14,465 research outputs found
Boundary States for AdS₂ Branes in AdS₃
We construct boundary states for the AdS₂ D-branes in AdS₃. We show that, in the semi-classical limit, the boundary states correctly reproduce geometric configurations of these branes. We use the boundary states to compute the one loop free energy of open string stretched between the branes. The result agrees precisely with the open string computation in hep-th/0106129
Elliptic Curve Variants of the Least Quadratic Nonresidue Problem and Linnik's Theorem
Let and be -nonisogenous, semistable
elliptic curves over , having respective conductors and
and both without complex multiplication. For each prime , denote
by the trace of Frobenius. Under the
assumption of the Generalized Riemann Hypothesis (GRH) for the convolved
symmetric power -functions where , we prove an explicit result that can be stated
succinctly as follows: there exists a prime such that
and
This improves and
makes explicit a result of Bucur and Kedlaya.
Now, if is a subinterval with Sato-Tate measure and if
the symmetric power -functions are functorial and
satisfy GRH for all , we employ similar techniques to prove an
explicit result that can be stated succinctly as follows: there exists a prime
such that and
Comment: 30 page
On Logarithmically Benford Sequences
Let be an infinite subset, and let
be a sequence of nonzero real numbers indexed by
such that there exist positive constants for which
for all . Furthermore, let be defined by for each , and suppose the 's are equidistributed in with
respect to a continuous, symmetric probability measure . In this paper, we
show that if is not too sparse, then the
sequence fails to obey Benford's Law with respect
to arithmetic density in any sufficiently large base, and in fact in any base
when is a strictly convex function of . Nonetheless,
we also provide conditions on the density of
under which the sequence satisfies Benford's Law
with respect to logarithmic density in every base.
As an application, we apply our general result to study Benford's Law-type
behavior in the leading digits of Frobenius traces of newforms of positive,
even weight. Our methods of proof build on the work of Jameson, Thorner, and
Ye, who studied the particular case of newforms without complex multiplication.Comment: 10 page
Linnik's Theorem for Sato-Tate Laws on Elliptic Curves with Complex Multiplication
Let be an elliptic curve with complex multiplication (CM), and
for each prime of good reduction, let
denote the trace of Frobenius. By the Hasse bound, for a unique . In this paper, we prove that
the least prime such that
satisfies where is
the conductor of and the implied constant and exponent are absolute
and effectively computable. Our result is an analogue for CM elliptic curves of
Linnik's Theorem for arithmetic progressions, which states that the least prime
for satisfies for an absolute
constant .Comment: 11 pages; made minor modification
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