Sato-Tate groups of 𝑦²=𝑥⁸+𝑐 and 𝑦²=𝑥⁷-𝑐𝑥.
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Publication:3187546
DOI10.1090/CONM/663/13351zbMATH Open1411.11090arXiv1412.0125OpenAlexW269114512MaRDI QIDQ3187546
Author name not available (Why is that?)
Publication date: 2 September 2016
Published in: (Search for Journal in Brave)
Abstract: We consider the distribution of normalized Frobenius traces for two families of genus 3 hyperelliptic curves over Q that have large automorphism groups: y^2=x^8+c and y^2=x^7-cx with c in Q*. We give efficient algorithms to compute the trace of Frobenius for curves in these families at primes of good reduction. Using data generated by these algorithms, we obtain a heuristic description of the Sato-Tate groups that arise, both generically and for particular values of c. We then prove that these heuristic descriptions are correct by explicitly computing the Sato-Tate groups via the correspondence between Sato-Tate groups and Galois endomorphism types.
Full work available at URL: https://arxiv.org/abs/1412.0125
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