### Math 678: Modular Forms : Syllabus (Fall 2012)

This syllabus records what the lectures covered in the course.

 # date material covered HW due dates problems 1 9/5 Intoduction, the modular jungle 2 9/7 Dolgachev: Binary Quadratic Forms and Lattices 3 9/10 Dolgachev: Binary Quadratic Forms and Lattices, 4 9/12 4.1 Oriented lattices and binary quadratic forms, 4.2 n-dimensional quadatic forms 5 9/14 4.3 Complex Tori, Isomorphism classes and modular surface (Dolgachev, Ch. 2) 5.1 Functional equations 6 9/17 5.1 Functional equations (cont.), 6.1 Functional equations-examples HW#1 due 7 9/19 6.2 Averaging to satisfy functional equations, 7.1 Modularity, 8 9/21 7.2. SL(2, R) magic, 8.1 Examples of modular forms 9 9/24 8.1 Examples (cont.) 9.1 New modular forms from old 10 9/26 10.1 Elliptic functions-properties, 10.2 Weierstrass P-function 11 9/28 11.1 Weierstrass P-function-construction 12 10/1 12.1 Laurent expansion of P-function, Eisenstein series. 12.2 Differential equation of P-function, 12.3 two-division points HW#2 due 13 10/3 13.1 Fourier expansion of P(z, \Lambda) in q= e^{2 \pi i \tau} and u= e^2\pi i z/\omega_2) variables 14 10/5 14.1 Field of elliptic functions generated by P, P' 14.2 Weierstrass zeta-function, quasi-periods, Legendre-Weierstrass relations. 15 10/8 15.1 E_2(z, \tau) is quasimodular form, correction term, 15.2 Weierstrass sigma-function 16 10/10 16.0 Weierstrass sigma function, 16.1 elliptic functions from sigma function; 16.2 Elliptic addtion formula, 16/3 Weierstrass P-function is Jacobi form, weight 2 17 10/12 17.1 Chow's theorem, 17.2 projective space and phase factors, 17.3 theta functions, [dolgachev, chap. 3] 10/15-10/16 Fall Break 18 10/17 18.0 theta factors, 18.1 classifying theta functions HW#3 due 19 10/19 19.0 Classifying theta functions (cont.), 19.1 theta functions with chararacteristics, 20 10/22 20.0 Theta characteristics, 201. Zeros of theta functions , 20.2 Projective embeddings of elliptic curves 21 10/24 21.0 projective embedding, cont. 21.1 Division points group action by translation, 21.2 Case k=3. Hesse cubic form of elliptic curve 22 10/26 22.0 Projective group order 9 versus linear group order 27, 22.1 Theta constant, -1/\tau transformation, 22.2 Poisson summation formula 23 10/29 23.0 Fast convergence of theta functions, 23.1 Product formula for Jacobi theta function, 23.2 Jacobi's theorem, \theta_{1/2, 1/2}' is product of three theta constants HW#4 due 24 10/31 24.1 Jacobi's theorem proved 25 11/2 25.1 Determining the multiplier Q(q), 25.2 Applications: Jacobi triple product, 25.3 (Weak) modular properties of \theta_[ab} 26 11/5 26.0 Importance of eta function, 26.1 Jacobi-form type transformation laws for Th(k,\tau). 27 11/7 27.0 Belonging to family, differential equations, 27.1 Proof of Jacobi-transformation law 28 11/9 28.1 Jacobi transformation law for \theta_[ab}, 28.2 Applications weak transformation laws for \delta_{1/2.1/2}^{'} 29 11/12 29.0 Modularity of \delta_{00} on theta group , 29.1 \delta_[ab}^4 weak modularity on \Gamma(2), 29.3 Principal congruence groups, generators \Gamma(2) 29.4 Theta group, generators HW#5 due 30 11/14 30.1 Theta group, cont'd. 30.2 Cusps and their widths, p 31 11/16 31.0 Cusp width examples, 31.1 Modular forms, q-series at cusps, examples 32 11/19 32.1 Modular forms: from Weierstrass P-function:holomorphic Eisenstein series 32.2 Weierstrass function and Jacobi forms, HW#6 due 33 11/21 33.1 Algebra of modular forms; 33.2 Finite dimensionality of holomorphic forms 11/22-11/23 Thanksgiving Break 34 11/26 34.0 Finite dimensionality; generating set 35 11/28 35.1 Modular invarant j(\tau) ; 35.2 Fourier coefficient bounds on Gamma(1) 36 11/30 36.1 Fourier coefficient bounds for cusp forms; 38.2 Modular identities for Eisenstein series 39 12/7 39.0 Branched covers (cont'd); 39.1 Hurwitz formula for branched covers; 39.2 Genus formula for general modular curves, finite index \Gamma ; 39.3 Genus of \Gamma_0(N), \Gamma(N) 40 12/10 40. 0 Elliptic andd parabolic points revisited; 40.1 Dimension of spaces of modular forms, general case HW#7 due 41 12/12 41.0 Jacobi elliptic functions sn, dn; 41.1 Lambda function; 41.2 Picard's little theorem