## Expander graphs, gonality, and variation of Galois representations

Suppose you have a 1-dimensional family of polarized abelian varieties — or, just to make things concrete, an abelian variety A over Q(t) with no isotrivial factor.

You might have some intuition that abelian varieties over Q don’t usually have rational p-torsion points — to make this precise you might ask that A_t[p](Q) be empty for “most” t.

In fact, we prove (among other results of a similar flavor) the following strong version of this statement.  Let d be an integer, K a number field, and A/K(t) an abelian variety.  Then there is a constant p(A,d) such that, for each prime p > p(A,d), there are only finitely many t such that A_t[p] has a point over a degree-d extension of K.

The idea is to study the geometry of the curve U_p parametrizing pairs (t,S) where S is a p-torsion point of A_t.  This curve is a finite cover of the projective line; if you can show it has genus bigger than 1, then you know U_p has only finitely many K-rational points, by Faltings’ theorem.

But we want more — we want to know that U_p has only finitely many points over degree-d extensions of K.  This can fail even for high-genus curves:  for instance, the curve

C:   y^2 = x^100000 + x + 1

has really massive genus, but choosing any rational value of x yields a point on C defined over a quadratic extension of Q.  The problem is that C is hyperelliptic — it has a degree-2 map to the projective line.  More generally, if U_p has a degree-d map to P^1,  then U_p has lots of points over degree-d extensions of K.  In fact, Faltings’ theorem can be leveraged to show that a kind of converse is true.

So the relevant task is to show that U_p admits no map to P^1 of degree less than d; in other words, its gonality is at least d.

Now how do you show a curve has large gonality?  Unlike genus, gonality isn’t a topological invariant; somehow you really have to use the geometry of the curve.  The technique that works here is one we learned from an paper of Abramovich; via a theorem of Li and Yau, you can show that the gonality of U_p is big if you can show that the Laplacian operator on the Riemann surface U_p(C) has a spectral gap.  (Abramovich uses this technique to prove the g=1 version of our theorem:  the gonality of classical modular curves increases with the level.)

We get a grip on this Laplacian by approximating it with something discrete.  Namely:  if U is the open subvariety of P^1 over which A has good reduction, then U_p(C) is an unramified cover of U(C), and can be identified with a finite-index subgroup H_p of the fundamental group G = pi_1(U(C)), which is just a free group on finitely many generators g_1, … g_n.  From this data you can cook up a Cayley-Schreier graph, whose vertices are cosets of H_p in G, and whose edges connect g H with g_i g H.  Thanks to work of Burger, we know that this graph is a good “combinatorial model” of U_p(C); in particular, the Laplacian of U_p(C) has a spectral gap if and only if the adjacency matrix of this Cayley-Schreier graph does.

At this point, we have reduced to a spectral problem having to do with special subgroups of free groups.  And if it were 2009, we would be completely stuck.  But it’s 2010!  And we have at hand a whole spray of brand-new results thanks to Helfgott, Gill, Pyber, Szabo, Breuillard, Green, Tao, and others, which guarantee precisely that Cayley-Schreier graphs of this kind, (corresponding to finite covers of U(C) whose Galois closure has Galois group a perfect linear group over a finite field) have spectral gap; that is, they are expander graphs. (Actually, a slightly weaker condition than spectral gap, which we call esperantism, is all we need.)

Sometimes you think about a problem at just the right time.  We would never have guessed that the burst of progress in sum-product estimates in linear groups would make this the right time to think about Galois representations in 1-dimensional families of abelian varieties, but so it turned out to be.  Our good luck.

## In the Land of Invented Languages

I had a thing for invented languages as a kid.  I took a correspondence course in Esperanto, and when I got tired of that, I started work (as one does) on my own ideal language, which was called Ilenga.  Later, when I was at Johns Hopkins, I spent a lot of time in Eisenhower Library looking at their collection of pamphlets, broadsides, and mimeographed polemics — and even the occasional published book — by language creators whose painstaking constructions never rose to the level of fame Esperanto enjoyed.  In the end, a lot of this stuff made its way into The Grasshopper King, which in some sense is about the question:  “What if a real language worked the way people who invent languages want languages to work, and what would happen to you if you tried to speak that language?”

It turns out Arika Okrent was looking at the same shelf of pamphlets.  And she now has a book, In the Land of Invented Languages, a kind of cultural history of the idea of the invented language.  You know how when you see the one-paragraph description of a book, and the premise is really great, and you say to yourself “I really hope this book is good, because if it isn’t,  it’ll be impossible for any future good book on this premise ever to published?”  That’s how I felt.  And I’m happy to report that Okrent’s book is everything I wanted it to be.  Partly because she’s a good, energetic writer.  Partly because she has a Ph.D. in linguistics and writes with an easy authority about the technicalities that vex her subjects.  And partly because she’s a hell of a researcher with an eye for the strange, decisive detail.  Three great facts I learned from this book:

• Grover Cleveland’s wife had a dog named Volapük.
• George Soros’s father was born with the surname Schwarz; he was a dedicated Esperantist and changed it to Soros, Esperanto for “will soar.”
• James Cooke Brown, the inventor of Loglan, had the time and disposable income to create a language because he also invented the boardgame Careers.  Brown, a lifelong socialist, intended Careers to counteract what he saw as Monopoly’s overemphasis on making money as the sole goal of life.  I was a major Careers fan as a kid and let me just say this point was utterly lost on me.

This book pulls off a very difficult trick.  Okrent is writing about people who are often strange and almost always, in one way or another, misguided.  She gives you the full measure of their strangeness, but never deviates from her posture of bemused respect for the audacity and technical difficulty of the tasks they’ve set themselves.  Good trick; good book.

Here’s Okrent on Klingon speakers in Slate. Here’s her blog, which right now is just a list of book events. Here’s her bagel recipe.

And here’s the longest text I ever wrote in Ilenga:  a translation of the first verse of “Shout,” by Tears for Fears.

Shautoc, shautoc

Jame relsoc

Pas o i cosas nu as ni nido, disoc

Loc za

A disoi tu ta

Loc za!