Let p$\ge$2 be a given prime number. We prove, for any number field kappa and any integer e$\ge$1, the p-rank $ε$-conjecture, on the p-class groups Cl\_F, for the family F\_kappa^p^e of towers F/kappa built as successive degree p cyclic extensions (without any other Galois conditions) such that F/kappa be of degree p^e, namely: #(Cl\_F[p])<<\_{kappa,p^e,$ε$}($\sqrt$D\_F)^$ε$, where D\_F is the absolute value of the discriminant (Theorem 3.6) and, more generally, #(Cl\_F[p^r])<<\_{kappa,p^e,$ε$}($\sqrt$D\_F)^$ε$, for any r$\ge$1 fixed. This Note generalizes the case of the family F\_Q^p (Genus theory and $ε$-conjectures on p-class groups, J. Number Theory 207, 423--459 (2020)), whose techniques appear to be "universal" for all relative degree p cyclic extensions and use the Montgomery--Vaughan result on prime numbers. Then we prove, for F\_kappa^p^e, the p-rank $ε$-conjecture on the cohomology groups H^2(G\_F,Z\_p) of Galois p-ramification theory over F (Theorem 4.3) and for some other classical finite p-invariants of F, as the Hilbert kernels and the logarithmic class groups.
Let $k$ denote an algebraically closed field. We revisit a construction of the author of families of elliptic curves over the rational function field $k(t)$. Combining a combinatorial analysis with a rank formula of Ulmer we prove that, for all but finitely many families of these curves, the Mordell-Weil groups over $k(t^{1/d})$ have rank zero, as $d$ ranges over positive integers prime to the characteristic of $k$.
We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Roh…
We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$ over a finite field of characteristic $p$, and consider the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower defined by $\bar f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified extension of $\mathbb{Z}_p$, which is abstractly isomorphic to $(\mathbb{Z}_p)^\ell$ as a topological group. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton …
Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an elliptic curve in the ${\bf{Z}}_p^2$-extension is finitely generated modulo Heegner points. The novelty of the approach here is to use the existence of a suitable $p$-adic $L$-function to reduce the problem to a minimal nonvanishing criterion, which should be applicable to a broader class of problems than considered here.
We study the variation of Mordell-Weil ranks in the Jacobians of curves in a pro-p tower over a fixed number field. In particular, we show that under mild conditions the Mordell-Weil rank of a Jacobian in the tower is bounded above by a constant multiple of its dimension. In the case of the tower of Fermat curves, we show that the constant can be taken arbitrarily close to 1. The main result is used in the forthcoming paper of Guillermo Mantilla-Soler on the Mordell-Weil rank of the modular Jacobian J(Np^m).
Let $E$ be an elliptic curve defined over a number field $F$ with good ordinary reduction at all primes above $p$, and let $F_\infty$ be a finitely ramified uniform pro-$p$ extension of $F$ containing the cyclotomic $\mathbb{Z}_p$-extension $F_{cyc}$. Set $F^{(n)}$ be the $n$-th layer of the tower, and $F^{(n)}_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of $F^{(n)}$. We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $λ$-invariant of the Selmer group over $F^{(n)}_{cyc}$ as $n\rightarrow \infty$. This method has its origins in work of A.Cuoco, who studied $\math…
Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.
Let $k$ be a field of characteristic zero and let $K=k(t)$ be the rational function field over $k$. In this paper we combine a formula of Ulmer for ranks of certain Jacobians over $K$ with strong upper bounds on endomorphisms of Jacobians due to Zarhin to give many examples of higher dimensional, absolutely simple Jacobians over $k(t)$ with bounded rank in towers $k(t^{1/p^r})$. In many cases we are able to compute the rank at every layer of the tower.
In 2000, based on his procedure for constructing explicit towers of modular curves, Elkies deduced explicit equations of rank-2 Drinfeld modular curves which coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. In 2015, Bassa, Beelen, Garcia, and Stichtenoth constructed a celebrated (recursive and good) tower (BBGS-tower for short) of curves and outlined a modular interpretation of the defining equations. Soon after that, Gekeler studied in depth the modular curves coming from sparse Drinfeld modules. In this paper, to establish a link between these …
To extend Iwasawa's classical theorem from ${\mathbb Z}_p$-towers to ${\mathbb Z}_p^d$-towers, Greenberg conjectured that the exponent of $p$ in the $n$-th class number in a ${\mathbb Z}_p^d$-tower of a global field $K$ ramified at finitely many primes is given by a polynomial in $p^n$ and $n$ of total degree at most $d$ for sufficiently large $n$. This conjecture remains open for $d\geq 2$. In this paper, we prove that this conjecture is true in the function field case. Further, we propose a series of general conjectures on $p$-adic stability of zeta functions in a $p$-adic Lie tower of funct…
Let $C$ be a smooth projective curve defined over a number field $k$, $X/k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(τ,B)$ the $k(C)/k$-trace of $J_X$. We estimate how the rank of $J_X(k(C))/τB(k)$ varies when we take an unramified abelian cover $π:C'\to C$ defined over $k$.
We classify all complex quadratic number fields with 2-class group of type (2,2^m) whose Hilbert 2-class fields have class groups of 2-rank equal to 2. These fields all have 2-class field tower of length 2. We still don't know examples of fields with 2-class field tower of length 3, but the smallest candidate is the field with discriminant -1015.
Inspired by the Odlyzko root discriminant and Golod--Shafarevich $p$-group bounds, Martinet (1978) asked whether an imaginary quadratic number field $K/\mathbb{Q}$ must always have an infinite Hilbert $2$-class field tower when the class group of $K$ has $2$-rank $4$, or equivalently when the discriminant of $K$ has $5$ prime factors. No negative results are known. Benjamin (2001, 2002) and Sueyoshi (2004, 2009, 2010) systematically established infinite $2$-towers for many $K$ in question, by casework on the associated Rédei matrices. Others, notably Mouhib (2010), have also made progress, but…
The genus of projective curves discretely separates decidedly different two variable algebraic relations. So, we can focus on the connected moduli M_g of genus g curves. Yet, modern applications require a data variable (function) on such curves. The resulting spaces are versions, depending on our need from this data variable, of Hurwitz spaces. A Nielsen class is a set defined by r \ge 3 conjugacy classes C in the data variable monodromy G. It gives a striking genus analog. Using Frattini covers of G, every Nielsen class produces a projective system of related Nielsen classes for any prime p d…
Let E --> C be an elliptic surface defined over a number field K. For each finite covering C' --> C defined over K, let E' --> C' be the pullback. We give a strong upper bound for the rank of E'(C'/K) in the case that C' --> C is an unramified abelian covering and under the assumption that the Tate conjecture is true for the surface E'/K. In the case that C is an elliptic curve and the map C'=C --> C is the multiplication-by-n map, the rank of E'(C'/K) is O(n^e) for every e > 0, which may be compared with the elementary bound of O(n^2).
With K=Q((3812377)^(1/2)) we give the first example of an algebraic number field possessing a 5-class tower of exact length L(5,K)=3. The rigorous proof is conducted by means of the p-group generation algorithm, showing the existence of a unique finite metabelian 5-group G with abelianization [5,5] having the kernels (M(1),G^5) and targets ([25,5,5,5],[5,5]^5) of Artin transfers T(i):G-->M(i)/M(i)' to its six maximal subgroups M(i), prescribed by arithmetical invariants of K. Thus, G must be the second 5-class group G(5,2,K) of the real quadratic field K but cannot be its 5-class tower group G…
Let $\ell$ be a rational prime. We show that an analogue of a conjecture of Greenberg in graph theory holds true. More precisely, we show that when $n$ is sufficiently large, the $\ell$-adic valuation of the number of spanning trees at the $n$th layer of a $\mathbb{Z}_{\ell}^{d}$-tower of graphs is given by a polynomial in $\ell^{n}$ and $n$ with rational coefficients of total degree at most $d$ and of degree in $n$ at most one.
Rotations $f_α$ of the one-dimensional torus (equipped with the normalized Lebesgue measure) by an irrational angle $α$ are known to be dynamical systems of rank one. This is equivalent to the property that the covering number $F^*(f_α)$ of the dynamical system is one. In other words, there exists a basis $B$ such that for arbitrarily high $h$ an arbitrarily large proportion of the unit torus can be covered by the Rokhlin tower $(f_α^kB)_{k=0}^{h-1}$. Although $B$ can be chosen with diameter smaller than any fixed $\varepsilon > 0$, it is not always possible to take an interval for $B$ but thi…
We prove an Equivariant Main Conjecture in Iwasawa Theory along any rank one, sign-normalized Drinfeld modular, split at infinity Iwasawa tower of a general function field of characteristic p, for the Iwasawa modules recently considered by Greither and Popescu, in their proof of the classical Equivariant Main Conjecture along the (arithmetic) cyclotomic Iwasawa tower.
Given a fixed prime number \(p\), the multiplet of abelian type invariants of the \(p\)-class groups of all unramified cyclic degree \(p\) extensions of a number field \(K\) is called its IPAD (index-\(p\) abelianization data). These invariants have proved to be a valuable information for determining the Galois group \(G_p^2\) of the second Hilbert \(p\)-class field and the \(p\)-capitulation type \(\varkappa\) of \(K\). For \(p=3\) and a number field \(K\) with elementary \(p\)-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated I…
For a fixed prime p, the p-class tower F(p,infinity,K) of a number field K is considered to be known if a pro-p presentation of the Galois group H = Gal( F(p,infinity,K)/K ) is given. In the last few years, it turned out that the Artin pattern AP(K) = (tau(K),kappa(K)) consisting of targets tau(K) = (Cl(p,L)) and kernels kappa(K) = (ker(J(L/K)) of class extensions J(L/K): Cl(p,K) --> Cl(p,L) to unramified abelian subfields L/K of the Hilbert p-class field F(p,1,K) only suffices for determining the two-stage approximation G = H/H" of H. Additional techniques had to be developed for identifying …
Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M) of the second Hilbert 5-class field M(5,2) of M are investigated, when the 5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already. The …
In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(γ(H)+o(1))\log x/\log\log x}$ where $γ(H)$ is an explicit constant computable from the (absolute) root system of $H$. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate $x^{c\log x}$. A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarev…
The isomorphism type of the Galois group of the 2-class field tower of quadratic number fields having a 2-class group with abelian type invariants (4,4) is determined by means of information on the transfer of 2-classes to unramified abelian 2-extensions, collected in the Artin pattern. In recent investigations by Benjamin and Snyder, the length of the tower of such fields has turned out to be dependent on the rank of the 2-class group of the first Hilbert 2-class field. Significant progress is achieved by extending the pool of possible metabelian 2-groups of the second Hilbert 2-class field f…
Let $p$ be a prime. We define the deficiency of a finitely-generated pro-$p$ group $G$ to be $r(G)-d(G)$ where $d(G)$ is the minimal number of generators of $G$ and $r(G)$ is its minimal number of relations. For a number field $K$, let $K_\emptyset$ be the maximal unramified $p$-extension of $K$, with Galois group $G_\emptyset = Gal(K_\emptyset/K)$. In the 1960s, Shafarevich (and independently Koch) showed that the deficiency of $G_\emptyset$ satisfies $$0\leq \mathrm{Def}({\rm G}_\emptyset) \leq dim (O_K^\times/(O_K^{\times })^p),$$ relating the deficiency of $G_\emptyset$ to the $p$-rank of …
We prove the rank of the group of signatures of the circular units (hence also the full group of units) of ${\mathbb Q}( ζ_m)^+$ tends to infinity with $m$. We also show the signature rank of the units differs from its maximum possible value by a bounded amount for all the real subfields of the composite of an abelian field with finitely many odd prime-power cyclotomic towers. In particular, for any prime $p$ the signature rank of the units of ${\mathbb Q}( ζ_{p^n})^+$ differs from $φ(p^n)/2$ by an amount that is bounded independent of $n$. Finally, we show conditionally that for general cyclo…
In this article we classify the complex quadratic number fields k with 2-class group of type (2,2,2) whose Hilbert 2-class fields have a 2-class group of rank 2, and then determine the length of their 2-class field towers.
For a finite abelian p-group A of rank d, we define its (logarithmic) mean exponent to be the base-p logarithm of the d-th root of its cardinality. We study the behavior of the mean exponent of p-class groups in towers of number fields. By combining techniques from group theory with the Tsfasman-Valdut generalization of the Brauer-Siegel Theorem, we construct infinite tamely ramified towers in which the mean exponent of class groups remains bounded. Several explicit examples with p=2 are given. We introduce an invariant M(G) attached to a finitely generated FAb pro-p group G which measures the…
Theoretical foundations of a new algorithm for determining the p-capitulation type kappa(K) of a number field K with p-class rank rho=2 are presented. Since kappa(K) alone is insufficient for identifying the second p-class group G=Gal(F(p,2,K) | K) of K, complementary techniques are developed for finding the nilpotency class and coclass of G. An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(tau(K),kappa(K)) of all 34631 real quadratic fields K=Q(squareroot(d)) with discriminants 0<d<100000000 and 3-class …
We study a natural question in the Iwasawa theory of algebraic curves of genus $>1$. Fix a prime number $p$. Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field $K$ of genus $g>1$, such that the Jacobian of $X$ has good ordinary reduction at the primes above $p$. Fix an odd prime $p$ and for any integer $n>1$, let $K_n^{(p)}$ denote the degree-$p^n$ extension of $K$ contained in $K(μ_{p^{\infty}})$. We prove explicit results for the growth of $\#X(K_n^{(p)})$ as $n\rightarrow \infty$. When the Jacobian of $X$ has rank zero and the associated adelic Galo…
In this paper we give an elementary proof of results on the structure of 4-class groups of real quadratic number fields originally due to A. Scholz. In a second (and independent) section we strengthen C. Maire's result that the 2-class field tower of a real quadratic number field is infinite if its ideal class group has 4-rank at least $4$, using a technique due to F. Hajir.
Infinitely many large Schur sigma-groups G with logarithmic order lo(G)=19+e, non-elementary bicyclic commutator quotient G/G' ~ C(3^e) x C(3), e >= 2, elevated rank distribution rho(G)=(3,3,3;3), punctured transfer kernel type kappa(G) ~ (144;4) and soluble length sl(G)=3 are constructed. Up to e <= 4, they are realized as 3-class field tower groups Gal(F(3,infty,K)/K) of imaginary quadratic number fields K=Q(d^1/2), d<0. Their metabelianizations M=G/G'' are BCF-groups with lo(M)=8+e and bicyclic third lower central factor gamma3(M)/gamma4(M) ~ C(3) x C(3).
Recently, Andrews, Chan, Kim and Osburn introduced a $q$-series $h(q)$ for the study of the first positive rank and crank moments for overpartitions. They conjectured that for all integers $m \geq 3$, \begin{equation*}\label{hqcon} \frac{1}{(q)_{\infty}} (h(q) - m h(q^{m})) \end{equation*} has positive power series coefficients for all powers of $q$. Byungchan Kim, Eunmi Kim and Jeehyeon Seo provided a combinatorial interpretation and proved it is asymptotically true by circle method. In this note, we show this conjecture is true if $m$ is any positive power of $2$, and we show that in order t…
Rank of Pólya Groups in Lecacheux Parametric Family of Quintic Fields
Indian Institute of Science Education and Research, Thiruvananthapuram（印度科学教育研究所，特里凡得琅）；Indian Institute of Science Education and Research, Berhampur（印度科学教育研究所，贝尔哈姆普尔）
Let $k$ be a field of characteristic $q$, $\cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(\cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$ or $>2d+1$. Let $p\ne q$ be a prime number and $\cac'\to\cac$ a finite geometrically \textsc{Galois} and étale cover defined over $k$ with function field $K':=k(\cac')$. Let $(τ',B')$ be the $K'/k$-trace of $A/K$. We give an upper bound for the $\bbz_p$-corank of the \textsc{Selmer} group $\text{Sel}_p(A\times_KK')$, defined in terms of the $p$-descent map. A…
