Frank Thorne

Orbital

We introduce the notion of orbital L-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from the intrinsic interest, results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper (arXiv:1102.2914).

L

For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For instance, we prove that if f is a classical holomorphic modular form whose L-function does not vanish for Re(s) > (k+1)/2, then f is a Hecke eigenform. Our proof adapts and extends work of Saias and Weingartner, who proved a similar result for degree 1 L-functions.

-functions for the Space of Binary Cubic Forms

We prove that the Shintani zeta function associated to the space of binary cubic forms cannot be written as a finite sum of Euler products. Our proof also extends to several closely related Dirichlet series. This answers a question of Wright in the negative.

Zeros of L-functions outside the critical strip

In [ 15 ], Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q . We generalize Shius theorem to imaginary quadratic fields, where we prove the existence of “bubbles” containing arbitrarily many primes which are all, up to units, congruent to a modulo q .

Shintani’s zeta function is not a finite sum of Euler products

In this note, we prove a power-saving remainder term for the function counting

Bubbles of Congruent Primes

S_3

An error estimate for counting S3-sextic number fields

-sextic number fields. We also give a prediction on the second main term. We also present numerical data on counting functions for

On the existence of large degree Galois representations for fields of small discriminant

S_3

Secondary terms in counting functions for cubic fields

-sextic number fields. Our data indicates that our prediction is likely to be correct, and it also suggests the existence of additional lower order terms which we have not yet been able to explain.

Bounded gaps between products of primes with applications to ideal class groups and elliptic curves

Let L=K be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of Gal.L=K/, the sharper of which is conditional on the Artin conjecture and the generalized Riemann hypothesis. Our bound is nontrivial when TK V QU is small and L has small root discriminant, and might be summarized as saying that such fields can’t be “too abelian”.