bstract Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword without having to access each of the remaining coordinates. A subfamily of LRC codes with hierarchical locality (H-LRC codes) provides added flexibility to the construction by introducing several tiers of recoverability for correcting different numbers of erasures. We present a general construction of codes with 2-level hierarchical locality from maps between algebraic curves and specialize it to several code families obtained from quotients of curves by a subgroup of the automorphism group, including rational, elliptic, Kummer, and Artin-Schreier curves. We further address the question of H-LRC codes with availability, and suggest a general construction of such codes from fiber products of curves. Detailed calculations of parameters for H-LRC codes with availability are performed for Reed-Solomon-and Hermitian-like code families. Finally, we construct asymptotically good families of H-LRC codes from curves related to the Garcia-Stichtenoth tower.:

Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof's algorithm. While we are currently missing the tools we need to generalize Elkies' methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this article, we prove Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves. IntroductionEfficiently computing the number of points on the Jacobian of a genus 2 curve over a finite field is an important problem in experimental number theory and numbertheoretic cryptography. When the characteristic of the finite field is small, Kedlaya's algorithm and its descendants provide an efficient solution (see [18], [13], and [12]), while in extremely small characteristic we have extremely fast AGM-style algorithms (see for example [25], [26], and [3]). However, the running times of these algorithms are exponential in the size of the field characteristic; the hardest case, therefore (and also the most important case for contemporary cryptographic applications) is where the characteristic is large, or even where the field is a prime field.So let q be a power of a large prime p, and let C be a genus-2 curve over F q . Our fundamental problem is to compute the number of F q -rational points on the Jacobian J C of C .

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