198 research outputs found
Simultaneous Integer Values of Pairs of Quadratic Forms
We prove that a pair of integral quadratic forms in 5 or more variables will
simultaneously represent "almost all" pairs of integers that satisfy the
necessary local conditions, provided that the forms satisfy a suitable
nonsingularity condition. In particular such forms simultaneously attain prime
values if the obvious local conditions hold. The proof uses the circle method,
and in particular pioneers a two-dimensional version of a Kloosterman
refinement.Comment: 63 page
On the size of p-adic Whittaker functions
In this paper we tackle a question raised by N. Templier and A. Saha
concerning the size of Whittaker new vectors appearing in infinite dimensional
representations of GL(2) over non-archimedean fields. We derive precise bounds
for such functions in all possible situations. Our main tool is the p-adic
method of stationary phase.Comment: 41 pages, v4: Minor corrections, including suggestions by the
anonymous Referee. Accepted for publication by the Transactions of the
American Mathematical Societ
Much ado about Mathieu
Eguchi, Ooguri and Tachikawa have observed that the elliptic genus of type II
string theory on K3 surfaces appears to possess a Moonshine for the largest
Mathieu group. Subsequent work by several people established a candidate for
the elliptic genus twisted by each element of M24. In this paper we prove that
the resulting sequence of class functions are true characters of M24, proving
the Eguchi-Ooguri-Tachikawa conjecture. We prove the evenness property of the
multiplicities, as conjectured by several authors. We also identify the role
group cohomology plays in both K3-Mathieu Moonshine and Monstrous Moonshine; in
particular this gives a cohomological interpretation for the non-Fricke
elements in Norton's Generalised Monstrous Moonshine conjecture. We investigate
the intriguing proposal of Gaberdiel-Hohenegger-Volpato that K3-Mathieu
Moonshine lifts to the Conway group Co1.Comment: 38 pages; references added; minor corrections and additions,
including more speculation
On moments of twisted -functions
We study the average of the product of the central values of two
-functions of modular forms and twisted by Dirichlet characters to a
large prime modulus . As our principal tools, we use spectral theory to
develop bounds on averages of shifted convolution sums with differences ranging
over multiples of , and we use the theory of Deligne and Katz to estimate
certain complete exponential sums in several variables and prove new bounds on
bilinear forms in Kloosterman sums with power savings when both variables are
near the square root of . When at least one of the forms and is
non-cuspidal, we obtain an asymptotic formula for the mixed second moment of
twisted -functions with a power saving error term. In particular, when both
are non-cuspidal, this gives a significant improvement on M.~Young's asymptotic
evaluation of the fourth moment of Dirichlet -functions. In the general
case, the asymptotic formula with a power saving is proved under a conjectural
estimate for certain bilinear forms in Kloosterman sums.Comment: final version; to appear in American Journal of Mat
Single-centered black hole microstate degeneracies from instantons in supergravity
We obtain holographic constraints on the microscopic degeneracies of black
holes by computing the exact macroscopic quantum entropy using localization,
including the effects of string worldsheet instantons in the supergravity
effective action. For -BPS black holes in type II string theory on , the constraints can be explicitly checked against expressions
for the microscopic BPS counting functions that are known in terms of certain
mock modular forms. We find that the effect of including the infinite sum over
instantons in the holomorphic prepotential of the supergravity leads to a sum
over Bessel functions with successively sub-leading arguments as in the
Rademacher expansion of Jacobi forms -- but begins to disagree with such a
structure near an order where the mock modular nature becomes relevant. This
leads to a systematic method to recover the polar terms of the microscopic
degeneracies from the degeneracy of instantons (the Gromov-Witten invariants).
We check explicitly that our formula agrees with the known microscopic answer
for the first seven values of the magnetic charge invariant.Comment: 32 pages, comments added in v2, submitted to JHE
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