Superelliptic curves with large Galois images

Abstract Let $$r&gt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> be primes. In this paper we study the mod $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> Galois representations attached to curves of the form $$y^r = f(x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> where f is monic and has coefficients belonging to the r th cyclotomic field. We provide conditions on the coefficients (and degree) of f which allow one to verify the mod $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> image is large outside of a (typically small) finite explicit set of primes. We allow all values of r for which the r th cyclotomic field has odd class number. This appears to be the first explicit result for abelian varieties of dimension greater than two and not of $${{\,\mathrm{\textrm{GL}}\,}}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>GL</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -type which allows the ground field to have unramified extensions. To determine the exact image we study the “endomorphism character”, a certain algebraic Hecke character which generalises the CM character. This is achieved in entirety when $$r=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . To the author’s knowledge, this is the first accurate description of the full image in the literature. Finally, we give several examples with genus ranging from 10 to 36. Applications to the Inverse Galois Problem are also included.