Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori An Introduction to Elliptic Functions A Walking Tour of the Elliptic Zoo David Grabovsky Berenstein High Energy Theory and Gravity Group May 18, 2021 Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori An Introduction to Elliptic Functions A Walking Tour of the Elliptic Zoo David Grabovsky Berenstein High Energy Theory and Gravity Group May 18, 2021 Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori An Introduction to Abelian Integrals A Safari Tour of the Elliptic Prairie David Grabovsky Berenstein Complex and Algebraic Geometry Group May 18, 2021 Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Outline 1 Complex Analysis: Local Theory 2 Functions on Riemann Surfaces First Example: The Riemann Sphere Second Example: The Complex Torus 3 The Abel Map and Abelian Integrals Holomorphic Forms and Jacobi's Theorem Abel's Theorem via Meromorphic Forms 4 Epilogue: Function Theory on Tori Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Outline 1 Complex Analysis: Local Theory 2 Functions on Riemann Surfaces First Example: The Riemann Sphere Second Example: The Complex Torus 3 The Abel Map and Abelian Integrals Holomorphic Forms and Jacobi's Theorem Abel's Theorem via Meromorphic Forms 4 Epilogue: Function Theory on Tori Definition (Holomorphic function) A function f :Ω −! C is holomorphic or analytic on Ω if df f(z + h) − f(z) f 0(z) = ≡ lim (1.1) dz z!h h exists for all z 2 Ω. Complex differentiability implies the Cauchy-Riemann equations, @f ≡ 0: (1.2) @z It turns out that complex differentiability is an extremely strong property. Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Holomorphic Functions Let Ω ⊂ C be an open, connected subset (i.e. a domain) of C. We set i2 = −1 and write z = x + iy 2 C for a complex number. Complex differentiability implies the Cauchy-Riemann equations, @f ≡ 0: (1.2) @z It turns out that complex differentiability is an extremely strong property. Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Holomorphic Functions Let Ω ⊂ C be an open, connected subset (i.e. a domain) of C. We set i2 = −1 and write z = x + iy 2 C for a complex number. Definition (Holomorphic function) A function f :Ω −! C is holomorphic or analytic on Ω if df f(z + h) − f(z) f 0(z) = ≡ lim (1.1) dz z!h h exists for all z 2 Ω. Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Holomorphic Functions Let Ω ⊂ C be an open, connected subset (i.e. a domain) of C. We set i2 = −1 and write z = x + iy 2 C for a complex number. Definition (Holomorphic function) A function f :Ω −! C is holomorphic or analytic on Ω if df f(z + h) − f(z) f 0(z) = ≡ lim (1.1) dz z!h h exists for all z 2 Ω. Complex differentiability implies the Cauchy-Riemann equations, @f ≡ 0: (1.2) @z It turns out that complex differentiability is an extremely strong property. For every domain D ⊂ Ω, f satisfies I f(z) dz = 0 (homotopy formula): (1.3) @D For every disk Dr ⊂ Ω and every z 2 Dr, f satisfies 1 I f(w) f(z) = dw (Cauchy integral formula): (1.4) 2πi @D z − w For every z0 2 Ω, there exists a disk Dr(z0) ⊂ Ω such that 1 X n f(z) = an(z − z0) (Taylor series) (1.5) n=0 ! converges uniformly for all z 2 Dr(z0). In particular, f 2 C (Ω). Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Holomorphic Miracles Theorem (Properties of holomorphic functions) The following are equivalent to the holomorphicity of f :Ω −! C on Ω: For every disk Dr ⊂ Ω and every z 2 Dr, f satisfies 1 I f(w) f(z) = dw (Cauchy integral formula): (1.4) 2πi @D z − w For every z0 2 Ω, there exists a disk Dr(z0) ⊂ Ω such that 1 X n f(z) = an(z − z0) (Taylor series) (1.5) n=0 ! converges uniformly for all z 2 Dr(z0). In particular, f 2 C (Ω). Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Holomorphic Miracles Theorem (Properties of holomorphic functions) The following are equivalent to the holomorphicity of f :Ω −! C on Ω: For every domain D ⊂ Ω, f satisfies I f(z) dz = 0 (homotopy formula): (1.3) @D For every z0 2 Ω, there exists a disk Dr(z0) ⊂ Ω such that 1 X n f(z) = an(z − z0) (Taylor series) (1.5) n=0 ! converges uniformly for all z 2 Dr(z0). In particular, f 2 C (Ω). Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Holomorphic Miracles Theorem (Properties of holomorphic functions) The following are equivalent to the holomorphicity of f :Ω −! C on Ω: For every domain D ⊂ Ω, f satisfies I f(z) dz = 0 (homotopy formula): (1.3) @D For every disk Dr ⊂ Ω and every z 2 Dr, f satisfies 1 I f(w) f(z) = dw (Cauchy integral formula): (1.4) 2πi @D z − w Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Holomorphic Miracles Theorem (Properties of holomorphic functions) The following are equivalent to the holomorphicity of f :Ω −! C on Ω: For every domain D ⊂ Ω, f satisfies I f(z) dz = 0 (homotopy formula): (1.3) @D For every disk Dr ⊂ Ω and every z 2 Dr, f satisfies 1 I f(w) f(z) = dw (Cauchy integral formula): (1.4) 2πi @D z − w For every z0 2 Ω, there exists a disk Dr(z0) ⊂ Ω such that 1 X n f(z) = an(z − z0) (Taylor series) (1.5) n=0 ! converges uniformly for all z 2 Dr(z0). In particular, f 2 C (Ω). Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Local Structure Near Zeros Let f :Ω −! C be a holomorphic function. Proposition (Zeros are isolated) If f is not identically zero, then the zeros of f are isolated: if f(z0) = 0, then there is a neighborhood U of z0 where f(z) 6= 0 for all z 2 U n fz0g. Proof. P n Expand f = n cn(z − z0) . Since f 6= 0, consider the smallest N for N which cN 6= 0. Write f(z) = (z − z0) g(z), where g(z0) = aN 6= 0. N.B. The order of a zero of f at z0 is N. Theorem (Local n-fold covering) Let f have a zero of order n at z0. Then for all v near 0 2 C, there are n points fzig near z0 2 Ω such that f(z1) = ··· f(zn) = v. Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Open Mapping and Maximum Modulus Theorem (Open mapping) If f :Ω −! C is a non-constant holomorphic function, then f maps open sets to open sets. In particular, f(Ω) ⊂ C is open. \Non-constant holomorphic maps are open." Theorem (Maximum modulus principle) If f :Ω −! C is a non-constant holomorphic function, then there cannot exist z0 2 Ω such that jf(z0)j ≥ jf(z)j for all z 2 Ω. \The maximum of jfj can only be attained on @Ω." Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Liouville, Picard, and More Definition (Entire function) A function holomorphic on the whole complex plane is called entire. Theorem (Liouville) Every bounded entire function is constant. Liouville kills off interesting global extensions of holomorphic maps. Corollary (Functions on P1) Every holomorphic function f : P1 = C [ f1g −! C is constant. Theorem (Little Picard) If f : C −! C is entire and non-constant, then the image of f is either the whole complex plane or the plane minus a single point. Definition (Meromorphic function) A function g :Ω −! C is meromorphic on Ω if, for every z0 2 Ω, g(z) can be expressed in terms of its Laurent series: 1 X n g(z) = an(z − z0) ;N 2 Z: (1.6) n=N If N < 0, z0 is a pole of g of order jNj. If N = −1, the pole is simple. If z0 is a pole of g, the residue of g at z0 is Res[g; z0] ≡ a−1. Theorem (Residue) If g :Ω −! C is meromorphic with poles z1; :::; zk inside D ⊂ Ω, then k 1 I X g(z) dz = Res[g; z ]: (1.7) 2πi i @D i=1 Complex Analysis: Local Theory Functions on Riemann Surfaces The Abel Map and Abelian Integrals Epilogue: Function Theory on Tori Meromorphic Functions Holomorphic functions are the world's nicest. The next best thing is..