation with the kernel of reduction modulo π.

• Lecture 8: Kummer pairing and the proof of Weak Mordell-Weil theorem. Descent theorem. Logarithmic height function. Proof of Mordell-Weil theorem over rational numbers.

• Lecture 9: Isogenies, examples: multiplication by m map, elliptic curves with complex multiplication, the Frobenius endomorphism. Ramification theory for isogenies. The action of isogenies on differentials and representation of endomorphisms.

• Lecture 10: The dual isogeny. Quadratic form induced by the degree map. Determining the group of m-torsion points, Tate module. Hasse theorem. Weil conjectures over elliptic curves.

• Lecture 11: Canonical height, Néron-Tate theorem, elliptic regulator and linear independence of rational points. The effective descent method via homogeneous spaces. Bounds on the rank.

• Lecture 12: Divisors on complex tori. Elliptic functions, Weierstrass ℘ function, Eisenstein series, Weierstrass σ function. Elliptic curve representation of a complex torus.

• Lecture 13: The upper-half plane and the full modular group. Weakly modular functions, q-expansions, modular forms, cusp forms. Examples: Eisenstein series, modular j function, ∆ function. The uniformization theorem, periods of elliptic curves, homological interpretation of Tate module.

• Lecture 14: Normalized j function, normalized Eisenstein series, rationality of coefficients. Fundamental domain, the algebra of modular forms is generated by E4 and E6. Congruence subgroups and their modular forms.

• Lecture 15: Eisenstein series of weight 2. Dedekind η function, product formula for ∆, Ramanujan τ function. Eisenstein series of higher level. The extended upper-half plane, modular curves, meromorphic differentials of higher degree.

• Lecture 16: Elliptic points, isotropy groups, periods, charts around elliptic points. Width of cusps, charts around cusps. GAGA principle, Hurwitz formula (analytic version), genus of modular curves. Automorphic forms, isomorphism with meromorphic differentials.

• Lecture 17: Regular and irregular cusps, divisor of an automorphic form. Identification of spaces of modular forms and cusp forms with Riemann-Roch spaces, dimension formulas. The Abel-Jacobi theorem and construction of non-trivial automorphic form of weight 1 for a congruence subgroup not containing −Id.

• Lecture 18: Hecke operators of level 1 and their Fourier expansions. First two parts of Ramanujan conjecture. Hecke eigenforms, commutativity of Hecke operators. Petersson inner product, adjointness of Hecke operators.

• Lecture 19: The order of Fourier coefficients of cusp forms and a weaker version of third Ramanujan conjecture. L-series attached to a cusp form for the full modular group, analytic continuation and functional equation. More examples of Dirichlet series, their Euler product formulas and functional equations. Hasse-Weil L-series attached to an elliptic curve and its functional equation. Birch and Swinnerton-Dyer conjecture and the parity conjecture.

• Lecture 20: Hecke operators for Γ1(N), nebentypus, q-expansion of Hecke operators. Diamond operators, Hecke eigenforms and commutativity of operators. Oldforms and newforms, Atkin-Lehner theorem on the multiplicity one property for newforms.

• Lecture 21: L-series attached to modular forms of higher level, Mellin transform. WN operator, analytic continuation of Mellin transform, functional equation for L-series attached to eigenforms of WN. Modularity theorem, functional equation and analytic continuation of Hasse-Weil L-series. Quasi-modular forms, Ramanujan relations of Eisenstein series. Calabi-Yau varieties, Gauss-Manin connection, enhanced elliptic curves, Movasati’s theorem on Ramanujan vector field, enhanced Calabi-Yau varieties, Calabi-Yau modular forms, Dwork family and Youkawa coupling.