Sato-Tate distributions of Catalan curves. (English) Zbl 1540.11126
Summary: For distinct odd primes \(p\) and \(q\), we define the Catalan curve \(C_{p, q}\) by the affine equation \(y^q = x^p-1\). In this article we construct the Sato-Tate groups of the Jacobians in order to study the limiting distributions of coefficients of their normalized \(L\)-polynomials. Catalan Jacobians are nondegenerate and simple with noncyclic Galois groups (of the endomorphism fields over \(\mathbb{Q}\)), thus making them interesting varieties to study in the context of Sato-Tate groups. We compute both statistical and numerical moments for the limiting distributions. Lastly, we determine the Galois endomorphism types of the Jacobians using both old and new techniques.
MSC:
| 11M50 | Relations with random matrices |
| 11G10 | Abelian varieties of dimension \(> 1\) |
| 11G20 | Curves over finite and local fields |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
Software:
CoCalcReferences:
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