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Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\\mathbb F_p(t)$, when $p$ is prime and $r\\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\\mathbb F_q(t^1/d)$.
eBook
Rising force : the magic of magnetic levitation
Learn about the force of magnetic levitation and how it can be used to perform illusionary tricks.
Book
Hypergeometric functions over finite fields
Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we
consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions
over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the
classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric
transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As
an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation
formulas, and evaluation formulas. We further apply these finite field formulas to compute the number of rational points of certain
hypergeometric varieties.
eBook
Time-reversal breaking in QCD4, walls, and dualities in 2 + 1 dimensions
A
bstract
We study SU(
N
) Quantum Chromodynamics (QCD) in 3+1 dimensions with
N
f
degenerate fundamental quarks with mass
m
and a
θ
-parameter. For generic
m
and
θ
the theory has a single gapped vacuum. However, as
θ
is varied through
θ
=
π
for large
m
there is a first order transition. For
N
f
= 1 the first order transition line ends at a point with a massless
η
′ particle (for all
N
) and for
N
f
>
1 the first order transition ends at
m
= 0, where, depending on the value of
N
f
, the IR theory has free Nambu-Goldstone bosons, an interacting conformal field theory, or a free gauge theory. Even when the 4
d
bulk is smooth, domain walls and interfaces can have interesting phase transitions separating different 3
d
phases. These turn out to be the phases of the recently studied 3
d
Chern-Simons matter theories, thus relating the dynamics of QCD
4
and QCD
3
, and, in particular, making contact with the recently discussed dualities in 2+1 dimensions. For example, when the massless 4
d
theory has an SU(
N
f
) sigma model, the domain wall theory at low (nonzero) mass supports a 3
d
massless
ℂ
ℙ
N
f
−
1
nonlinear
σ
-model with a Wess-Zumino term, in agreement with the conjectured dynamics in 2+1 dimensions.
Journal Article
An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert’s 17th problem
We prove an elementary recursive bound on the degrees for Hilbert’s 17th problem. More precisely we express a nonnegative polynomial
as a sum of squares of rational functions, and we obtain as degree estimates for the numerators and denominators the following tower of
five exponentials
eBook
Low-energy effective field theory below the electroweak scale: operators and matching
A
bstract
The gauge-invariant operators up to dimension six in the low-energy effective field theory below the electroweak scale are classified. There are 70 Hermitian dimension-five and 3631 Hermitian dimension-six operators that conserve baryon and lepton number, as well as Δ
B
= ±Δ
L
= ±1, Δ
L
= ±2, and Δ
L
= ±4 operators. The matching onto these operators from the Standard Model Effective Field Theory (SMEFT) up to order 1
/
Λ
2
is computed at tree level. SMEFT imposes constraints on the coefficients of the low-energy effective theory, which can be checked experimentally to determine whether the electroweak gauge symmetry is broken by a single fundamental scalar doublet as in SMEFT. Our results, when combined with the one-loop anomalous dimensions of the low-energy theory and the one-loop anomalous dimensions of SMEFT, allow one to compute the low-energy implications of new physics to leading-log accuracy, and combine them consistently with high-energy LHC constraints.
Journal Article