Book chapters will be unavailable on Saturday 24th August between 8am-12pm BST. This is for essential maintenance which will provide improved performance going forwards. Please accept our apologies for any inconvenience caused.
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 obtain a non-trivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short intervals, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However, the underlying technique is different and allows us to obtain non-trivial results starting from much shorter ranges.
We improve some previously known deterministic algorithms for finding integer solutions
$x,y$
to the exponential equation of the form
$af^{x}+bg^{y}=c$
over finite fields.
We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato–Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ⩽ n11/2(logn)−1/2+o(1) for a set of n of asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n.
In addition, we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato–Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally and might be within reach unconditionally using the currently established potential automorphy.
$$\begin{eqnarray}\mathfrak{P}_{n}=\mathop{\prod }_{\substack{ p \\ s_{p}(n)\geqslant p}}p,\end{eqnarray}$$
where
$p$
runs over primes and
$s_{p}(n)$
is the sum of the base
$p$
digits of
$n$
. For all
$n$
we prove that
$\mathfrak{P}_{n}$
is divisible by all “small” primes with at most one exception. We also show that
$\mathfrak{P}_{n}$
is large and has many prime factors exceeding
$\sqrt{n}$
, with the largest one exceeding
$n^{20/37}$
. We establish Kellner’s conjecture that the number of prime factors exceeding
$\sqrt{n}$
grows asymptotically as
$\unicode[STIX]{x1D705}\sqrt{n}/\text{log}\,n$
for some constant
$\unicode[STIX]{x1D705}$
with
$\unicode[STIX]{x1D705}=2$
. Further, we compare the sizes of
$\mathfrak{P}_{n}$
and
$\mathfrak{P}_{n+1}$
, leading to the somewhat surprising conclusion that although
$\mathfrak{P}_{n}$
tends to infinity with
$n$
, the inequality
$\mathfrak{P}_{n}>\mathfrak{P}_{n+1}$
is more frequent than its reverse.
We show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J.139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.
We improve a recent result of B. Hanson [Estimates for character sums with various convolutions. Preprint, 2015, arXiv:1509.04354] on multiplicative character sums with expressions of the type
$a+b+cd$
and variables
$a,b,c,d$
from four distinct sets of a finite field. We also consider similar sums with
$a+b(c+d)$
. Our new bounds rely on some recent advances in additive combinatorics.
where
$\,\mathbf{e}_{p}(z)$
is a non-trivial additive character of the prime finite field
$\mathbb{F}_{p}$
of
$p$
elements, with integers
$U$
,
$V$
, a polynomial
$f\in \mathbb{F}_{p}[X]$
and some complex weights
$\{\unicode[STIX]{x1D6FC}_{u}\}$
,
$\{\unicode[STIX]{x1D6FD}_{v}\}$
. In particular, for
$f(X)=aX+b$
, we obtain new bounds of bilinear sums with Kloosterman fractions. We also obtain new bounds for similar sums with multiplicative characters of
$\mathbb{F}_{p}$
.
modulo a prime
$p$
, with variables
$1\leq x_{i}\leq h$
,
$i=1,\ldots ,{\it\nu}$
and arbitrary integers
$s_{j},{\it\lambda}_{j}$
,
$j=1,\ldots ,m$
, for a parameter
$h$
significantly smaller than
$p$
. We also mention some applications of this bound.
Given a finite field of q elements, we consider a trajectory of the map associated with a polynomial ]. Using bounds of character sums, under some mild condition on f, we show that for an appropriate constant C > 0 no N ⩾ Cq½ distinct consecutive elements of such a trajectory are contained in a small subgroup of , improving the trivial lower bound . Using a different technique, we also obtain a similar result for very small values of N. These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.
For an elliptic curve
$E/\mathbb{Q}$
without complex multiplication we study the distribution of Atkin and Elkies primes
$\ell$
, on average, over all good reductions of
$E$
modulo primes
$p$
. We show that, under the generalized Riemann hypothesis, for almost all primes
$p$
there are enough small Elkies primes
$\ell$
to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in
$(\log p)^{4+o(1)}$
expected time.
We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which previously known methods do not apply. In particular, in the case of multiplicative characters modulo a prime
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$
we break the barrier of
$p^{1/4}$
for ranges of individual variables.
with a multiplicative character
${\it\chi}$
modulo
$p$
where
${\mathcal{I}}=\{1,\dots ,H\}$
and
${\mathcal{G}}$
is a subgroup of order
$T$
of the multiplicative group of the finite field of
$p$
elements. A nontrivial upper bound on
$S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$
can be derived from the Burgess bound if
$H\geq p^{1/4+{\it\varepsilon}}$
and from some standard elementary arguments if
$T\geq p^{1/2+{\it\varepsilon}}$
, where
${\it\varepsilon}>0$
is arbitrary. We obtain a nontrivial estimate in a wider range of parameters
$H$
and
$T$
. We also estimate double sums
We give an upper bound for the number of elliptic Carmichael numbers
$n\,\le \,x$
that were recently introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non
$\text{CM}$
). We also discuss several possible further improvements.
We obtain an asymptotic formula for the number of square-free integers in
$N$
consecutive values of polynomials on average over integral polynomials of degree at most
$k$
and of height at most
$H$
, where
$H\,\ge \,{{N}^{k-1+\varepsilon }}$
for some fixed
$\varepsilon \,>\,0$
. Individual results of this kind for polynomials of degree
$k\,>\,3$
, due to A. Granville (1998), are only known under the
$ABC$
-conjecture.
We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime
$p$
. In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below
${p}^{1/ 2} $
, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.
We obtain nontrivial estimates of quadratic character sums of division polynomials
${{\text{ }\!\!\psi\!\!\text{ }}_{n}}\left( P \right)$
,
$n\,=\,1,\,2,\,\ldots $
, evaluated at a given point
$P$
on an elliptic curve over a finite field of
$q$
elements. Our bounds are nontrivial if the order of
$P$
is at least
${{q}^{{}^{1}/{}_{2+\varepsilon }}}$
for some fixed
$\varepsilon \,>\,0$
. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author.
We recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f(X) ∈ ℤ[X] can be detected by a finite algorithm; this property is closely related to the stability of f(X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.