Gamma Factors, Root Numbers, and Distinction

Root numbers

Neighborhood Density and the Root-Affix Distinction

Work in phonology and morphology consistently recognizes that affixes show a strong tendency to be less marked than roots crosslinguistically. Within Optimality Theory, this observation has been dealt with through the imposition of a universally-fixed ranking, known as the Root-Affix Faithfulness Metaconstraint (McCarthy and Prince 1995), which states that faithfulness to root material universa...

Galois Theory, Elliptic Curves, and Root Numbers

The inverse problem of Galois theory asks whether an arbitrary finite group G can be realized as Gal(K/Q) for some Galois extension K of Q. When such a realization has been given for a particular G then a natural sequel is to find arithmetical realizations of the irreducible representations of G. One possibility is to ask for realizations in the Mordell-Weil groups of elliptic curves over Q: Gi...

Root Numbers and Ranks in Positive Characteristic

For a global fieldK and an elliptic curve Eη overK(T ), Silverman’s specialization theorem implies rank(Eη(K(T ))) ≤ rank(Et(K)) for all but finitely many t ∈ P(K). If this inequality is strict for all but finitely many t, the elliptic curve Eη 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 ...

Euler Factors and Local Root Numbers for Symmetric Powers of Elliptic Curves

For any elliptic curve E over a number field, there is, for each n ≥ 1, a symmetric n-power L-function, defined by an Euler product, and conjecturally having a meromorphic continuation and satisfying a precise functional equation. The sign in the functional equation is conjecturally a product of local signs. Given an elliptic curve over a finite extension of some Qp, we calculate the associated...