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Boundedness results for 2‐adic Galois images associated to hyperelliptic Jacobians
Author(s) -
Yelton Jeffrey
Publication year - 2021
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800244
Subject(s) - mathematics , monic polynomial , discriminant , pure mathematics , galois group , algebraic number field , prime power , elliptic curve , jacobian matrix and determinant , prime (order theory) , discrete mathematics , polynomial , combinatorics , mathematical analysis , artificial intelligence , computer science
Let K be a number field, and let C be a hyperelliptic curve over K with Jacobian J . Suppose that C is defined by an equation of the formy 2 = f ( x ) ( x − λ )for some irreducible monic polynomial f ∈ O K [ x ]of discriminant Δ and some element λ ∈ O K . Our first main result says that if there is a prime p of K dividing ( f ( λ ) ) but not (2Δ), then the image of the natural 2‐adic Galois representation is open in GSp ( T 2 ( J ) ) and contains a certain congruence subgroup of Sp ( T 2 ( J ) ) depending on the maximal power of p dividing ( f ( λ ) ) . We also present and prove a variant of this result that applies when C is defined by an equation of the formy 2 = f ( x ) ( x − λ ) ( x − λ ′ )for distinct elements λ , λ ′ ∈ K . We then show that the hypothesis in the former statement holds for almost all λ ∈ O Kand prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.