A number of constructions in function field arithmetic involve extensions from linear objects using digit expansions. This technique is described here as a method of constructing orthonormal bases in spaces of continuous functions. We illustrate several examples of orthonormal bases from this viewpoint, and we also obtain a concrete model for the continuous functions on the integers of a local field as a quotient of a Tate algebra in countably many variables. Academic Press
Abstract. The initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve L-function at s = 1. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of √ 2. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general √ 2 phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seems much deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.
Abstract. For a prime polynomial f (T ) ∈ Z[T ], a classical conjecture predicts how often f has prime values. For a finite field κ and a prime polynomial f (T ) ∈ κ [u][T ], the natural analogue of this conjecture (a prediction for how often f takes prime values on κ [u]) is not generally true when f (T ) is a polynomial in T p (p the characteristic of κ). The explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero Möbius values µ(f (g)) as g varies. We prove the surprising fact that this "Möbius average," which can be defined without reference to any conjectures, has a periodic behavior governed by the geometry of the plane curve f = 0.The periodic Möbius average behavior implies in specific examples that a polynomial in κ [u][T ] does not take prime values as often as analogies with Z[T ] suggest, and it leads to a modified conjecture for how often prime values occur.
For a global field K and an elliptic curve E over K(T ), Silverman's specialization theorem implies rank(E (K(T ))) rank(E t (K)) for all but finitely many t ∈ P 1 (K). If this inequality is strict for all but finitely many t, the elliptic curve E is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T ) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K = (u) over any finite field with characteristic = 2, we construct an explicit 2-parameter family E c,d of non-isotrivial elliptic curves over K(T ) (depending on arbitrary c, d ∈ × ) such that, under the parity conjecture, each E c,d has elevated rank.
We examine a q-analogue of Mahler expansions for continuous functions in p-adic analysis, replacing binomial coefficient polynomials ( x n ) with a q-analogue ( x n ) q for a p-adic variable q with |q&1| p <1. Mahler expansions are recovered at q=1 and we consider the p-adic q-Gamma function 1 p, q of Koblitz relative to its q-Mahler expansion. Academic Press
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