We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We prove that for every sufficiently large integer
$n$
, the polynomial
$1+x+x^{2}/11+x^{3}/111+\cdots +x^{n}/111\ldots 1$
is irreducible over the rationals, where the coefficient of
$x^{k}$
for
$1\leqslant k\leqslant n$
is the reciprocal of the decimal number consisting of
$k$
digits which are each
$1$
. Similar results following from the same techniques are discussed.
We study various families of Artin
$L$
-functions attached to geometric parametrizations of number fields. In each case we find the Sato–Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros.
A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz’s
$2$
-variable
$p$
-adic
$L$
-functions) and algebraic objects (two ‘everywhere unramified’ Iwasawa modules) involving codimension two cycles in a
$2$
-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field
$K$
(where an odd prime
$p$
splits) of an elliptic curve
$E$
, defined over
$\mathbb{Q}$
, with good supersingular reduction at
$p$
. On the analytic side, we consider eight pairs of
$2$
-variable
$p$
-adic
$L$
-functions in this setup (four of the
$2$
-variable
$p$
-adic
$L$
-functions have been constructed by Loeffler and a fifth
$2$
-variable
$p$
-adic
$L$
-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the
$\mathbb{Z}_{p}^{2}$
-extension of
$K$
. We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.
Let
$n$
be a positive integer and
$a$
an integer prime to
$n$
. Multiplication by
$a$
induces a permutation over
$\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$
. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo
$p$
and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary
$k$
th power residues modulo
$p$
and primitive roots modulo a power of
$p$
.
Let
$p$
be an odd prime number and
$E$
an elliptic curve defined over a number field
$F$
with good reduction at every prime of
$F$
above
$p$
. We compute the Euler characteristics of the signed Selmer groups of
$E$
over the cyclotomic
$\mathbb{Z}_{p}$
-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above
$p$
and mixed signs in the definition of the signed Selmer groups.
We generalize our previous method on the subconvexity problem for
$\text{GL}_{2}\times \text{GL}_{1}$
with cuspidal representations to Eisenstein series, and deduce a Burgess-like subconvex bound for Hecke characters, that is, the bound
$|L(1/2,\unicode[STIX]{x1D712})|\ll _{\mathbf{F},\unicode[STIX]{x1D716}}\mathbf{C}(\unicode[STIX]{x1D712})^{1/4-(1-2\unicode[STIX]{x1D703})/16+\unicode[STIX]{x1D716}}$
for varying Hecke characters
$\unicode[STIX]{x1D712}$
over a number field
$\mathbf{F}$
with analytic conductor
$\mathbf{C}(\unicode[STIX]{x1D712})$
. As a main tool, we apply the extended theory of regularized integrals due to Zagier developed in a previous paper to obtain the relevant triple product formulas of Eisenstein series.
Such a sequence is eventually periodic and we denote by
$P(n)$
the maximal period of such sequences for given
$n$
. We prove a new upper bound in the case where
$n$
is a power of a prime
$p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$
for which
$2$
is a primitive root and the Pellian equation
$x^{2}-py^{2}=-4$
has no solutions in odd integers
$x$
and
$y$
.
We study the number of non-trivial simple zeros of the Dedekind zeta-function of a quadratic number field in the rectangle
$\{\unicode[STIX]{x1D70E}+\text{i}t:0<\unicode[STIX]{x1D70E}<1,0<t<T\}$
. We prove that such a number exceeds
$T^{6/7-\unicode[STIX]{x1D700}}$
if
$T$
is sufficiently large. This improves upon the classical lower bound
$T^{6/11}$
established by Conrey et al [Simple zeros of the zeta function of a quadratic number field. I. Invent. Math.86 (1986), 563–576].
For
$\unicode[STIX]{x1D6FC}$
an algebraic integer of any degree
$n\geqslant 2$
, it is known that the discriminants of the orders
$\mathbb{Z}[\unicode[STIX]{x1D6FC}^{k}]$
go to infinity as
$k$
goes to infinity. We give a short proof of this result.
We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.
In this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke
$L$
-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.
In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let
$\ell$
be a prime,
$q$
a prime power and consider the ensemble
${\mathcal{H}}_{g,\ell }$
of
$\ell$
-cyclic covers of
$\mathbb{P}_{\mathbb{F}_{q}}^{1}$
of genus
$g$
. We assume that
$q\not \equiv 0,1~\text{mod}~\ell$
. If
$2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$
, then
${\mathcal{H}}_{g,\ell }$
is empty. Otherwise, the number of rational points on a random curve in
${\mathcal{H}}_{g,\ell }$
distributes as
$\sum _{i=1}^{q+1}X_{i}$
as
$g\rightarrow \infty$
, where
$X_{1},\ldots ,X_{q+1}$
are independent and identically distributed random variables taking the values
$0$
and
$\ell$
with probabilities
$(\ell -1)/\ell$
and
$1/\ell$
, respectively. The novelty of our result is that it works in the absence of a primitive
$\ell$
th root of unity, the presence of which was crucial in previous studies.
We study the variation of
$\unicode[STIX]{x1D707}$
-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the
$p$
-adic zeta function. This lower bound forces these
$\unicode[STIX]{x1D707}$
-invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When
$U_{p}-1$
generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the
$p$
-adic
$L$
-function is simply a power of
$p$
up to a unit (i.e.
$\unicode[STIX]{x1D706}=0$
). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.
A cyclotomic polynomial
$\unicode[STIX]{x1D6F7}_{k}(x)$
is an essential cyclotomic factor of
$f(x)\in \mathbb{Z}[x]$
if
$\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$
and every prime divisor of
$k$
is less than or equal to the number of terms of
$f.$
We show that if a monic polynomial with coefficients from
$\{-1,0,1\}$
has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].
In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products
$f_{d}(x)=\prod _{n=0}^{\infty }(1-x^{-d^{n}})$
,
$d\in \mathbb{N}$
,
$d\geq 2$
, which generalize the generating function
$f_{2}(x)$
of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of
$x^{-d+1}f_{d}(x)$
have a regular structure. We also address the question of whether the corresponding Mahler numbers
$f_{d}(a)\in \mathbb{R}$
,
$a,d\in \mathbb{N}$
,
$a,d\geq 2$
, are badly approximable.
J. Bellaïche and M. Dimitrov showed that the
$p$
-adic eigencurve is smooth but not étale over the weight space at
$p$
-regular theta series attached to a character of a real quadratic field
$F$
in which
$p$
splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over
$F$
at the overconvergent cuspidal Eisenstein points, being the base change lift for
$\text{GL}(2)_{/F}$
of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.
Let
$K$
be a two-dimensional global field of characteristic
$\neq 2$
and let
$V$
be a divisorial set of places of
$K$
. We show that for a given
$n\geqslant 5$
, the set of
$K$
-isomorphism classes of spinor groups
$G=\operatorname{Spin}_{n}(q)$
of nondegenerate
$n$
-dimensional quadratic forms over
$K$
that have good reduction at all
$v\in V$
is finite. This result yields some other finiteness properties, such as the finiteness of the genus
$\mathbf{gen}_{K}(G)$
and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups
$H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$
for
$i\geqslant 1$
established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type
$\mathsf{G}_{2}$
.
Let
$E$
be an elliptic curve over
$\mathbb{Q}$
without complex multiplication. Let
$p\geq 5$
be a prime in
$\mathbb{Q}$
and suppose that
$E$
has good ordinary reduction at
$p$
. We study the dual Selmer group of
$E$
over the compositum of the
$\text{GL}_{2}$
extension and the anticyclotomic
$\mathbb{Z}_{p}$
-extension of an imaginary quadratic extension as an Iwasawa module.
Suppose that
$f(x)=x^{n}+A(Bx+C)^{m}\in \mathbb{Z}[x]$
, with
$n\geq 3$
and
$1\leq m<n$
, is irreducible over
$\mathbb{Q}$
. By explicitly calculating the discriminant of
$f(x)$
, we prove that, when
$\gcd (n,mB)=C=1$
, there exist infinitely many values of
$A$
such that the set
$\{1,\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{2},\ldots ,\unicode[STIX]{x1D703}^{n-1}\}$
is an integral basis for the ring of integers of
$\mathbb{Q}(\unicode[STIX]{x1D703})$
, where
$f(\unicode[STIX]{x1D703})=0$
.
Schertz conjectured that every finite abelian extension of imaginary quadratic fields can be generated by the norm of the Siegel–Ramachandra invariants. We present a conditional proof of his conjecture by means of the characters on class groups and the second Kronecker limit formula.