A permutation of the abstracts for Midwest Algebraic Number Theory Day

Optimal elliptic curves, discriminants and the degree conjecture over function fields
Mihran Papikian (Michigan)

We discuss the function field analogues of some well-known conjectures over number fields. In particular, for one dimensional subvarieties of certain Drinfeld Jacobians we prove the analogue of the degree conjecture (i.e., bounds on the degree of the optimal modular parametrization), and improve the bound on the degree of the minimal discriminant in terms of the degree of the conductor due to Szpiro. The main technical tool used in the proofs is Grothendieck's monodromy pairing and its rigid analytic realization.

Sharper ABC-based bounds for congruent polynomials
Daniel J. Bernstein (UIC)

Agrawal, Kayal, and Saxena recently introduced a new method of proving that an integer is prime. The speed of the Agrawal-Kayal-Saxena method depends on proven lower bounds for the size of the group generated by several elements of a finite field. I will discuss an intriguing idea introduced by Voloch for using ABC to obtain such lower bounds.

On Simplest CM abelian varieties over imaginary quadratic fields
T. H. Yang (Wisconsin)

In this talk, we associate canonically to every imaginary quadratic field $K=\Bbb Q(\sqrt{-D})$ one or two isogenous classes of CM (complex multiplication) abelian varieties over $K$, depending on whether $D$ is odd or even ($D \ne 4$). These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to $\Bbb Q$. When $D$ is odd or divisible by 8, they are the scalar restriction of `canonical' elliptic curves first studied by Gross and Rohrlich. We prove that these abelian varieties have the striking property that the vanishing order of their $L$-function at the center is dictated by the root number of the associated Hecke character. We also prove that the smallest dimension of a CM abelian variety over $K$ is exactly the ideal class number of $K$ and classify when a CM abelian variety over $K$ has the smallest dimension.

Non-linear bialgebras: a structural approach to the Witt vectors
Jim Borger (U of Chicago)

I will discuss a kind of algebraic object -- a "plethory" -- which is a non-linear analogue of the notion of algebra. It can also be seen as a possibly non-linear generalization of the notion of cocommutative bialgebra. I will also show that if one begins with the bialgebra generated by Frobenius and performs an analogue of an affine blow-up, then the resulting plethory is the object that, in a certain sense, gives rise to the Witt vectors (of any commutative ring) and that the various strange structures on them are quite clear from this point of view. This is joint work with Ben Wieland.

$P$-adic modular forms over Shimura Curves
Payman Kassaei (MSU)

We will introduce $P$-adic modular forms (and the corresponding Hecke algebras) defined over unitary Shimura curves associated to certain quaternion algebras, and use the rigid geometry of Shimura curves to study $P$-adic variations of modular forms over these curves. In particular, we will discuss Gouvea-Mazure type results for these modular forms.

On the p-adic interpolation of automorphic forms
Matt Emerton (Northwestern)

In this talk we will explain a construction of eigenvarieties that provide a p-adic interpolation of classical Hecke eigenforms on reductive groups, that are of finite slope at p. We work in rather large generality (any reductive group that is quasi-split at p), and use representation-theoretic, rather than geometric methods. The goal of the talk will be to both to state the theorem on the existence of eigenvarieties, and to give some explanation of the representation-theoretic techniques required in its proof.