More Like this

1. Third Galois cohomology group of function fields of curves over number fields

   https://doi.org/10.2140/ant.2020.14.701  

   Suresh, Venapally (January 2020, Algebra & Number Theory)

   null (Ed.)
2. Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves

   https://doi.org/10.1090/proc/13818  

   Kedlaya, Kiran S.; Temkin, Michael (February 2018, Proceedings of the American Mathematical Society)
3. The size of wild Kloosterman sums in number fields and function fields

   https://doi.org/10.1007/s11854-023-0325-9  

   Sawin, Will (December 2023, Journal d'Analyse Mathématique)

   We study p-adic hyper-Kloosterman sums, a generalization of the Kloosterman sum with a parameter k that recovers the classical Kloosterman sum when k = 2, over general p-adic rings and even equal characteristic local rings. These can be evaluated by a simple stationary phase estimate when k is not divisible by p, giving an essentially sharp bound for their size. We give a more complicated stationary phase estimate to evaluate them in the case when k is divisible by p. This gives both an upper bound and a lower bound showing the upper bound is essentially sharp. This generalizes previously known bounds in the case of ℤ/p. The lower bounds in the equal characteristic case have two applications to function field number theory, showing that certain short interval sums and certain moments of Dirichlet L-functions do not, as one might hope, admit square-root cancellation. 

   more » « less
4. Ray class groups and ray class fields for orders of number fields

   https://doi.org/10.2140/ent.2025.4.1  

   Kopp, Gene S; Lagarias, Jeffrey C (February 2025, Essential Number Theory)

   We contribute to the theory of orders of number fields. We define a notion of ray class group associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order (including Archimedean data), constructed using invertible fractional ideals of the order. We show existence of ray class fields corresponding to the class groups. These ray class groups (resp., ray class fields) specialize to classical ray class groups (resp., fields) of a number field in the case of the maximal order, and they specialize to ring class groups (resp., fields) of orders in the case of trivial modulus. We give exact sequences for simultaneous change of order and change of modulus. As a consequence, we identify the ray class field of an order with a given modulus as a specific subfield of a ray class field of the maximal order with a larger modulus. We also uniquely describe each ray class field of an order in terms of the splitting behavior of primes. 

   more » « less
5. Maximal immediate extensions of valued differential fields: MAXIMAL IMMEDIATE EXTENSIONS OF VALUED DIFFERENTIAL FIELDS

   https://doi.org/10.1112/plms.12128  

   Aschenbrenner, Matthias; van den Dries, Lou; van der Hoeven, Joris (August 2018, Proceedings of the London Mathematical Society)