### Conversations on Mathematics

#### Posted by John Baez

Now that I’ve retired, I have more time for pure math. So after a roughly decade-long break, James Dolan and I are talking about math again. Here are our conversations. Some are in email, but mainly these are our weekly 2-hour-long Zoom sessions, which I’ve put on YouTube. They focus on algebraic geometry — especially abelian varieties and motives — but also ‘doctrines’ and their applications to algebraic geometry, group representation theory, combinatorics and other subjects.

They may not be easy to follow, but maybe a few people will get something out of them. I have not corrected all the mistakes, some of which we eventually catch. I’ve added lots of links to papers and Wikipedia articles.

These conversations are continuing, but I won’t keep putting links to them here on $n$-Category Café, so if you want more of them you can either check out my webpage at your leisure, or subscribe to my YouTube channel. I’ll probably fall behind in putting up videos, and then catch up, and then fall behind, etc. — so please don’t expect one to show up each week.

The conversations on YouTube are labelled ‘conversation’.

Minkowski and lattices - February 11, 2022.

The kid who learned math on the street - February 18, 2022. On calculating the Néron–Severi group of an abelian surface.

**February 21, 2022 conversation**. Topics include:How to describe complex line bundles, or holomorphic line bundles, on a complex torus (e.g. an abelian variety) using Riemann forms.

The square tiling honeycomb {4,4,3} and its relation to the abelian surface that’s the product of two copies of the elliptic curve given by the complex numbers mod the Gaussian integers.

The hexagonal tiling honeycomb {6,3,3} and its relation to the abelian surface that’s the product of two copies of the elliptic curve given by the complex numbers mod the Eisenstein integers.

Endomorphism rings of abelian varieties - February 24, 2022.

Chern connection - February 24, 2022.

**February 28, 2022 conversation**. Topics include:Classifying holomorphic line bundles on an abelian variety using their Riemann forms.

Endomorphisms of abelian varieties, the Rosati involution, how to describe polarizations on an abelian variety as ‘positive’ endomorphisms, and the structure of the Néron–Severi group tensored with $\mathbb{Q}$ as a Jordan algebra. For this it is useful to read:

Christina Birkenhake and Herbert Lange,

*Complex Abelian Varieties*.James Milne,

*Abelian Varieties*.

The Kronecker–Weber theorem and Kronecker’s Jugendtraum: the relation between elliptic curves with complex multiplication and abelian extensions of imaginary quadratic fields.

Digression on Jordan cone of 3-by-3 hermitian complex matrixes - March 3, 2022.

Stacks - March 5, 2022.

**March 7, 2022 conversation**. Topics include:The factor of automorphy of an arbitrary holomorphic line bundle over a complex abelian variety.

Jordan algebras and the Rosati involution

Paracompact hyperbolic Coxeter groups - March 8, 2022.

John Baez, Line bundles on complex tori (part 1),

*The n-Category Café*, March 13, 2022.**March 14, 2022 conversation**. Topics include:Hyperbolic Coxeter groups coming from rings of algebraic integers:

Norman Johnson and Asia Ivić Weiss, Quadratic integers and Coxeter groups.

The integers $\mathbb{Z}$, the modular group and $\mathrm{PGL}(2,\mathbb{Z})$, and the closely related Coxeter group $\{\infty,3\}$ which acts as symmetries of the infinite-order triangular tiling of the hyperbolic plane.

The Gaussian integers $\mathbb{Z}[i]$, the groups $\mathrm{PSL}(2,\mathbb{Z}[i])$ and $\mathrm{PGL}(2,\mathbb{Z}[i])$, and the closely related Coxeter group $\{4,4,3\}$ which acts as symmetries of the square tiling honeycomb in hyperbolic 3-space.

The Eisenstein integers $\mathbb{Z}[\omega]$, the groups $\mathrm{PSL}(2,\mathbb{Z}[\omega])$ and $\mathrm{PGL}(2,\mathbb{Z}[\omega])$, and the closely related Coxeter group $\{6,3,3\}$ which acts as symmetries of the honeycomb tiling honeycomb in hyperbolic 3-space.

The classification of holomorphic line bundles on complex tori.

Pic

_{0}(the Picard variety) - March 14, 2022.John Baez, Line bundles on complex tori (part 2),

*The n-Category Café*, March 19, 2022.Hodge conjecture - March 24, 2022.

John Baez, Holomorphic gerbes (part 1),

*The n-Category Café*, March 27, 2022.John Baez, Holomorphic gerbes (part 2),

*The n-Category Café*, April 10, 2022.**March 28, 2022 conversation**. Topics include:The Appell–Humbert theorem classifying holomorphic line bundles on a complex torus, and Ben-Bassat’s analogous result for holomorphic gerbes:

Eran Assaf, The Appell–Humbert theorem.

Oren Ben-Bassat, Gerbes and the holomorphic Brauer group of complex tori.

The ‘belief method’ for describing algebras of monads on the bicategory of locally presentable $k$-enriched categories, where $k$ is a symmetric monoidal locally presentable category.

Belief method - March 28, 2022.

**April 4, 2022 conversation**. Topics include:Ben-Bassat’s version of the Néron–Severi group for holomorphic gerbes:

- Oren Ben-Bassat, Gerbes and the holomorphic Brauer group of complex tori.

Generalizations of the Jacobian, the Picard group and the Néron–Severi group from holomorphic line bundles to holomorphic $n$-gerbes:

Holomorphic gerbes (part 1),

*The n-Category Café*, March 27, 2022.Holomorphic gerbes (part 2),

*The n-Category Café*, April 10, 2022.

An application of the belief method to commutative quantales (which are cocomplete symmetric monoidal posets).

**April 11, 2022 conversation**. Topics include:Using sheaf cohomology to generalize the Jacobian, the Picard group and the Néron–Severi group from holomorphic line bundles to holomorphic $n$-gerbes:

Holomorphic gerbes (part 1),

*The n-Category Café*, March 27, 2022.Holomorphic gerbes (part 2),

*The n-Category Café*, April 10, 2022.

The belief method applied to the doctrine of symmetric monoidal locally presentable $k$-linear categories, which gives ‘algebro-geometric theories’.

The ‘theory of an object’ is the category of $k$-linear species with its Cauchy tensor product.

The theory of an object whose exterior cube is the initial object. The representations of $\mathrm{GL}(2)$, $\mathrm{SL}(2)$ and related groups.

**April 18, 2022 conversation**. Topics include:Young diagrams and representations of:

the general linear group $\mathrm{GL}(2)$,

the special linear group $\mathrm{SL}(2)$,

the projective general linear group $\mathrm{PGL}(2)$,.

**May 16, 2022 conversation**. Topics include:John Baez, Motivating motives, May 25, 2022.

J. S. Milne, Motives: Grothendieck’s dream.

J. S. Milne, The Riemann Hypothesis over finite fields: from Weil to the present day.

**May 23, 2022 conversation**. Topics include:Counting points on elliptic curves, and motives.

Supersingular elliptic curves over finite fields, which are those with a quaternion algebra of endomorphisms.

Getting abelian varieties from cyclotomic fields, using the example of the 20th cyclotomic field.

**June 6, 2022 conversation**. Topics include:The moduli stack of real elliptic curves.

Generalizing Kronecker’s Jugendtraum from elliptic curves to abelian varieties.

Manin’s “Alterstraum” and elliptic curves with real multiplication.

**June 13, 2022 conversation**. Topics include:The category of pure numerical motives (with rational coefficients) over the field with $q$ elements, $\mathbb{F}_q$.

The classification of simple objects in this category. The absolute Galois group of the rationals acts on the set of Weil $q$-numbers, and assuming the Tate Conjecture, simple objects correspond to orbits of this action. This is Proposition 2.6. in Milne’s Motives over finite fields. I got the definition of Weil $q$-number wrong. Given a prime power $q$, a Weil $q$-number is a complex number such that:

for some integer $n$, $q^n z$ is an algebraic integer,

for some integer $m$, $|g z| = q^{m/2}$ for every element $g$ in the absolute Galois group of the rationals.

**June 20, 2022 conversation**. Topics include:Correcting the definition of Weil $q$-number, which is explained in Definition 2.5 of Milne’s Motives over finite fields.

Toposes in number theory.

The topos of species.

Dirichlet species:

John Baez and James Dolan, Dirichlet species and the Hasse–Weil zeta function.

Set-valued functors on the groupoid or category of finite fields of characteristic $p$.

The topos of actions of the Galois group.

**June 27, 2022 conversation**. Topics include:The exponential sheaf sequence of a complex abelian surface $X$, and how the corresponding long exact sequence in sheaf cohomology lets us describe the Néron–Severi group as the image of $H^2(X,\mathcal{O}^\ast)$ in $H^2(X,\mathbb{Z})$, or equivalently the kernel of the map from $H^2(X,\mathbb{Z}$) to $H^2(X,\mathcal{O})$. When the rank of the Néron–Severi group is maximal the image of the map from $H^2(X,\mathbb{Z})$ to $H^2(X,\mathcal{O})$ is a lattice in a complex vector space — but generically it appears to be a dense subgroup.

How the classification of simple objects in the category of pure numerical motives changes when we use the algebraic closure of the rationals as coefficients, rather than the rationals. This is Proposition 2.21 in Milne’s Motives over finite fields. (I said tensoring irreps of $\mathbb{R}$ amounts to multiplication of numbers, but it’s really addition.)

**July 4, 2022 conversation**. Topics include:The exponential sheaf sequence and a detailed study of the map from $H^2(X,\mathbb{Z})$ to $H^2(X,\mathcal{O})$ when $X$ is an abelian surface. We can work out this map and its image explicitly in examples. Here we do the easiest case, when $X$ the product of two identical elliptic curves, each being the complex numbers mod the lattice of Gaussian integers. We see that in this case the image of the map from $H^2(X,\mathbb{Z})$ to $H^2(X,\mathcal{O})$ is a lattice in a 2d complex vector space.

A polarization on an abelian variety $X = V/L$ puts a positive definite hermitian form on $V$, and this lets us describe the Néron–Severi group in terms of self-adjoint operators on $V$. This clarifies how a polarization puts a Jordan algebra structure on the Néron-Severi group tensored with $\mathbb{R}$.

**July 11, 2022 conversation**. Topics include:Classifying complex abelian surfaces in order of genericity, and the ranks of their Néron–Severi group:

The generic case gives a Néron–Severi group of rank 1,

the cartesian product of two distinct elliptic curves gives rank 2,

the cartesian square of a generic elliptic curve gives rank 3, and

the cartesian square of an elliptic curve with complex multiplication gives rank 4.

The moduli stack of elliptic curves and the associated Coxeter group. How can we generalize this to higher-dimensional principally polarized abelian varieties? How do modular curves generalize to the higher-dimensional case, giving Siegel modular varieties?

## Re: Conversations on Mathematics

Well, I said I wouldn’t update these, but I couldn’t resist it this time, because we just figured out something that we’d been leading up to in many conversations. James was puzzled by how the map sending smooth complex line bundles on a complex variety $X$ to elements of the vector space $H^2(X,\mathcal{O})$ could have an image that’s “dust-like” — dense, but not the whole space. I finally did the calculation and showed that this really happens in an example! We still need to think about the consequences.