{"paper":{"title":"L-functions of symmetric powers of the generalized Airy family of exponential sums: ell-adic and p-adic methods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonio Rojas-Leon, C. Douglas Haessig","submitted_at":"2009-08-09T15:18:53Z","abstract_excerpt":"For \\psi a nontrivial additive character on the finite field F_q, the map t \\mapsto \\sum_{x \\in F_q} \\psi(f(x)+tx) is the Fourier transform of the map t \\mapsto \\psi(f(t))$. As is well-known, this has a cohomological interpretation, producing a continuous ell-adic Galois representation. This paper studies the L-function attached to the k-th symmetric power of this representation using both ell-adic and p-adic methods. Using ell-adic techniques, we give an explicit formula for the degree of this L-function and determine the complex absolute values of its roots. Using p-adic techniques, we study"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.1240","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}