Timo Keller's blogResearch announcements in number theory and arithmetic geometry, with non-technical summaries and introductory entries for a general audience.https://www.timo-keller.de/blog/2026-07-24T00:00:00ZTimo KellerHarmonizing Bach chorales with a Transformer (a machine-learning side project)https://www.timo-keller.de/blog/july-24-2026-harmonizing-bach-chorales-with-a-transformer-a-machine-learning-side-project.html2026-07-24T00:00:00Z2026-07-24T00:00:00ZA short detour from number theory. In my spare time I built ChoraleHarmonizer, a small program that takes a soprano melody and writes the three lower voices (alto, tenor, and bass) in the style of Johann Sebastian Bach’s four-part chorales. Bach left several hundred of these harmonizations, and reconstructing the lower voices from a given melody is a classic music-theory exercise, so there is …How many point counts determine an abelian variety?https://www.timo-keller.de/blog/july-4-2026-how-many-point-counts-determine-an-abelian-variety.html2026-07-04T00:00:00Z2026-07-04T00:00:00ZWith Shiva Chidambaram, we proved that for an abelian variety A of dimension g over a finite field 𝐅q, the first g point counts #A(𝐅qi) for 1≤i≤g already determine its zeta function, and hence, by Tate’s theorem, its isogeny class, once q>Q(g) for an explicit (but far-from-optimal) constant. Since Kedlaya proved in 2006 that 2g counts suffice, this halves the number needed.Rational points on X_0(N)^* when N is non-squarefreehttps://www.timo-keller.de/blog/july-5-2025-rational-points-on-x_0n-when-n-is-non-squarefree.html2025-07-05T00:00:00Z2025-07-05T00:00:00ZWith Sachi Hashimoto and Samuel Le Fourn, we proved integrality properties of rational points on the Atkin–Lehner quotient X0(N)* when N is non-squarefree. This is the quotient of the modular curve X0(N) by the group of all Atkin–Lehner involutions.Inverse Galois theory for 17T7 over the rationalshttps://www.timo-keller.de/blog/november-13-2024-inverse-galois-theory-for-17t7-over-the-rationals.html2024-11-13T00:00:00Z2024-11-13T00:00:00ZWhy this matters. The inverse Galois problem asks which finite groups occur as the Galois group of a polynomial with rational coefficients. It is wide open, and even when a group is known to occur, an explicit polynomial realizing it is often out of reach. We realize the group 17T7 over 𝐐 and, going beyond a pure existence statement, exhibit an explicit degree-17 polynomial with this Galois group.p-converse theorems for odd Eisenstein primes of potentially good ordinary reductionhttps://www.timo-keller.de/blog/november-12-2024-p-converse-theorems-for-odd-eisenstein-primes-of-potentially-good-ordinary-reduction.html2024-11-12T00:00:00Z2024-11-12T00:00:00ZThis is a follow-up to my blog entry from February 26, 2024. Mulun Yin (University of California Santa Barbara) managed to generalize our p-converse theorem for odd Eisenstein primes from the good ordinary to the potentially good ordinary case, i.e., allowing additive reduction at p. Note that the construction of p-adic L-functions for additive p is an open problem in general.The (strong) Birch–Swinnerton-Dyer conjecture: an informal introduction (for a general audience)https://www.timo-keller.de/blog/may-13-2024-the-strong-birchswinnerton-dyer-conjecture-an-informal-introduction-for-a-general-audience.html2024-05-13T00:00:00Z2024-05-13T00:00:00ZThe focus of the project [K–Stoll 2023] was the conjecture of Birch and Swinnerton-Dyer (BSD for short) for abelian surfaces. Abelian surfaces are two-dimensional abelian varieties, and abelian varieties are higher-dimensional analogues of elliptic curves. An elliptic curve is an algebraic curve that carries a group structure. This means that we can add two points on the curve to get another …Congruent numbers, elliptic curves, the Birch–Swinnerton-Dyer rank conjecture, and applications (for a general audience)https://www.timo-keller.de/blog/may-13-2024-congruent-numbers-elliptic-curves-the-birchswinnerton-dyer-rank-conjecture-and-applications-for-a-general-audience.html2024-05-13T00:00:00Z2024-05-13T00:00:00ZCongruent numbers. We say that a number n∈𝐙≥1 is a congruent number if there are x,y,z∈𝐐 such that y2=x2+n and z2=y2+n, i.e., if the distance of successive numbers x2,y2,z2 is n. (One can prove that n is congruent if and only if there is a right triangle with rational side lengths and area n.)The Birch–Swinnerton-Dyer conjecturehttps://www.timo-keller.de/blog/may-13-2024-the-birchswinnerton-dyer-conjecture.html2024-05-13T00:00:00Z2024-05-13T00:00:00ZThe Conjecture of Birch and Swinnerton-Dyer (“BSD” for short), originally formulated by Birch and Swinnerton-Dyer in the 1960s for elliptic curves over 𝐐, is one of the most important open conjectures in number theory. For example, it is one of the seven “Millennium Problems”, for whose solution the Clay Foundation is offering a million dollars each. It relates in a surprising way analytic …Article on the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reductionhttps://www.timo-keller.de/blog/february-26-2024-article-on-the-anticyclotomic-iwasawa-theory-of-newforms-at-eisenstein-primes-of-semistable-reduction.html2024-02-26T00:00:00Z2024-02-26T00:00:00ZWhy this matters. Iwasawa theory studies arithmetic invariants of an elliptic curve simultaneously along an infinite tower of number fields, which often reveals structure invisible at a single level. Its Main Conjecture predicts that an analytic object (built from a p-adic L-function) and an algebraic one (built from Galois cohomology) coincide. Establishing it has strong consequences: as the …Article on the verification of strong BSD for many modular abelian surfaces over \mathbf{Q}https://www.timo-keller.de/blog/december-24-2023-article-on-the-verification-of-strong-bsd-for-many-modular-abelian-surfaces-over-mathbfq.html2023-12-24T00:00:00Z2023-12-24T00:00:00ZIn our recent preprint “Complete verification of strong BSD for many modular abelian surfaces over 𝐐” Michael Stoll and I verify the strong Birch–Swinnerton-Dyer (BSD) conjecture for the first time for abelian surfaces over 𝐐 in cases where it cannot immediately be reduced to elliptic curves, i.e., such that the surface is absolutely simple.Introduction to arithmetic geometry: plane quadrics (for a general audience)https://www.timo-keller.de/blog/december-24-2023-introduction-to-arithmetic-geometry-plane-quadrics-for-a-general-audience.html2023-12-24T00:00:00Z2023-12-24T00:00:00ZThis is my first blog entry for the mathematically interested public.New preprint on quadratic points on X_0(N)https://www.timo-keller.de/blog/may-14-2023-new-preprint-on-quadratic-points-on-x_0n.html2023-05-14T00:00:00Z2023-05-14T00:00:00ZWhy this matters. A point of X0(N) over a number field K is the same as an elliptic curve over K with a cyclic N-isogeny, so finding such points classifies these curves. After the rational case (K=𝐐), the first genuinely new problem is that of quadratic points, those defined over a degree-2 field. We determine all of them for a large list of modular curves where they were previously unknown, and …Two articles acceptedhttps://www.timo-keller.de/blog/february-13-2023-two-articles-accepted.html2023-02-13T00:00:00Z2023-02-13T00:00:00ZRecently, two articles have been accepted:My quadratic Chabauty conferencehttps://www.timo-keller.de/blog/august-26-2022-my-quadratic-chabauty-conference.html2022-08-26T00:00:00Z2022-08-26T00:00:00ZI have been back to Bayreuth to my quadratic Chabauty conference (it had to take place there because of my funding).Last day of ANTShttps://www.timo-keller.de/blog/august-12-2022-last-day-of-ants.html2022-08-12T00:00:00Z2022-08-12T00:00:00ZANTS-XV. Here are the slides of my 25 minutes talk I gave at ANTS-XV (Fifteenth Algorithmic Number Theory Symposium), taking place at the University of Bristol this year. This is the largest international conference on this topic.First blog entry, PCMIhttps://www.timo-keller.de/blog/july-30-2022-first-blog-entry-pcmi.html2022-07-30T00:00:00Z2022-07-30T00:00:00ZThis is the first entry of my blog started as announced in my Marie Skłodowska-Curie fellowship starting in June 2023 with Steffen Müller in Groningen.