Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting 2020

The second annual meeting of the collaboration drew about 90 people to the Simons Foundation to discuss progress over the previous two years and map out directions for future work. The meeting was prefaced by a public lecture by a leading expert in computational number theory: “Escher and the Droste Effect” by Hendrik Lenstra (Leiden). Lenstra described mathematical structures underlying Escher’s contradictory shifting of scales in works like the Print Gallery.

Speakers at the meeting played several roles. First, we asked collaboration members to report on recent research accomplishments. In “(1, 3)-polarized abelian surfaces and their moduli,” PI Noam Elkies (Harvard) reported on the fine geometric structure of these moduli spaces with a view toward arithmetic applications (e.g., the frequency of sporadic Q-rational points). C´eline Maistret (Boston University) joins the collaboration this year. She spoke on “Compatibility of Some Local Invariants of Abelian Varieties” on geometric approaches to local invariants of hyperelliptic Jacobians with a view toward the Birch–Swinnerton–Dyer conjecture. In work with Vladimir Dokchitser, Maistret proved the parity conjecture for principally polarized abelian surfaces, assuming suitable local constraints and finiteness of the Shafarevich–Tate group. PI Andrew Sutherland (MIT) told a fascinating story in “Sums of cubes: Which integers N may be expressed as a sum of three cubes and in how many ways?” Since this was popularized by Mordell in the 1950s, many mathematicians have tackled this question using both theory and massive computation. Sutherland described how he and Andrew Booker resolved three of most vexing cases (N = 3, 33, 42).

We also sought lectures from researchers working on topics close to the themes of the collaboration but not officially part of the group. Henri Darmon (McGill) lectured on “The Dedekind–Rademacher Cocycle and its RM Values”; he discussed using rigid analytic geometry and p-adic deformations of Hilbert modular forms to shed light on class groups of real quadratic fields. Melanie Matchett-Wood (Berkeley) spoke on “Distributions of Unramified Extensions of Global Fields,” presenting conjectures about the statistical distribution of Galois groups of maximal unramified extensions of number fields, and the evidence for function fields of finite fields. “f-adic Monodromy Groups of Abelian Varieties” by David Zywina (Cornell) discussed algorithmic approaches to Galois representations via Mumford–Tate groups and their root systems, with a view toward computing Sato–Tate groups. His techniques apply to examples of large genus that are not practical with existing software.

The third group consisted of leading experts working on topics with more speculative connections to the themes of the collaboration. Brian Lawrence (University of Chicago) spoke on “Diophantine Problems and a p-adic Period Map.” His new proof of Faltings’ Theorem on the finiteness of rational points raises hope for a computationally effective approach to Diophantine questions on curves of genus greater than one. And Kirsten Wickelgren (Duke) discussed refinements of enumerative problems in “An Arithmetic Count of Rational Plane Curves.” Her counts take place in Grothendieck–Witt groups of quadratic forms and have elegant interpretations over finite fields and the real numbers.

Escher and the Droste Effect
Hendrik Lenstra, Universiteit Leiden

Wednesday, January 8, 2020
Tea 4:15-5:00 PM
Lecture 5:00-6:15 PM

In 1956, the Dutch graphic artist M.C. Escher made an unusual lithograph. Titled “Prentententoonstelling” (or, “Print Gallery”), the piece shows a young man standing in an exhibition gallery viewing a print of a Mediterranean seaport. Among the buildings depicted in the twisting print, the man paradoxically sees the very same gallery in which he is standing. Curiously, Escher left the middle of the lithograph blank, filling it with only his monogram and signature.

In this lecture, Hendrik W. Lenstra will discuss interactions between mathematics and M.C. Escher’s artwork. A mathematical analysis of the methods used by Escher leads to a series of hallucinating computer animations that show, among others, what’s in the blurry blank hole in the middle of the piece.

Participation is optional; separate registration is required.

See the lecture page for further information.

Henri Darmon
McGill University

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.
 

Noam Elkies
Harvard University

(1,3)-polarized abelian surfaces and their moduli

We study \((1,3)\)-polarized abelian surfaces \(A\) with full level-2 structure (i.e.\ a choice of identification of \(A[2]\) with \(({\bf Z}/2{\bf Z})^4\)) via their Kummer surfaces \({\rm Km}(A)\). Barth and Nieto gave a model of the generic such \({\rm Km}(A)\) as a smooth quartic surface with \(32\) lines, and used this to birationally identify the moduli space of \(A\) with the double cover

$$ N’: u^2 = -t_1 t_2 t_3 t_4 t_5 t_6 $$

of the quintic threefold \(N\) with birational projective model

$$ t_1 + \cdots + t_6 = 1/t_1 + \cdots + 1/t_6 = 0. $$

We find several other nice models of \({\rm Km}(A)\): another quartic model with \(42\) lines, and a few of the \(148\) elliptic fibrations (counted up to automorphisms of the Calabi-Yau threefold \(N’\)). It was also known that A contains a hyperelliptic genus-4 curve, call it \(C\)\/; we write \(C\) in terms of the \(t_i\), letting us exhibit examples over \(\bf Q\) such as

$$ y^2 = x (x-1) (25x-12) (3x-1) (5x-9) (2x-1) (4x-3) (5x-2) (13x-8). $$

We then show that the Jacobian \(J(C)\) is isogenous to \(A \times B\) for some other \((1,3)\) surface \(B\) with full level-2 structure, and that the points on \(N’\) corresponding to \(A\) and \(B\) are related by an exotic involution of \(N’\). Future applications will include searching the rational moduli space \(N’ / S_6\) of \((1,3)\) surfaces (with no conditions on \(A[2]\)) for examples of low conductor with \({\rm End}(A) = {\bf Z}\) that we cannot yet access in any other way.
 

Brian Lawrence
University of Chicago

Diophantine Problems and a P-adic Period Map

Lawrence will outline a proof of Mordell’s conjecture/Faltings’s theorem using p-adic Hodge theory (joint with Akshay Venkatesh) and discuss some of the issues involved in making the proof computationally effective.
 

Celine Maistret
Boston University

Compatibility of Some Local Invariants of Abelian Varieties

Let A/K be an abelian variety over a number field. If A/K admits an isogeny of a particular type, it is known how to control the parity of its rank. On the other hand, the Birch and Swinnerton-Dyer conjecture (BSD) also provides a way to compute the parity of the rank, but these two expressions for the parity of the rank are not obviously compatible. By proving this compatibility, one proves the so-called parity conjecture. In other words, one proves that BSD correctly predicts the parity of the rank of abelian varieties.

The key to proving the parity conjecture in this setting is to find an expression for the local discrepancy between specific local arithmetic invariants of A/K. In this talk, Maistret will discuss several properties of this local discrepancy for elliptic curves and some specific abelian varieties and present a new expression for the local discrepancy in the case of Jacobians of genus two curves.
 

Andrew Sutherland
Massachusetts Institute of Technology

Sums of Cubes

Sutherland will report on some recent computational investigations of a conjecture of Heath-Brown.
 

Melanie Matchett Wood
University of California, Berkeley

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.
 

Kirsten Wickelgren
Duke University

An Arithmetic Count of Rational Plane Curves

There are finitely many degree d rational plane curves passing through 3d-1 points, and over the complex numbers, this number is independent of (generically) chosen points. For example, there are 12 degree 3 rational curves through 8 points, one conic passing through 5, and one line passing through 2. Over the real numbers, this number can vary. For example, there can be 8, 10 or 12 real degree 3 curves through 8 real points. Welschinger obtained a fixed number by weighting real rational curves by behavior of their nodes, and work of Solomon identifies this invariant with a local degree. It is a feature of A1-homotopy theory that analogous real and complex results can indicate the presence of a common generalization, valid over a general field. In the work covered in this talk, Wickelgren develops and computes an A1-degree, following Morel, of the evaluation map on Kontsevich moduli space to obtain an arithmetic count of rational plane curves, which is valid for any field k of characteristic not 2 or 3. This shows independence of the count on the choice of generically chosen points with fixed residue fields, strengthening a count of Marc Levine. This is joint work with Jesse Kass, Marc Levine and Jake Solomon.
 

David Zywina
Cornell University

\(\ell\)-adic Monodromy Groups of Abelian Varieties

Consider an abelian variety \(A\) defined over a number field. For a fixed prime \(\ell\), the Galois action on the \(\ell\)-power torsion points of \(A\) can be described in terms of an \(\ell\)-adic Galois representation. The Zariski closure \(G_\ell\) of the image is called the \(\ell\)-adic monodromy group of \(A\). The group \(G_\ell\) encodes many important arithmetic/geometric properties of \(A\). For example, the (conjectural) Sato–Tate distribution of \(A\) can be determined from \(G_\ell\). We will discuss approaches to studying and computing monodromy groups.