Counting abelian extensions by Artin–Schreier conductor
F. Gundlach, Proceedings of the American Mathematical Society (2025).
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Abstract
<p>
Let
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<mml:mi>G</mml:mi>
<mml:annotation encoding="application/x-tex">G</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
be a finite abelian
<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>
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-group. We count étale
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G">
<mml:semantics>
<mml:mi>G</mml:mi>
<mml:annotation encoding="application/x-tex">G</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-extensions of global rational function fields
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q Baseline left-parenthesis upper T right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="double-struck">F</mml:mi>
</mml:mrow>
<mml:mi>q</mml:mi>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>T</mml:mi>
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</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbb F_q(T)</mml:annotation>
</mml:semantics>
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</inline-formula>
of characteristic
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<mml:mi>p</mml:mi>
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by the degree of what we call their Artin–Schreier conductor. The corresponding (ordinary) generating function turns out to be rational. This gives an exact answer to the counting problem, and seems to beg for a geometric interpretation.
</p>
<p>This is in contrast with the generating functions for the ordinary conductor (from class field theory) and the discriminant, which in general have no meromorphic continuation to the entire complex plane.</p>
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Cite this
Gundlach F. Counting abelian extensions by Artin–Schreier conductor. Proceedings of the American Mathematical Society. Published online 2025. doi:10.1090/proc/17440
Gundlach, F. (2025). Counting abelian extensions by Artin–Schreier conductor. Proceedings of the American Mathematical Society. https://doi.org/10.1090/proc/17440
@article{Gundlach_2025, title={Counting abelian extensions by Artin–Schreier conductor}, DOI={10.1090/proc/17440}, journal={Proceedings of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Gundlach, Fabian}, year={2025} }
Gundlach, Fabian. “Counting Abelian Extensions by Artin–Schreier Conductor.” Proceedings of the American Mathematical Society, 2025. https://doi.org/10.1090/proc/17440.
F. Gundlach, “Counting abelian extensions by Artin–Schreier conductor,” Proceedings of the American Mathematical Society, 2025, doi: 10.1090/proc/17440.
Gundlach, Fabian. “Counting Abelian Extensions by Artin–Schreier Conductor.” Proceedings of the American Mathematical Society, American Mathematical Society (AMS), 2025, doi:10.1090/proc/17440.
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