计算数论

Motivating questions:

1) How do we compute rational points on curves in practice?

2) How do we predict the random distribution of class groups and rational points?

3) What is unknown about the current framework of number theory?

Henri Darmon The Dedekind-Rademacher Cocycle and its RM Values

The Dedekind-Rademacher cocycle is a class in the first cohomology of the Ihara group SL(2, Z[1/p]) with values in the multiplicative group of nowhere vanishing rigid analytic functions on Drinfeld’s p-adic upper half plane. Samit Dasgupta and Henri Darmon conjectured around 2002 that its special values at ‘real multiplication points’ of Drinfeld’s p-adic upper half plane are (often nontrivial) global p-units in narrow ring class fields of real quadratic fields. An important recent breakthrough of Dasgupta and Mahesh Kakde, building on earlier work of Dasgupta, Robert Pollack and Darmon, has led to the proof of a large part of this conjecture. In this talk, Darmon will describe an independent approach to the theorem of Dasgupta and Kakde that rests on the p-adic deformation theory of Hilbert modular Eisenstein series and the properties of their diagonal restrictions, developed in collaboration with Alice Pozzi and Jan Vonk. Time permitting, computational questions surrounding the efficient numerical calculation of the Dedekind-Rademacher cocycle and its RM values will be addressed.

Melanie Matchett Wood Distributions of Unramified Extensions of Global Fields

Every number field K has a maximal unramified extension K^un, with Galois group Gal(K^un/K) (whose abelianization is the class group of K). As K varies, we ask about the distribution of the groups Gal(K^un/K). In this talk, Wood gives a conjecture about this distribution, which she also conjectures in the function field analog. Wood and colleagues will give some results about Gal(K^un/K) that motivate them to build certain random groups whose distributions appear in their conjectures. She gives theorems in the function field case (as the size of the finite field goes to infinity) that support these new conjectures. In particular, their distributions abelianize to the Cohen-Lenstra-Martinet distributions for class groups, and so their function-field theorems give support to (suitably modified) versions of the Cohen-Lenstra-Martinet heuristics. This talk is on joint work with Yuan Liu and David Zureick-Brown.

Andrew Booker and Andrew Sutherland

Sums of cubes

Kiran S. Kedlaya, Alexander Kolpakov, Bjorn Poonen, and Michael Rubinstein

Vectors in space forming rational angles

Lassina Demb ́el ́e, Alexei Panchishkin, John Voight, and Wadim Zudilin

Formulas for π

Francesc Fit ́e, Kiran Kedlaya, and Andrew Sutherland

Sato-Tate distributions

Balakrishnan–Dogra–Mu ̈ller–Tuitman–Vonk

−x3y+2x2y2−xy3−x3z+x2yz+xy2z−2xyz2+2y2z2+xz3−3yz3 = 0

There are precisely 7 rational points on the cursed curve.

LMDFB

Nina Zubrilina Root Number Correlation Bias of Fourier Coefficients of Modular Forms

In a recent machine learning based study, He, Lee, Oliver and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a broader class of arithmetic L-functions when split by root number.

Adam Logan Computing Shadows of Ceresa Cycles The Ceresa cycle on the Jacobian of a curve was introduced by Ceresa in his 1982 dissertation to illuminate the distinction between algebraic and homological equivalence on a variety. He proved by a nonconstructive Hodge-theoretic argument that for a very general curve the Ceresa cycle is not torsion in the group of cycles modulo algebraic equivalence.

In recent years there has been a lot of interest in proving that the Ceresa cycle is or is not torsion for explicit curves. One type of obstruction to the cycle being torsion lies in the Jacobian of the reduction of the curve to a finite field. Adam Logan will describe this obstruction, some extensions of it, and calculations of these for a set of plane quartics. This is joint work with J. Ellenberg, P. Srinivasan, and A. Venkatesh.

Akshay Venkatesh Quadratic Functions on Class Groups

Hecke proved that the different of a number field is always a square in the ideal class group. In work in progress with collaborator Artane Siad, Akshay Venkatesh will show that a choice of a square root induces an interesting algebraic structure, a quadratic function on the class group, which clarifies the origin of some peculiar class group statistics.