Welcome to the *Algebra and its Applications* seminar
home page, which has been copied from Ettore's old webpage!

The seminar is organized by
Amod
Agashe. Please email
`agashe@math.fsu.edu` to contact me.

The meeting time and place is the standard one: Thursdays, at 2:00 p.m. in 104 LOV

Sept 6 | Payman Kassaei (King's College, London) | Canonical Subgroups of abelian varieties |
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Sept 13 | Andy Novocin (FSU) | Factoring Polynomials over the Rationals |

Sept 27 | Mark van Hoeij (FSU) | The LLL Lattice Reduction Algorithm and Applications |

Oct 4 | Mark van Hoeij (FSU) | Using LLL to compute the subfields of an algebraic number field |

Oct 18 | Eriko Hironaka (FSU) | Cyclotomic factors of Coxeter polynomials |

Oct 25 | Andy Novocin(FSU) | The van Hoeij algorithm for factoring polynomials over the rationals |

Nov 1 | Amod Agashe (FSU) | Modular forms and L-functions |

Nov 8 | Amod Agashe (FSU) | Modular forms and L-functions II |

Nov 15 | Amod Agashe (FSU) | Modular forms and L-functions III |

Nov 29 | Quan Yuan (FSU) | Exponential solutions of linear differential operators |

Lubin and Katz proved that certain elliptic curves which are ``not too supersingular" modulo a prime p have a canonical subgroup which varies (p-adically) analytically when the elliptic curves does. The existence and properties of canonical subgroups turned out to be a crucial ingredient in the theory of p-adic modular forms. Lubin-Katz's proof relies crucially on the one-dimensionality of (the formal group) of the ellpitic curve. In this talk I will survey the classical theory and show how one can prove similar results for other abelian varieties.

I intend to explore how computer algebra systems factor polynomials. This will begin with the early algorithms which were fast in practice but not in theory and conclude with our new result which is the current fastest algorithm both in theory and in practice. Other topics of interest which will be utilized: Lattice Reduction, Factoring over Finite Fields, Hensel Lifting.

As in important interlude to factoring polynomials I decided that a talk dedicated to this wonderful algorithm would be of interest. The algorithm will find small vectors in a lattice with a bound on how far they can be from the smallest vectors. By creating clever lattices this algorithm can be used to solve Diophantine Equations, Compute Minimal Polynomials, Find Algebraic Relationships, and more.

The talk will present a new application of LLL lattice reduction, namely an algorithm to compute the subfields of an algebraic number field. If time remains, another application of LLL will be presented as well.

The E_n Coxeter systems have been well-studied both by number theorists and by low-dimensional topologists. For n greater than or equal to 7, the E_n Coxeter polynomial is a Salem polynomial. The Salem number associated to E_7 is Lehmer's number, and as n increases the associated Salem numbers of E_n increase monotonically, converging to the smallest Pisot number. The E_n Coxeter polynomials appear as the Alexander polynomial of (-2,3,n)-pretzel links, and as the denominator of polygonal reflection groups studied by Cannon, Floyd, Plotnick and others. Recently, the Salem numbers associated to the E_n have shown up as the topological entropies of a family of automorphisms of the complex projective plane in work of Bedford and our own Kyounghi Kim. In this talk, we will focus on the cyclotomic factors of the Coxeter polynomials of E_n, and describe some joint work with Dick Gross and Curt McMullen. As shown by Bedford and Kim, these appear periodically with n. We show that this periodic phenomenon results from a decomposition of E_n into spherical "parts", and apply this observation to general Coxeter systems.

In this series, we will discuss modular forms and L-functions associated to them, without assuming any previous background. These talks are intended to be expository and as such should be accessible to graduate students in any area. Modular forms are certain functions that arise in various areas of mathematics and L-functions are a generalization of the Riemann zeta function. In the first talk, we will start by defining various modular forms. In the subseqent talks, we will define the L-function of a modular form and prove some of its properties, in analogy with the Riemann zeta function. Finally, we will discuss the significance of L-functions of modular forms in number theory and representation theory.

I intend to introduce how computer algebra systems find the exponential solutions of linear differential operator. For L belongs to C(x)[Dx], we will give a algorithm to computer a basis of solutions which have form exp(int(r)). We will introduce generalized exponent of linear differential operator and some properties of the ring C(x)[Dx] to explain our algorithem.