563 research outputs found
The generalized Catalan equation in positive characteristic
Let be a finitely generated field over . In this article we study the generalized Catalan equation in and integers coprime with .Our main result corrects earlier work of Silverman
Malle's conjecture for fair counting functions
We show that the naive adaptation of Malle's conjecture to fair counting
functions is not true in general.Comment: 21 page
Bounds for moments of -torsion in class groups
Fix a number field , integers , and a prime . For all
, we prove strong unconditional upper bounds on the -th moment of
-torsion in the ideal class groups of degree extensions of and of
degree -extensions of , improving upon results of Ellenberg, Pierce
and Wood as well as GRH-conditional results of Frei and Widmer. For large ,
our results are comparable with work of Heath-Brown and Pierce for imaginary
quadratic extensions of . When , our results are new even for
the family of all quadratic extensions of , leading to an improved
upper bound for the count of degree -extensions over
(where is the dihedral group of order ).Comment: 12 page
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