Formalization of a Deontic Logic Theorem in the Isabelle Proof Assistant

<a href=”https://math.hawaii.edu/home/depart/theses/MA_2019_Fennick.pdf”>Draft project report</a>

Title: Single- and Multivariable (φ, Γ)-Modules and Galois Representations

Abstract: I will introduce the notion of a (single-variable) (φ, Γ)-module and explain the relationship of (φ, Γ)-modules to representations of Gal(Q̅_p | Q_p), the absolute Galois group of the p-adic numbers. I will then describe joint work with Kiran Kedlaya and Gergely Zábrádi which extends this relationship to multivariate (φ, Γ)-modules.

Let $K$ be a number field. We will show that any bicritical polynomial $f(z) in K[z]$ is conjugate to a polynomial of the form $amathcal{B}_{d,k}(z) +c in K[z]$ where $mathcal{B}_{d,k}(z)$ is a normalized single-cycle Belyi map with combinatorial type $(d; d-k, k+1, d)$. We use results of Ingram to determine height bounds on pairs $(a,c)$ such that $amathcal{B}_{d,k}(z) +c$ is post-critically finite. Using these height bounds, we completely describe the set of post-critically finite cubic polynomials over $Q$, up to conjugacy over $Q$. We give partial results for post-critically finite polynomials over $Q$ of arbitrary degree $d>3$.

Alina Bucur of the University of California, San Diego

Title: Effective Sato-Tate under GRH

Abstract: Based on the Lagarias-Odlyzko effectivization of the Chebotarev density theorem, Kumar Murty gave an effective version of the Sato-Tate conjecture for an elliptic curve conditional on the analytic continuation and the Riemann hypothesis for all the symmetric power L-functions. Using similar techniques, Kedlaya and I obtained a similar conditional effectivization of the generalized Sato-Tate conjecture for an arbitrary motive. As an application, we obtained a conditional upper bound of the form O((logN)^2(loglogN)^2) for the smallest prime at which two given rational elliptic curves with conductor at most N have Frobenius traces of opposite sign. In this talk, I will discuss how to improve this bound to the best possible in terms of N and under slightly weaker assumptions. Our new approach extends to abelian varieties. This is joint work with Kiran Kedlaya and Francesc Fite.

Kiran Kedlaya of the University of California, San Diego

Title: Frobenius structures on hypergeometric equations

Abstract: Hypergeometric equations are a class of ordinary differential equations with strong ties to geometry and arithmetic. In particular, each hypergeometric equation parametrizes a family of motives with associated L-functions; the minimal example of this is the Gaussian hypergeometric equation corresponding to the Legendre family of elliptic curves. We sketch an algorithm, based on work of Dwork, to compute these L-functions using the existence of p-adic analytic “Frobenius structures” on the equation. This is expected to be useful for building tables of hypergeometric L functions for the LMFDB.

For a positive integer $n$, we compute the shape of a totally real multiquadratic extension of degree $2^n$ in which the prime $2$ does not ramify. From this calculation, we see that the shape of such a number field is parametrized by the generators of its $2^n-1$ quadratic subfields. Restricting to the case $n=3$, we use this parametrization to count the number of triquadratic extensions of bounded discriminant and bounded shape parameters. We then show that, as the discriminant goes to infinity, these shapes become equidistributed in a regularized sense in the subset of the space of shapes of rank $7$ lattices that contains them.

Taylor Markham of the University of Calgary

Title: Integer Factorization

Abstract: The security of many modern day cryptosystems are impacted by the fact that it is computationally difficult to factor large numbers. This talk will give an introduction to the general number field sieve, which is currently the most efficient algorithm for factoring large integers.