Timo Keller

Arithmetic geometry · abelian varieties, modular curves, rational points

How many rational solutions does an equation like y2 = x3 − x have, and is there an algorithm that finds them all? Surprisingly, this is open in general. The missing piece is one of the Clay Millennium Prize Problems!

I work in the cryptography team of IBM Deutschland Research & Development and continue to conduct research in arithmetic geometry. Before that I was an associate professor substitute of mathematics at Universität Würzburg, a Marie Skłodowska-Curie postdoctoral fellow at Rijksuniversiteit Groningen, and a postdoctoral researcher at Leibniz Universität Hannover and Universität Bayreuth.

Research topics

Arithmetic geometry, especially arithmetic and computational aspects of curves and abelian varieties over arithmetic fields, their rational points (especially of ), , , , and cohomology.

Real points of the elliptic curve y squared equals x cubed minus x: a small oval and an unbounded branch
Real points of E: y2 = x3 − x.
The Birch–Swinnerton-Dyer (BSD) conjecture states a surprising and deep relation between algebraic and analytic, local and global invariants of elliptic curves over Q, and more generally of abelian varieties over global fields. There are many important consequences, for example the existence of an algorithm to compute the Mordell–Weil group.
Modular curves are moduli spaces for elliptic curves with additional data like level structure, e.g. isogenies of degree N. Knowing their rational points is important for many other questions in arithmetic geometry. For example, they allow one to classify the possible torsion subgroups of elliptic curves over Q.
A modular form is a holomorphic function on the upper half plane that transforms in a prescribed way under the action of a congruence subgroup of SL2(Z). By modularity (Wiles, Breuil–Conrad–Diamond–Taylor) every elliptic curve over Q corresponds to a weight-2 newform.
A Galois representation is a continuous homomorphism from the absolute Galois group of a number field into a matrix group. Galois representations encode arithmetic information, for example, the action of Galois on the points of finite order of an elliptic curve.
An L-function is an analytic function attached to an arithmetic object (an elliptic curve, a modular form, a Galois representation) that packages its local data at every prime into a single complex-analytic gadget. Many of the deepest conjectures in number theory predict that special values of L-functions encode global arithmetic invariants, and BSD is one example.
The Mordell–Weil theorem says that for an abelian variety A over a number field K, the group A(K) of K-rational points is finitely generated. Its rank, the number of independent points of infinite order, is the central mystery: BSD predicts it equals the order of vanishing of L(A, s) at s = 1.
The Tate–Shafarevich group Ш(A/K) measures the failure of the local-global (Hasse) principle for principal homogeneous spaces under A. It is conjectured to be finite. Its predicted order appears in the strong BSD formula, but in general we know neither that it is finite nor how to compute it without conjectural input.
Iwasawa theory studies arithmetic invariants like class groups, Selmer groups, L-values in towers of number fields. The main conjectures of Iwasawa theory link an algebraic object (a characteristic ideal) to an analytic object (a p-adic L-function), and proving them yields cases of BSD.
The Chabauty–Coleman method bounds the rational points on a curve of genus g ≥ 2 using p-adic integration on the Jacobian, when the Mordell–Weil rank is small. The Chabauty–Kim programme generalises this by replacing the Jacobian with deeper, non-abelian unipotent quotients of the étale fundamental group; in practice this leads to quadratic Chabauty, which has settled rational points on many modular curves.
  • Abelian varieties and the

    Over the rationals, BSD is a bridge between counting solutions of an equation (algebra) and the behavior of an associated complex function at a single point (analysis). More precisely: the rank of the equals the order of vanishing of the L-function. The strong form of the conjecture moreover predicts the leading coefficient in terms of the order of the , the regulator, and the Tamagawa product.

    • over number fields: explicit methods and
    • in positive characteristic: over higher-dimensional bases
  • Rational points on

    Modular curves classify elliptic curves with extra structure; their rational points control which torsion subgroups, isogeny degrees, or Galois images can occur over Q. The challenge is that classical Chabauty fails when the Mordell–Weil rank meets or exceeds the genus. This is where the and its quadratic refinement come in.

    • explicit and theoretical methods, including quadratic Chabauty

The strong BSD formula

For an elliptic curve E/Q of analytic rank r, the strong BSD conjecture predicts

L(r)(E, 1) / r!  =  ΩE · RE · |Ш(E/Q)| · ∏p cp  /  |E(Q)tors|2.

For E: y2 = x3 − x, there is no zero at s = 1, hence the analytic rank is 0, so the formula reads L(E, 1) / ΩE = |Ш| · ∏p cp / |E(Q)tors|2. This consistent with the (algebraic) rank being 0.

Domain-coloured complex plot of L(E, s): hue encodes the argument, brightness the modulus, with dark zeros along Re(s) = 1/2
Domain-coloured plot of L(E, s) for E: y2 = x3 − x. Hue encodes the argument, brightness the modulus; zeros appear as dark points where every hue meets.
“This remarkable conjecture relates the behavior of a function L at a point where it is not at present known to be defined to the order of a group Ш which is not known to be finite!” — John Tate, 1974

Selected results

Rational points on X0(N)* when N is non-squarefree (with S. Hashimoto and S. Le Fourn).
Preprint, 2025. integrality result for j-invariants on X0(N)*, conjecturally complete classification of the rational points for genus 1–5, and previously unknown exceptional rational points on X0(147)* and X0(75)*. Step towards Elkies' conjecture.

Selected slides and exposition

More

Code accompanying my articles is on GitHub. I also keep a blog, partially intended for the interested mathematical public.