Timo Keller
I work in the cryptography team of IBM Deutschland Research & Development on post-quantum cryptography and confidential computing, while continuing my research in arithmetic geometry.
Arithmetic geometry abelian varieties, modular curves, rational points
How many rational solutions does an equation like 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!
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Selected results
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Golod–Shafarevich for arithmetic surfaces
Golod–Shafarevich towers of arithmetic surfaces over , giving an arithmetic surface with a section and infinite geometric étale fundamental group (answering Bost and Charles), curves over with infinite geometric étale towers in which a rational point splits completely (answering Ihara and Frey–Kani–Völklein), and curves of growing genus whose logarithmic conductor grows at most linearly in the genus (answering Venkatesh). Comes with a Lean formalization.
Why this matters
By Minkowski’s theorem, every number field other than is ramified at some prime, so has no nontrivial unramified covers (in topological language, it is simply connected). The rings of integers of other number fields can have many such covers. Golod and Shafarevich showed in 1964 that some number fields even have an infinite class field tower, i.e., an infinite chain of unramified extensions such that the degree grows and the root discriminant stays bounded. We go one dimension up, from rings of integers to curves over . Here the unramified covers are classified by the étale fundamental group, and the analogue of the root discriminant is the conductor of a curve (which measures its bad reduction) relative to its genus.
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Semi-stable models, local heights and quadratic Chabauty for
Explicit semi-stable minimal regular models of at the primes of bad reduction, and a global -adic height with vanishing local contributions at those primes for every squarefree level , which simplifies . Determines the rational points on several that were previously out of reach.
Why this matters
Elkies conjectured that for large enough level , all -points on the modular curve quotient are cusps or CM points. These points correspond to certain multiquadratic points on , so the conjecture is about cyclic isogenies of -curves, i.e., elliptic curves that are isogenous to all their Galois conjugates.
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Rational points on when is non-squarefree
Integrality result for -invariants on , conjecturally complete classification of the rational points for genus 1–5, and previously unknown exceptional rational points on and . Step towards Elkies’ conjecture.
Why this matters
The modular curve classifies elliptic curves together with a cyclic isogeny of degree , so understanding its points means understanding all elliptic curves of this kind. The -rational points of every were determined in famous work of Mazur and Kenku. Over larger number fields, much less is known. There is no such that one knows the points of over all number fields of degree and for all . Our work goes significantly beyond this, since we determine certain points over multiquadratic fields.
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Complete verification of strong BSD for many modular abelian surfaces over
First unconditional and exact verification of for absolutely simple abelian varieties of dimension .
Why this matters
The Birch–Swinnerton-Dyer conjecture is one of the Clay Millennium Problems. Its strong form predicts not only the rank, but also the order of the Tate–Shafarevich group, and it had been verified only for elliptic curves and for abelian varieties that reduce to them. We verify strong BSD for the first time for absolutely simple abelian surfaces, including the computation of the order of their Tate–Shafarevich groups, which was not possible before.
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On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction
Proves the -part of the strong BSD conjecture for elliptic curves of analytic rank 0 or 1 over at ordinary Eisenstein primes, allowing non-trivial -torsion in the , including -converse theorems to Gross–Zagier–Kolyvagin.
Why this matters
Iwasawa theory studies arithmetic invariants of an elliptic curve along an infinite tower of number fields at the same time, and this often shows structure that one cannot see at a single level. Its Main Conjecture predicts that an analytic object (built from a -adic -function) and an algebraic object (built from Galois cohomology) are equal. Proving it has strong consequences. It gives a -converse theorem, new cases of the Birch–Swinnerton-Dyer conjecture, and better proportions for Goldfeld’s conjecture on the distribution of ranks in families. What is new here is that we work at Eisenstein primes, where the mod- Galois representation is reducible and the standard tools do not work.
Research topics
Arithmetic geometry, especially arithmetic and computational aspects of curves and abelian varieties over arithmetic fields, their rational points (especially of ), , , , and cohomology.
- 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 -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 . 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
Modular curves
Modular curves are moduli spaces for elliptic curves with additional data like level structure, e.g. isogenies of degree . 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 .
Modular forms
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 . By modularity (Wiles, Breuil–Conrad–Diamond–Taylor) every elliptic curve over corresponds to a weight-2 newform.
Galois representations
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.
-functions
An -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 -functions encode global arithmetic invariants, and BSD is one example.
Birch–Swinnerton-Dyer conjecture
The Birch–Swinnerton-Dyer (BSD) conjecture states a surprising and deep relation between algebraic and analytic, local and global invariants of elliptic curves over , 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.
Mordell–Weil group
The Mordell–Weil theorem says that for an abelian variety over a number field , the group of -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 at .
Tate–Shafarevich group
The Tate–Shafarevich group measures the failure of the local-global (Hasse) principle for principal homogeneous spaces under . 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
Iwasawa theory studies arithmetic invariants like class groups, Selmer groups, -values in towers of number fields. The main conjectures of Iwasawa theory link an algebraic object (a characteristic ideal) to an analytic object (a -adic -function), and proving them yields cases of BSD.
Chabauty–Kim method
The Chabauty–Coleman method bounds the rational points on a curve of genus using -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.
Previous positions and studies
Before joining IBM, 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. I studied mathematics and computer science at the universities of Kaiserslautern, Bonn, and Regensburg. I received my doctorate in mathematics (Dr. rer. nat., equivalent to a PhD) from Universität Regensburg and completed my habilitation, the German qualification for a professorship, at Universität Bayreuth. Both were on the Birch–Swinnerton-Dyer conjecture.