Mathematics / research
Elliptic curves and arithmetic geometry
Connect cubic curves to group laws, rational points, isogenies, local reduction, and arithmetic evidence.
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Module 01 research
Elliptic curves and arithmetic geometry
Connect cubic curves to group laws, rational points, isogenies, local reduction, and arithmetic evidence.
- 01
Weierstrass models and nonsingular cubics
Identify the smooth projective curve behind a cubic equation and distinguish an elliptic curve from a singular drawing.
34 min - 02
The chord–tangent group law
Compute sums on a smooth cubic while separating rational formulas from the proof of associativity.
35 min - 03
Rational functions, divisors and differentials
Track zeros and poles with local parameters and recover the group law through degree-zero divisor classes.
38 min - 04
Isogenies and torsion subgroups
Relate finite morphisms, function-field degree and geometric kernels without confusing separable and inseparable behavior.
39 min - 05
Reduction modulo primes and Hasse's bound
Count points on a smooth finite-field reduction and distinguish singular equations, local models and intrinsic bad reduction.
38 min - 06
Formal groups and local points
Describe points near infinity with a single p-adic parameter and connect reduction kernels to a convergent formal group law.
39 min - 07
Heights and the Mordell–Weil theorem
Use arithmetic size, quadratic normalization and descent to explain finite generation over a number field.
41 min - 08
Descent and Selmer groups
Use local conditions to bound rational-point rank while keeping the remaining Tate–Shafarevich obstruction explicit.
41 min - 09
Complex uniformization and lattices
Build a complex elliptic curve from a rank-two lattice and interpret complex-linear maps through their lattice indices.
40 min - 10
L-functions and the Birch–Swinnerton-Dyer conjecture
Build the rational elliptic L-function from local counts and state exactly which links to rational rank remain conjectural.
43 min