Farshid Hajir

Algebraic Properties of a Family of Generalized Laguerre Polynomials

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral valuesof the parameter. For integersr,n ≥ 0, we conjecture thatL ( 1 n r) n (x) = P n j=0 `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 1950s 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 islarge 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.

Tamely Ramified Towers and Discriminant Bounds for Number Fields

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim infmR2m. One knows that α0 ≥ 4πeγ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α0 ≥ 8πeγ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α0 < 83.9.

Modular Forms and Elliptic Curves over the Field of Fifth Roots of Unity

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.

Generating Functions, Polynomials and Vortices with Alternating Signs in Bose-Einstein Condensates

In this work, we construct suitable generating functions for vortices of alternating signs in the realm of Bose-Einstein condensates. In addition to the vortex-vortex interaction included in earlier fluid dynamics constructions of such functions, the vortices here precess around the center of the trap. This results in the generating functions of the vortices of positive charge and of negative charge satisfying a modified, so-called, Tkachenko differential equation. From that equation, we reconstruct collinear few-vortex equilibria obtained in earlier work, as well as extend them to larger numbers of vortices. Moreover, particular moment conditions can be derived e.g. about the sum of the squared locations of the vortices for arbitrary vortex numbers. Furthermore, the relevant differential equation can be generalized appropriately in the two-dimensional complex plane and allows the construction e.g. of polygonal vortex ring and multi-ring configurations, as well as ones with rings surrounding a vortex at the center that are again connected to earlier bibliography.

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.