Number Theory in Function Fields

By M. Rosen. 2002

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The hyperring of adèle classes (2011, background): “The Galois group Gal(K : K) is a compact profinite group which acts transitively on the finite sets of valuations extending w over finite algebraic Galois extensions of K ([38], § 9 Proposition 9.”
Polynomial representation growth and the congruence subgroup problem (2004, background): “(b) This is a consequence of the Prime Number Theorem (see [ Ros02 , Theorem 5.12] for the function field case).”
Heights and preperiodic points of polynomials over function fields (2005, methodology): “For more details on function fields and their absolute values, we refer the reader to Chapter 5 of [ 19 ], as well as Chapters 1–2 of [12], Section B.1 of [9], and Section 4.4 of [17].”
The mean value of L(12,χ) in the hyperelliptic ensemble (2012, neutral): “For a general reference, see [16].”
THE ARITHMETIC OF NUMBER RINGS (2008, background): “The analogy between number rings and algebraic curves usually transpires from our wording, but we do not formally treat the case of subrings of function elds [ 29 ].”
Traces of high powers of the Frobenius class in the hyperelliptic ensemble (2008, methodology): “[16] M. Rosen, Number theory in function fields.”
Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field (2008, neutral): “We review some generalities about Dirichlet L-functions for the rational function field; see [14] for details.”