A Scrapbook of Complex Curve Theory

Spectral Problems for Quasinormal Modes of Black Holes

“…Such ODEs for contour integrals are known as Picard-Fuchs equations [53]. To find the Picard-Fuchs Equation ( 136), the partial differential Equation (135) for p 0 is crucial.…”

Special Hypergeometric Motives and Their L-Functions: Asai Recognition

“…A hypergeometric motive over Q arises from a parametric family of varieties with certain periods (conjecturally) satisfying a hypergeometric differential equation; the construction of this family was made explicit by Beukers-Cohen-Mellit [3] based on work of Katz [29]. Following the analogy between periods and point counts (Manin's "unity of mathematics" [7]), counting points on the reduction of these varieties over finite fields is accomplished via finite field hypergeometric functions, a notion originating in work of Greene [17] and Katz [29]. These finite sums are analogous to truncated hypergeometric series in which Pochhammer symbols are replaced with Gauss sums, and they provide an efficient mechanism for computing the L-functions of hypergeometric motives.…”

Formulae for null curves deriving from elliptic curves

“…Thus the curve S may be brought to any one of the standard forms; cf. [2,7]. The corresponding bundle automorphism of T induces via the adjoint representation an SO(3, C)-rotation of H 0 (P 1 , O(T)) [10].…”

“…Here M * is obtained from M by deleting a finite number of points, and d f /dg = f /g , d 2 f /dg 2 = (d f /dg) /g , etc.…”