This thematic session offers a broad perspective on number-theoretical topics in geometry, spanning from algebraic to analytic methods, and including excursions into Computational Number Theory. The proposed topics ensure numerous and engaging interactions with other areas. For instance, algebraic geometers may be interested in talks given by experts working on the arithmetic of abelian varieties, while the use of potential theory in Arakelov geometry could appeal to the more analytically minded participants attending the event.
16:00h Arithmetic of singular CM points on the eigencurve, by Victor Rotger (Polytechnic University of Catalonia)
16:40h On Artin formalism for triple product p-adic L-functions, by Carlos De Vera Piquero (University of Zaragoza)
17:20h Evaluation of Bianchi rigid meromorphic cocycles at big ATR points, by Xevi Guitart (University of Barcelona)
12:20h On the Markov Uniqueness Conjecture, by Paloma Bengoechea (University of Barcelona)
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ABSTRACTS
Arithmetic of singular CM points on the eigencurve, by Victor Rotger
In this talk I will report on joint on-going work with Rohit Kumar (Duke) on the elliptic Stark conjecture at singular CM points on the eigencurve.
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On Artin formalism for triple product p-adic L-functions, by Carlos De Vera Piquero
We will state a conjectural factorization for certain triple product p-adic L-functions, and we will explain partial progress towards its proof. The strategy depends crucially on a comparison of Euler systems, and is based on a series of joint works with K. Büyükboduk, D. Casazza, and A. Pal.
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Evaluation of Bianchi rigid meromorphic cocycles at big ATR points, by Xevi Guitart
I will discuss joint work with Lennart Gehrmann and Marc Masdeu, in which we develop the tools needed to effectively evaluate the Bianchi rigid meromorphic cocycles introduced by Darmon–Gehrmann–Lipnowski at big ATR points.
We use these methods to obtain the first numerical evidence for the conjectured algebraicity of the resulting special values. Moreover, our computations suggest that these values exhibit behaviour analogous to that of special values of Borcherds products on Hilbert modular surfaces at big CM points.
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The integral Chebtshev problem and a conjecture of Montgomery, by Binggang Qu
Let K be a compact set in the complex plane. We first recall the Chebyshev constant of K which is very well understood through potential theory. We then introduce an arithmetic variant, the «integer Chebyshev constant» of K (Hilbert worked on this!). The goal of this talk is to embed this problem into Arakelov theory.
1. We prove the integer Chebyshev constant is nothing but the essential minimum of a height function on $\overline{\mathbb{Q}}$,
2. We prove the Hilbert–Fekete bound is nothing but Zhang’s theorem of successive minima,
3. We introduce a conjecture of Montgomery, we translate it to Arakelov theory, and we solve it using Arakelov theory techniques.
This is based my joint work arXiv:2609.39505 with Chengyuan Yang (Peking University).
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Linear programming and the essential minimum of height functions, by Martín Sombra
The determination of the essential minimum of a height function is a challenging problem. We consider classical methods to obtain lower and upper bounds, and using an infinite dimensional linear programming scheme we show that both methods agree in the limit, by the principle of strong duality.
As an application we obtain that, under natural hypothesis, the essential minimum is a computable real number.
This is joint work with J. Burgos Gil, R. Menares and B. Qu.
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On the Markov Uniqueness Conjecture, by Paloma Bengoechea
I will revise the Markov Uniqueness Conjecture (stated by Frobenius in 1913) in different contexts, such as binary quadratic forms, arithmetic and hyperbolic geometry, and diophantine approximation. I will discuss the (little) progress there has been.
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