On universal elliptic curves over Igusa curves
Let E be the universal elliptic curve over an Igusa curve. Arithmetic properties of E are studied aiming toward the Birch and Swinnerton-Dyer conjecture. Let K denote the function field of an Igusa curve, and let E be the universal elliptic curve over an Igusa curve. Weierstrass models for E/K, the discriminant and the Hasse invariant of E over \({\mathbb{F}}_ p(j)\) are explicitly determined. Then the Hasse-Weil L-function of E over \({\mathbb{F}}_ p(j)\) is computed in terms of modular forms: Proposition. \(L(E/{\mathbb{F}}_ p(j),s)=H_ q(S_ 3(\Gamma_ 0(p),(\frac{.}{p})),q^{-s})\) where q is a power of p, \((\frac{.}{p})\) the Legendre symbol, and \(H_ q\) is the Hecke polynomial. Further the invariants of E over \({\mathbb{F}}_ p(j)\) such as torsion points, local invariants and Tamagawa numbers are explicitly computed. The functional equation for E is proved: Proposition. Let \(\Lambda(s)=N_ E^{s/2}D^{2s}_{{\mathbb{F}}_ q(j)}L(E/{\mathbb{F}}_ q(j),s)\). Then \(\Lambda(s)= \pm \Lambda(2-s).\) The Tate-Shafarevich group \(\text{Russian {Sh}}({\mathbb{F}}_ p(j),E)\) is computed for some primes p, and the Birch and Swinnerton-Dyer conjecture is verified numerically for these examples.
universal elliptic curve over an Igusa curve
scientific article; zbMATH DE number 4156632
scientific article