129 research outputs found
Higher rank case of Dwork's conjecture
This is the final version of ANT-0142 ("An embedding approach to Dwork's
conjecture"). It reduces the higher rank case of the conjecture over a general
base variety to the rank one case over the affine space. The general rank one
case is completed in ANT-0235 "Rank one case of Dwork's conjecture". Both
papers will appear in JAMS
T-adic exponential sums over finite fields
-adic exponential sums associated to a Laurent polynomial are
introduced. They interpolate all classical -power order exponential sums
associated to . The Hodge bound for the Newton polygon of -functions of
-adic exponential sums is established. This bound enables us to determine,
for all , the Newton polygons of -functions of -power order
exponential sums associated to an which is ordinary for . Deeper
properties of -functions of -adic exponential sums are also studied.
Along the way, new open problems about the -adic exponential sum itself are
discussed.Comment: new version, 21 pages, title is changed to
Slopes of modular forms and the ghost conjecture
We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any
p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a
conjecture of Buzzard on classical slopes and more recent conjectures on slopes
"at the boundary of weight space".Comment: 17 pages. 2 figures. Minor changes from v1. Final version. To appear
in IMRN. arXiv admin note: text overlap with arXiv:1607.0465
Slopes of Modular Forms and the Ghost Conjecture
We formulate a conjecture on slopes of overconvergent p role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-family: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: normal; vertical-align: baseline; display: inline-table; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative; \u3ep-adic cusp forms of any p role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-family: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: normal; vertical-align: baseline; display: inline-table; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative; \u3ep-adic weight in the Γ0(N) role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-family: inherit; font-variant-caps: inherit; font-stretch: inherit; line-height: normal; vertical-align: baseline; display: inline-table; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative; \u3eΓ0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes “at the boundary of weight space.
Moment Zeta Functions for Toric Calabi-Yau Hypersurfaces
We study in detail the family of Calabi-Yau hypersurfaces defined by
x_1+...+x_n+1/(x_1...x_n)=t over a finite field k. We determine its local and
global monodromy and the trivial factors of its moment zeta function and
Dwork's unit root zeta function
Random eigenvalues of graphenes and the triangulation of plane
We analyse the numbers of closed paths of length on two
important regular lattices: the hexagonal lattice (also called
in chemistry) and its dual triangular lattice. These
numbers form a moment sequence of specific random variables connected to the
distance of a position of a planar random flight (in three steps) from the
origin. Here, we refer to such a random variable as a of the underlying lattice. Explicit formulas for the probability
density and characteristic functions of these random eigenvalues are given for
both the hexagonal and the triangular lattice. Furthermore, it is proven that
both probability distributions can be approximated by a functional of the
random variable uniformly distributed on increasing intervals as
. This yields a straightforward method to simulate these random
eigenvalues without generating graphene and triangular lattice graphs. To
demonstrate this approximation, we first prove a key integral identity for a
specific series containing the third powers of the modified Bessel functions
of th order, . Such series play a crucial role in
various contexts, in particular, in analysis, combinatorics, and theoretical
physics
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