We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm.Peer ReviewedPostprint (author’s final draft
We study the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for cyclotomic number fields. When the number of primes dividing the conductor is fixed and the size of the coefficients is polynomial we can give a polynomial asymptotic bound for the condition number of the Vandermonde matrix. We refine our bound in the case where the conductor is divisible by at most three primes and we give an asymptotic subexponential formula for the condition number valid for arbitrary degree.
We propose and justify a generalized approach to prove the polynomial reduction of the RLWE to the PLWE problem attached to the ring of integers of a monogenic number field. We prove such equivalence in the case of the maximal totally real subextension of the [Formula: see text]th cyclotomic field, with [Formula: see text] arbitrary prime.
Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over Fp in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve C of genus g ≥ 2 has level 2. We provide a good number of examples and raise a conjecture.
Let
$L/F$
be a quadratic extension of totally real number fields. For any prime
$p$
unramified in
$L$
, we construct a
$p$
-adic
$L$
-function interpolating the central values of the twisted triple product
$L$
-functions attached to a
$p$
-nearly ordinary family of unitary cuspidal automorphic representations of
$\text{Res}_{L\times F/F}(\text{GL}_{2})$
. Furthermore, when
$L/\mathbb{Q}$
is a real quadratic number field and
$p$
is a split prime, we prove a
$p$
-adic Gross–Zagier formula relating the values of the
$p$
-adic
$L$
-function outside the range of interpolation to the syntomic Abel–Jacobi image of generalized Hirzebruch–Zagier cycles.
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