# An Introduction to Sage: Mathematical Software for Teaching and Research

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The Sage mathematics software project (http://www.sagemath.org) aims to "Create a viable free open source alternative to Magma, Maple, Mathematica and Matlab."

This hands-on introduction to Sage will get new users solving their computational problems quickly. Emphasis will be placed on using Sage for current research and for using Sage in teaching calculus to undergraduate students.

We will use Sage on the web (http://www.sagenb.org); please bring your laptop if you have one.

# Split reductions of simple abelian varieties

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# Twists of elliptic curves and Hilbert's Tenth Problem

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In joint work with Barry Mazur, we show that over every number field there are many elliptic curves of rank zero, and (assuming the finiteness of Shafarevich-Tate groups) many elliptic curves of rank one.

Combining our results about ranks of twists with ideas of Poonen and Shlapentokh, we show that if one assumes the finiteness of Shafarevich-Tate groups of elliptic curves, then Hilbert's Tenth Problem is undecidable (i.e., has a negative answer) over the ring of integers of every number field.

# Hodge groups of superelliptic jacobians

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The Hodge group (aka special Mumford-Tate group) of a complex abelian variety $X$ is a certain linear reductive algebraic group over the rationals that is closely related to the endomorphism ring of $X$. (For example, the Hodge group is commutative if and only if $X$ is an abelian variety of CM-type.) In this talk I discuss" lower bounds" for the center of Hodge groups of superelliptic jacobians. (This is a joint work with Jiangwei Xue.)

# The Overconvergent de Rham-Witt Complex

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The aim of the talk is to describe the overconvergent de Rham-Witt complex. It is a subcomplex of the de Rham-Witt complex and it can be used to compute Monsky-Washnitzer cohomology for affine varieties, and rigid cohomology in general. (All our varieties are over a perfect field of characteristic $p$.)

We will begin by reviewing Monsky-Washnitzer cohomology and the de Rham-Witt complex. Next we will define overconvergent Witt vectors and then the overconvergent de Rham-Witt complex. As time permits, we will say something about the proof of the comparison theorem between Monsky-Washnitzer cohomology and overconvergent de Rham-Witt ohomology.

This is joint work with Andreas Langer and Thomas Zink.

# Divisibility properties of values of partial zeta functions at non-positive integers

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The values of the partial zeta functions for an abelian extension of number fields at non-positive integers are rational numbers with known bounds on their denominators. David Hayes conjectured that when the associated fields satisfy certain algebraic conditions, the bound at s=0 can be sharpened. I will present a counterexample to Hayes's conjecture. I will then propose a new conjecture sharpening the bounds at arbitrary non-positive integers that implies a weaker version of Hayes conjecture at s=0. I will conclude by proving that the new conjecture is a consequence of the Coates-Sinnott conjecture.

# Refined class number formulas and Kolyvagin systems

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In this talk we will discuss refined class number formulas conjectured by Gross and by Darmon. We will prove (a slight variant of) Darmon's conjecture, using the theory of Kolyvagin systems. This is joint work with Barry Mazur.