Two-dimensional families of hyperelliptic jacobians with big monodromy

Yuri G. Zarhin

doi:10.1090/tran/6579

Two-dimensional families of hyperelliptic jacobians with big monodromy

Yuri G. Zarhin

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Let K be a global field of characteristic different from 2 and u(x) ∈ K[x] be an irreducible polynomial of even degree 2g ≥ 6 whose Galois group over K is either the full symmetric group (formula presented) or the alternating group A_2g. We describe explicitly how to choose (infinitely many) pairs of distinct t₁, t₂∈ K such that the g-dimensional Jacobian of a hyperelliptic curve y²= (x−t₁)(x−t₂))u(x) has no nontrivial endomorphisms over an algebraic closure of K and has big monodromy.

Original language	English (US)
Pages (from-to)	3651-3672
Number of pages	22
Journal	Transactions of the American Mathematical Society
Volume	368
Issue number	5
DOIs	https://doi.org/10.1090/tran/6579
State	Published - May 2016

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1090/tran/6579

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abstract = "Let K be a global field of characteristic different from 2 and u(x) ∈ K[x] be an irreducible polynomial of even degree 2g ≥ 6 whose Galois group over K is either the full symmetric group (formula presented) or the alternating group A2g. We describe explicitly how to choose (infinitely many) pairs of distinct t1, t2∈ K such that the g-dimensional Jacobian of a hyperelliptic curve y2= (x−t1)(x−t2))u(x) has no nontrivial endomorphisms over an algebraic closure of K and has big monodromy.",

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N2 - Let K be a global field of characteristic different from 2 and u(x) ∈ K[x] be an irreducible polynomial of even degree 2g ≥ 6 whose Galois group over K is either the full symmetric group (formula presented) or the alternating group A2g. We describe explicitly how to choose (infinitely many) pairs of distinct t1, t2∈ K such that the g-dimensional Jacobian of a hyperelliptic curve y2= (x−t1)(x−t2))u(x) has no nontrivial endomorphisms over an algebraic closure of K and has big monodromy.

AB - Let K be a global field of characteristic different from 2 and u(x) ∈ K[x] be an irreducible polynomial of even degree 2g ≥ 6 whose Galois group over K is either the full symmetric group (formula presented) or the alternating group A2g. We describe explicitly how to choose (infinitely many) pairs of distinct t1, t2∈ K such that the g-dimensional Jacobian of a hyperelliptic curve y2= (x−t1)(x−t2))u(x) has no nontrivial endomorphisms over an algebraic closure of K and has big monodromy.

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Two-dimensional families of hyperelliptic jacobians with big monodromy

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