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Obstructions to reciprocity laws related to division fields of Drinfeld modules.
- Source :
-
Proceedings of the American Mathematical Society . Apr2023, Vol. 151 Issue 4, p1379-1393. 15p. - Publication Year :
- 2023
-
Abstract
- Let q be an odd prime power, denote by \mathbb {F}_q the finite field with q elements, and set A ≔\mathbb {F}_q[T], F ≔\mathbb {F}_q(T). Let \psi : A \to F\{\tau \} be a Drinfeld A-module over F, of rank r \geq 2, with End_{\overline {F}}(\psi) = A. For a non-zero ideal \mathfrak {n} of A, denote its unique monic generator by n, denote the degree of n as a polynomial in T by \deg n, and denote the \mathfrak {n}-division field of \psi by F(\psi [\mathfrak {n}]). A reciprocity law for \psi asserts that, if \gcd (charF, r) = 1 or if \mathfrak {n} is prime, then a non-zero prime ideal \mathfrak {p} \nmid \mathfrak {n} of A splits completely in F(\psi [\mathfrak {n}]) if and only if the Frobenius trace a_{1, \mathfrak {p}}(\psi) of \psi at \mathfrak {p} and the first component b_{1, \mathfrak {p}}(\psi) of the Frobenius index of \psi at \mathfrak {p} satisfy the congruences a_{1, \mathfrak {p}}(\psi) \equiv -r \pmod n and b_{1, \mathfrak {p}}(\psi) \equiv 0 \pmod n. We find the Dirichlet density of the set of non-zero prime ideals \mathfrak {p} for which the latter congruence never holds, that is, for which b_{1, \mathfrak {p}}(\psi) = 1. Using similar methods, we prove an asymptotic formula for the function of x defined by the average \frac {1}{\#\{\mathfrak {p}: \ \deg p = x \}} \sum _{\mathfrak {p}: \ \deg p = x} \tau _A(b_{1, \mathfrak {p}}(\psi)), where \mathfrak {p} = A p denotes an arbitrary non-zero prime ideal of A whose monic generator p \in A has degree x and where \tau _A(b_{1, \mathfrak {p}}(\psi)) denotes the number of monic divisors of b_{1, \mathfrak {p}}(\psi). [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 151
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 161956682
- Full Text :
- https://doi.org/10.1090/proc/16157