Hajir, Farshid

Job Title

Professor & Department Head, Department of Mathematics and Statistics

Last Name

Hajir

First Name

Farshid

Discipline

Number Theory

Expertise

Algebraic number theory

Introduction

My specialty is algebraic number theory. I study unit and ideal class groups of number fields, the Galois theory of extensions with restricted ramification, the arithmetic of elliptic curves, and special values of L-functions. My past PhD students are Mairead Greene (Rockhurst University) and Laura Hall-Seelig ( Merrimack College). I'm currently working on a monograph on asymptotically good families of curves, codes, number fields, and graphs.

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Now showing 1 - 6 of 6

Algebraic Properties of a Family of Generalized Laguerre Polynomials
(2009-01-01) Hajir, F
We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r,n≥0 , we conjecture that L(−1−n−r)n(x)=∑nj=0(n−j+rn−j)xj/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 1950'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≥5 . Here we verify it in three situations: i) when n is large with respect to r , ii) when r≤8 , and iii) when n≤4 . The main tool is the theory of p -adic Newton Polygons.
Specializations of one-parameter families of polynomials
(2006-01-01) Hajir, F; Wong, S
Let K be a number field, and suppose λ(x,t)∈K[x,t] is irreducible over K(t). Using algebraic geometry and group theory, we describe conditions under which the K-exceptional set of λ, i.e. the set of α∈K for which the specialized polynomial λ(x,α) is K-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed n≥10, all but finitely many K-specializations of the degree n generalized Laguerre polynomial L n (t) (x) are K-irreducible and have Galois group S n . Second, we study specializations of the modular polynomial Φ n (x,t) (which vanishes on the j-invariants of pairs of elliptic curves related by a cyclic n-isogeny), and show that for any n≥53, all but finitely many of the K-specializations of Φ n (x,t) are K-irreducible and have Galois group containing SL 2 (ℤ/n)/{±I}. Third, for a simple branched cover π:Y→ℙ K 1 of degree n≥7 and of genus at least 2, all but finitely many K-specializations are K-irreducible and have Galois group S n .
Modular forms and elliptic curves over the field of fifth roots of unity
(2010-01-01) Gunnells, PE; Hajir, F; Yasaki, Dan
Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F
Grant Proposal Narrative
(2004-01-01) Sternheim, Morton; Feldman, Allan; Hajir, Farshid
On the Galois Group of generalized Laguerre polynomials
(2005-01-01) Hajir, F
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∈ℚ-ℤ <0 , Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L n (α) (x)=∑ j=0 n 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 L n (α) (x) is either the alternating or symmetric group on n letters, generalizing results of Schur for α=0,1,±1 2,-1-n.
Finitely ramified iterated extensions
(2005-01-01) Aitken, W; Hajir, F; Maire, C
Let K be a number field, t a parameter, F = K(t), and φ(x)∈ K [x] a polynomial of degree d ≥ 2. The polynomial Φn(x,t) = φ^n (x) − t ∈ F[x], where φ^ n = φ ^ φ ^ … ^ φ is the n-fold iterate of φ, is irreducible over F; we give a formula for its discriminant. Let F be the field obtained by adjoining to F all roots (in a fixed ) of Φn(x,t) for all n ≥ 1; its Galois group Gal(Fφ/F) is the iterated monodromy group of φ. The iterated extension Fφ is finitely ramified over F if and only if φ is postcritically finite. We show that, moreover, for post-critically finite φ, every specialization of Fφ/F at t = t0 ∈ K is finitely ramified over K, pointing to the possibility of studying Galois groups of number fields with restricted ramification via tree representations associated to iterated monodromy groups of postcritically finite 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 monogène number fields that arise from the construction.