The subconvexity of L-functions aims to refine estimates of central values, going beyond mere convexity. This is important in analytic number theory, especially in the study of the distribution of prime numbers. Researchers seek to establish more precise bounds for these L-functions to better understand prime numbers, particularly by exploring connections with automorphic forms. This approach offers an enriching perspective for understanding the deep structure of L-functions and also provides insights into advanced conjectures such as the Riemann hypothesis.[-]

The subconvexity of L-functions aims to refine estimates of central values, going beyond mere convexity. This is important in analytic number theory, especially in the study of the distribution of prime numbers. Researchers seek to establish more precise bounds for these L-functions to better understand prime numbers, particularly by exploring connections with automorphic forms. This approach offers an enriching perspective for understanding the deep structure of L-functions and also provides insights into advanced conjectures such as the Riemann hypothesis.[-]

The subconvexity of L-functions aims to refine estimates of central values, going beyond mere convexity. This is important in analytic number theory, especially in the study of the distribution of prime numbers. Researchers seek to establish more precise bounds for these L-functions to better understand prime numbers, particularly by exploring connections with automorphic forms. This approach offers an enriching perspective for understanding the deep structure of L-functions and also provides insights into advanced conjectures such as the Riemann hypothesis.[-]

The subconvexity of L-functions aims to refine estimates of central values, going beyond mere convexity. This is important in analytic number theory, especially in the study of the distribution of prime numbers. Researchers seek to establish more precise bounds for these L-functions to better understand prime numbers, particularly by exploring connections with automorphic forms. This approach offers an enriching perspective for understanding the deep structure of L-functions and also provides insights into advanced conjectures such as the Riemann hypothesis.[-]

In my talk, I will discuss an application of the theory of motives to transcendence theory, concentrating on the formal aspects. The Period Conjecture predicts that all relations between period numbers are induced by properties of the category of motives. It is a theorem for motives of points and curves, but wide open in general. The Period Conjecture also implies fullness of the Hodge-de Rham realization on Nori motives. If time permits, I will discuss how this generalizes (conjecturally) to triangulated motives and thus to motivic cohomology.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

The preceding Etats de la recherche on this topic happened in Rennes, 2006, and the organizers of the present edition asked me to make the bridge between these two sessions. My 2006 talks were devoted to the theory of heights and equidistribution theorems for algebraic dynamical systems. I will start from there by presenting the framework allowed by Arakelov geometry, and explaining the recent manuscript of X. Yuan and S.-W. Zhang who provide a birational perspective to these concepts. The theory is a bit complex and technical but I will try to emphasize the parallel between those ideas and the ones that lie at the ground of pluripotential theory in complex analysis, or in the theory of b-divisors in algebraic geometry.[-]

The degree of a dominant rational map $f: \mathbb{P}^n \rightarrow \mathbb{P}^n$ is the common degree of its homogeneous components. By considering iterates of $f$, one can form a sequence $\operatorname{deg}\left(f^n\right)$, which is submultiplicative and hence has the property that there is some $\lambda \geq 1$ such that $\left(\operatorname{deg}\left(f^n\right)\right)^{1 / n} \rightarrow \lambda$. The quantity $\lambda$ is called the first dynamical degree of $f$. We'll give an overview of the significance of the dynamical degree in complex dynamics and describe an example of a birational self-map of $\mathbb{P}^3$ in which this dynamical degree is provably transcendental. This is joint work with Jeffrey Diller, Mattias Jonsson, and Holly Krieger.[-]

The exponential period conjecture predicts how the Galois group of an exponential motive governs all polynomial relations among its periods. For classical motives (which are special exponential motives) this conjecture specialises to the classical period conjecture. My aim is to present some elementary, yet elucidative examples of exponential motives and periods which illustrate how the exponential period conjecture implies certain popular transcendence conjectures, and how its non classical part is related to the Siegel-Shidlovskii theorem.[-]