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 |
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| 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 |
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| 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) |
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| 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 |