An infinite family of Artin-Schreier curves with minimal 𝑎-number

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an odd prime and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an algebraically closed field with characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Booher and Cais [Algebra Number Theory 14 (2020), pp. 587–641] showed that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -number of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z slash p double-struck upper Z"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb Z/p \mathbb Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Galois cover of curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon upper Y right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi> ϕ </mml:mi> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi : Y \to X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must be greater than a lower bound determined by the ramification of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> ϕ </mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper, we provide evidence that the lower bound is optimal by finding examples of Artin-Schreier curves that have <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -number equal to its lower bound for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Furthermore we use formal patching to generate infinite families of Artin-Schreier curves with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -number equal to the lower bound in any characteristic.