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We investigate a novel geometric Iwasawa theory for ${\mathbf Z}_p$-extensions of function fields over a perfect field k of characteristic $p>0$ by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over k associated with a ${\mathbf Z}_p$-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty $. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$-module structure of the space $M_n:=H^0(X_n, \Omega ^1_{X_n/k})$ of global regular differential forms as $n\rightarrow \infty .$ For example, for each tower in a basic class of ${\mathbf Z}_p$-towers, we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + \lambda _r n + c_r(n)$ for all n sufficiently large, where $a_r, \lambda _r$ are rational constants and $c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$ is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on ${\mathbf Z}_p$-towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$.
Let $p$ be a prime number. Kęstutis Česnavičius proved that for an abelian variety $A$ over a global field $K$, the $p$-Selmer group $\mathrm {Sel}_{p}(A/L)$ grows unboundedly when $L$ ranges over the $(\mathbb {Z}/p\mathbb {Z})$-extensions of $K$. Moreover, he raised a further problem: is $\dim _{\mathbb {F}_{p}} \text{III} (A/L) [p]$ also unbounded under the above conditions? In this paper, we give a positive answer to this problem in the case $p \neq \mathrm {char}\,K$. As an application, this result enables us to generalize the work of Clark, Sharif and Creutz on the growth of potential $\text{III}$ in cyclic extensions. We also answer a problem proposed by Lim and Murty concerning the growth of the fine Tate–Shafarevich groups.
Given a singular modulus
$j_0$
and a set of rational primes S, we study the problem of effectively determining the set of singular moduli j such that
$j-j_0$
is an S-unit. For every
$j_0 \neq 0$
, we provide an effective way of finding this set for infinitely many choices of S. The same is true if
$j_0=0$
and we assume the Generalised Riemann Hypothesis. Certain numerical experiments will also lead to the formulation of a “uniformity conjecture” for singular S-units.
Let
$m>1$
and
$\mathfrak {d} \neq 0$
be integers such that
$v_{p}(\mathfrak {d}) \neq m$
for any prime p. We construct a matrix
$A(\mathfrak {d})$
of size
$(m-1) \times (m-1)$
depending on only of
$\mathfrak {d}$
with the following property: For any tame
$ \mathbb {Z}/m \mathbb {Z}$
-number field K of discriminant
$\mathfrak {d}$
, the matrix
$A(\mathfrak {d})$
represents the Gram matrix of the integral trace-zero form of K. In particular, we have that the integral trace-zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant.
A class of exotic
$_3F_2(1)$
-series is examined by integral representations, which enables the authors to present relatively easier proofs for a few remarkable formulae. By means of the linearization method, these
$_3F_2(1)$
-series are further extended with two integer parameters. A general summation theorem is explicitly established for these extended series, and several sample summation identities are highlighted as consequences.
By analogy with the trace of an algebraic integer
$\alpha $
with conjugates
$\alpha _1=\alpha , \ldots , \alpha _d$
, we define the G-measure
$ {\mathrm {G}} (\alpha )= \sum _{i=1}^d ( |\alpha _i| + 1/ | \alpha _i | )$
and the absolute
${\mathrm G}$
-measure
${\mathrm {g}}(\alpha )={\mathrm {G}}(\alpha )/d$
. We establish an analogue of the Schur–Siegel–Smyth trace problem for totally positive algebraic integers. Then we consider the case where
$\alpha $
has all its conjugates in a sector
$| \arg z | \leq \theta $
,
$0 < \theta < 90^{\circ }$
. We compute the greatest lower bound
$c(\theta )$
of the absolute G-measure of
$\alpha $
, for
$\alpha $
belonging to
$11$
consecutive subintervals of
$]0, 90 [$
. This phenomenon appears here for the first time, conforming to a conjecture of Rhin and Smyth on the nature of the function
$c(\theta )$
. All computations are done by the method of explicit auxiliary functions.
Let E be an elliptic curve with positive rank over a number field K and let p be an odd prime number. Let
$K_{\operatorname {cyc}}$
be the cyclotomic
$\mathbb {Z}_p$
-extension of K and
$K_n$
its nth layer. The Mordell–Weil rank of E is said to be constant in the cyclotomic tower of K if for all n, the rank of
$E(K_n)$
is equal to the rank of
$E(K)$
. We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in this sense. We then indicate the potential applications to Hilbert’s tenth problem for number rings.
Let
$\alpha $
be a totally positive algebraic integer of degree d, with conjugates
$\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$
. The absolute
$S_k$
-measure of
$\alpha $
is defined by
$s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$
. We compute the lower bounds
$\upsilon _k$
of
$s_k(\alpha )$
for each integer in the range
$2\leq k \leq 15$
and give a conjecture on the results for integers
$k>15$
. Then we derive the lower bounds of
$s_k(\alpha )$
for all real numbers
$k>2$
. Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.
Let
$F_{2^n}$
be the Frobenius group of degree
$2^n$
and of order
$2^n ( 2^n-1)$
with
$n \ge 4$
. We show that if
$K/\mathbb {Q} $
is a Galois extension whose Galois group is isomorphic to
$F_{2^n}$
, then there are
$\dfrac {2^{n-1} +(-1)^n }{3}$
intermediate fields of
$K/\mathbb {Q} $
of degree
$4 (2^n-1)$
such that they are not conjugate over
$\mathbb {Q}$
but arithmetically equivalent over
$\mathbb {Q}$
. We also give an explicit method to construct these arithmetically equivalent fields.
The Euler–Mascheroni constant
$\gamma =0.5772\ldots \!$
is the
$K={\mathbb Q}$
example of an Euler–Kronecker constant
$\gamma _K$
of a number field
$K.$
In this note, we consider the size of the
$\gamma _q=\gamma _{K_q}$
for cyclotomic fields
$K_q:={\mathbb Q}(\zeta _q).$
Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that
In other words, under EH, the
$\gamma _q /\!\log q$
in these ranges converge to the one point distribution at
$1$
. This theorem refines and extends a previous result of Ford, Luca and Moree for prime
$q.$
The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.
Iwasawa theory of elliptic curves over noncommutative
$GL(2)$
extension has been a fruitful area of research. Over such a noncommutative p-adic Lie extension, there exists a structure theorem providing the structure of the dual Selmer groups for elliptic curves in terms of reflexive ideals in the Iwasawa algebra. The central object of this article is to study Iwasawa theory over the
$PGL(2)$
extension and connect it with Iwasawa theory over the
$GL(2)$
extension, deriving consequences to the structure theorem when the reflexive ideal is the augmentation ideal of the center. We also show how the dual Selmer group over the
$GL(2)$
extension being torsion is related with that of the
$PGL(2)$
extension.
We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.
Let f be an elliptic modular form and p an odd prime that is coprime to the level of f. We study the link between divisors of the characteristic ideal of the p-primary fine Selmer group of f over the cyclotomic
$\mathbb {Z}_p$
extension of
$\mathbb {Q}$
and the greatest common divisor of signed Selmer groups attached to f defined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at p to the context of modular forms.
Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of $n$ points of an algebraic surface is algebraic at a CM point $\tau$ and rational numbers $z_1$ and $z_2$. Our result gives a refinement of the algebraicity on Betti numbers.
The cusped hyperbolic n-orbifolds of minimal volume are well known for
$n\leq 9$
. Their fundamental groups are related to the Coxeter n-simplex groups
$\Gamma _{n}$
. In this work, we prove that
$\Gamma _{n}$
has minimal growth rate among all non-cocompact Coxeter groups of finite covolume in
$\textrm{Isom}\mathbb H^{n}$
. In this way, we extend previous results of Floyd for
$n=2$
and of Kellerhals for
$n=3$
, respectively. Our proof is a generalization of the methods developed together with Kellerhals for the cocompact case.
Let F be a system of polynomial equations in one or more variables with integer coefficients. We show that there exists a univariate polynomial
$D \in \mathbb {Z}[x]$
such that F is solvable modulo p if and only if the equation
$D(x) \equiv 0 \pmod {p}$
has a solution.
We prove a new irreducibility result for polynomials over
${\mathbb Q}$
and we use it to construct new infinite families of reciprocal monogenic quintinomials in
${\mathbb Z}[x]$
of degree
$2^n$
.
In this paper, we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over ${\mathbb {Q}}$ satisfying $\max \{|a_2|, |a_1|, |a_0|\} \leq X$ is bounded from above by $O(X (\log X)^2)$. We also count the number of abelian monic binary cubic forms with integer coefficients up to a natural equivalence relation ordered by the so-called Bhargava–Shankar height. Finally, we prove an assertion characterizing the splitting field of 2-torsion points of semi-stable abelian elliptic curves.
We give effective finiteness results for the power values of polynomials with coefficients composed of a fixed finite set of primes; in particular, of Littlewood polynomials.
We use circulant matrices and hyperelliptic curves over finite fields to study some arithmetic properties of certain determinants involving Legendre symbols and kth power residues.