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Farshid Hajir

Publications and source records attributed to Farshid Hajir.

At least 19 recordsLinked to original sources
arXiv2026

Hasse-Witt invariants for trace forms of Jacobi polynomials

In \cite{feit}, Feit used the Generalized Laguerre Polynomials (GLP) to prove that the groups $\widetilde{A}_{5}$ and $\widetilde{A}_{7}$ occur as Galois groups over $\Q$. Hajir, in \cite{hajir} extended these results to prove that $\widetilde{A}_{n}$ is Galois over $\Q$ whenever $n \equiv 1 \pmod{8}$. A key ingredient of both proofs is the explicit determination of the Hasse-Witt invariant of (the diagonalization of) the trace form of the root fields of the GLP, which relies on the calculation of a certain determinant, $\Delta_t$. The explicit formula for $\Delta_t$ used in \cite{feit} and \cite{hajir} was derived using properties specific to the GLP which do not generalize to other polynomials. In this paper we revisit Feit's original calculation of $\Delta_t$ and situate it in the context of Hankel determinants. We give an alternate derivation of $\Delta_t$ using standard combinatorial arguments and then apply these results to the Jacobi polynomials, a two-parameter family of orthogonal polynomials encompassing the GLP as a special case. We compute an explicit formula for the $\Delta_t$ of the Jacobi polynomials as well as the associated Hasse-Witt invariant. The techniques used in this paper are not specific to the Jacobi polynomials and are widely applicable.

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arXiv2024

On tamely ramified infinite Galois extensions

For a number field $K$, we consider $K^{\rm ta}$ the maximal tamely ramified algebraic extension of~$K$, and its Galois group $G^{\rm ta}_K= Gal(K^{ta}/K)$. Choose a prime $p$ such that $μ_p \not \subset K$. Our guiding aim is to characterize the finitely generated pro-$p$ quotients of~$G^{\rm ta}$. We give a {unified point of view} by introducing the notion of {\it stably inertially generated} pro-$p$ groups~$G$, for which linear groups are archetypes. This key notion {is compatible} with local {\it tame liftings} as used in the Scholz-Reichardt Theorem. We realize every finitely generated pro-$p$ group~$G$ which is stably inertially generated as a quotient of $G^{\rm ta}$. Further examples of groups that we realize as quotients of $G^{\rm ta}$ include congruence subgroups of special linear groups over ${\mathbb Z}_p[[ T_1,\cdots, T_n ]]$. Finally, we give classes of groups which cannot be realized as quotients of $G^{\rm ta}_{\mathbb Q}$.

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arXiv2022

On tame ${\mathbb Z}/p{\mathbb Z}$ extensions with prescribed ramification

The tame Gras-Munnier Theorem gives a criterion for the existence of a ${\mathbb Z}/{\mathbb Z}$-extension of a number field $K$ ramified at exactly a set $S$ of places of $K$ prime to $p$ (allowing real Archimedean places when $p=2$) in terms of the existence of a dependence relation on the Frobenius elements of these places in a certain governing extension. We give a new and simpler proof of this theorem that also relates the set of such extensions of $K$ to the set of these dependence relations. After presenting this proof, we then reprove the key Proposition 3 using the more sophisticated Wiles-Greenberg formula based on global duality.

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arXiv2022

On Ozaki's theorem realizing prescribed $p$-groups as $p$-class tower groups

We give a streamlined and effective proof of Ozaki's theorem that any finite $p$-group $Γ$ is the Galois group of the $p$-Hilbert class field tower of some number field $\rm F$. Our work is inspired by Ozaki's and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field ${\rm k}_0$ with class number prime to $p$. We construct ${\rm F}/{\rm k}_0$ by a sequence of ${\mathbb Z}/p$-extensions ramified only at finite tame primes and also give explicit bounds on $[{\rm F}:{\rm k}_0]$ and the number of ramified primes of ${\rm F}/{\rm k}_0$ in terms of $\# Γ$.

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arXiv2021

Deficiency of p-Class Tower Groups and Minkowski Units

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 the unit group $O_K^\times$ of the ring of integers $O_K$ of $K$. In this work, we further explore connections between relations of the group $G_\emptyset$ and the units in the tower $K_\emptyset/K$, especially their Galois module structure. In particular, under the assumption that $K$ does not contain a primitive $p$th root of unity, we give an exact formula for $\mathrm{Def}({\rm G}_\emptyset)$ in terms of the number of independent Minkowski units in the tower. The method also allows us to infer more information about the relations of G$_\emptyset$, such as their depth in the Zassenhaus filtration, which in certain circumstances makes it easier to show that G$_\emptyset$ is infinite. We illustrate how the techniques can be used to provide evidence for the expectation that the Shafarevich-Koch upper bound is "almost always" sharp.

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arXiv2019

On the Shafarevich Group of Restricted Ramification Extensions of Number Fields in the Tame Case

Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $\Sha_S$ of maps $H^2(G_S,\fq_p) \rightarrow \oplus_{v\in S} H^2(\G_v,\fq_p)$. There is a natural injection $\Sha_S \hookrightarrow \CyB_S$, into the dual $\CyB_S$ of a certain readily computable Kummer group $V_S$, which is always an isomorphism in the wild case. The tame case is much more mysterious. Our main result is that given a finite $X$ coprime to $p$, there exists a finite set of places $S$ coprime to $p$ such that $\Sha_{S\cup X} \stackrel{\simeq}{\hookrightarrow} \CyB_{S\cup X} \stackrel{\simeq}{\twoheadleftarrow} \CyB_X \hookleftarrow \Sha_X$. In particular, we show that in the tame case $\Sha_Y$ can {\it increase} with increasing $Y$. This is in contrast with the wild case where $\Sha_Y$ is nonincreasing in size with increasing $Y$.

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arXiv2019

Cutting towers of number fields

Given a prime $p$, a number field $\K$ and a finite set of places $S$ of $\K$, let $\K_S$ be the maximal pro-$p$ extension of $\K$ unramified outside $S$. Using the Golod-Shafarevich criterion one can often show that $\K_S/\K$ is infinite. In both the tame and wild cases we construct infinite subextensions with bounded ramification using the refined Golod-Shafarevich criterion. In the tame setting we achieve new records on Martinet constants (root discriminant bounds) in the totally real and totally complex cases. We are also able to answer a question of Ihara by producing infinite asymptotically good extensions in which infinitely many primes split completely.

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arXiv2018

Heuristics for p-class towers of real quadratic fields

Let $p$ be an odd prime. For a number field $K$, we let $K_\infty$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\mathrm{Gal}(K_\infty/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite $p$-group occurs as the $p$-class tower group of an imaginary quadratic field. Here we do the same for an arbitrary real quadratic field $K$ as base. As before, the action of $\mathrm{Gal}(K/\mathbb{Q})$ on the $p$-class tower group of $K$ plays a crucial role; however, the presence of units of infinite order in the ground field significantly complicates the possibilities for groups that can occur. We also sharpen our results in the imaginary quadratic field case by removing a certain hypothesis, using ideas of Boston and Wood. In an appendix, we show how the probabilities introduced for finite $p$-groups can be extended in a consistent way to the infinite pro-$p$ groups which can arise in both the real and imaginary quadratic settings.}

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arXiv2017

Analytic lie extensions of number fields with cyclic fixed points and tame ramification

- Let p be a prime number and K an algebraic number field. What is the arithmetic structure of Galois extensions L/K having p-adic analytic Galois group $Γ$ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such extensions are either deeply ramified (at some prime dividing p) or ramified at an infinite number of primes. In this work, we take up a study (initiated by Boston) of this type of question under the assumption that L is Galois over some subfield k of K such that [K : k] is a prime = p. Letting $σ$ be a generator of Gal(K/k), we study the constraints posed on the arithmetic of L/K by the cyclic action of $σ$ on $Γ$, focusing on the critical role played by the fixed points of this action, and their relation to the ramification in L/K. The method of Boston works only when there are no non-trivial fixed points for this action. We show that even in the presence of arbitrarily many fixed points, the action of $σ$ places severe arithmetic conditions on the existence of finitely and tamely ramified uniform p-adic analytic extensions over K, which in some instances leads us to be able to deduce the non-existence of such extensions over K from their non-existence over k.

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arXiv2016

Prime Decomposition and the Iwasawa mu-invariant

For $Γ=\mathbb{Z}_p$, Iwasawa was the first to construct $Γ$-extensions over number fields with arbitrarily large $μ$-invariants. In this work, we investigate other uniform pro-$p$ groups which are realizable as Galois groups of towers of number fields with arbitrarily large $μ$-invariant. For instance, we prove that this is the case if $p$ is a regular prime and $Γ$ is a uniform pro-$p$ group admitting a fixed-point-free automorphism of odd order dividing $p-1$. Both in Iwasawa's work, and in the present one, the size of the $μ$-invariant appears to be intimately related to the existence of primes that split completely in the tower.

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arXiv2015

On the invariant factors of class groups in towers of number fields

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 asymptotic growth of the mean exponent of abelianizations of subgroups of index n with n going to infinity. When G=Gal(L/K), M(G) measures the asymptotic behavior of the mean exponent of class groups in L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

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arXiv2014

Heuristics for $p$-class towers of imaginary quadratic fields, with an Appendix by Jonathan Blackhurst

Cohen and Lenstra have given a heuristic which, for a fixed odd prime $p$, leads to many interesting predictions about the distribution of $p$-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field $K$, the Galois group of the $p$-class tower of $K$, i.e. $G_K:=\mathrm{Gal}(K_\infty/K)$ where $K_\infty$ is the maximal unramified $p$-extension of $K$. By class field theory, the maximal abelian quotient of $G_K$ is isomorphic to the $p$-class group of $K$. For integers $c\geq 1$, we give a heuristic of Cohen-Lentra type for the maximal $p$-class $c$ quotient of $\G_K$ and thereby give a conjectural formula for how frequently a given $p$-group of $p$-class $c$ occurs in this manner. In particular, we predict that every finite Schur $σ$-group occurs as $G_K$ for infinitely many fields $K$. We present numerical data in support of these conjectures.

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arXiv2004

Finitely ramified iterated extensions

Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold iterate of f, is absolutely irreducible over F; we compute a recursion for its discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in a fixed algebraic closure, of P_n for all n; its Galois group Gal(L/F) is the iterated monodromy group of f. The iterated extension L/F is finitely ramified if and only if f is post-critically finite (pcf). We show that, moreover, for pcf polynomials f, every specialization of L/F at t=t_0 in K is finitely ramified over K, pointing to the possibility of studying Galois groups with restricted ramification via tree representations associated to iterated monodromy groups of pcf polynomials. We discuss the wildness of ramification in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogene number fields.

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arXiv2004

Algebraic properties of a family of Generalized Laguerre Polynomials

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\geq 0$, we conjecture that $L_n^{(-1-n-r)}(x) = \sum_{j=0}^n \binom{n-j+r}{n-j}x^j/j!$ is a $\Q$-irreducible polynomial whose Galois group contains the alternating group on $n$ letters. That this is so for $r=n$ was conjectured in the 50's by Grosswald and proven recently by Filaseta and Trifonov. It follows from recent work of Hajir and Wong that the conjecture is true when $r$ is large with respect to $n\geq 5$. Here we verify it in three situations: i) when $n$ is large with respect to $r$, ii) when $r \leq 8$, and iii) when $n\leq 4$. The main tool is the theory of $p$-adic Newton Polygons.

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arXiv2004

On the Galois group of Generalized Laguerre Polynomials

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be ``large.'' For a fixed $α\in \Q - \Z_{<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre Polynomial $L_n^{(α)}(x) = \sum_{j=0}^n \binom{n+α}{n-j}(-x)^j/j!$ is irreducible for all large enough $n$. We use our criterion to show that, under these conditions, the Galois group of $\La$ is either the alternating or symmetric group on $n$ letters, generalizing results of Schur for $α=0,1$.

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arXiv2004

Specializations of one-parameter families of polynomials

Let K be a number field, and let lambda(x,t)\in K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of alpha\in K for which the specialized polynomial lambda(x,alpha) is K-reducible. We apply this to show that for any fixed n>=10 and for any number field K, all but finitely many K-specializations of the degree n generalized Laguerre polynomial are K-irreducible and have Galois group S_n. In conjunction with the theory of complex multiplication, we also show that for any K and for any n>=53, all but finitely many of the K-specializations of the modular equation Phi_n(x, t) are K-irreducible and have Galois group containing PSL_2(Z/n).

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