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# Documents  Howe, Everett | enregistrements trouvés : 5

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## Computing classical modular forms as orthogonal modular forms Voight, John | CIRM H

Post-edited

Algebraic and Complex Geometry;Number Theory

Birch gave an extremely efficient algorithm to compute a certain subspace of classical modular forms using the Hecke action on classes of ternary quadratic forms. We extend this method to compute all forms of non-square level using the spinor norm, and we exhibit an implementation that is very fast in practice. This is joint work with Jeffery Hein and Gonzalo Tornaria.

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## Brauer-Siegel theorem and analogues for varieties over global fields Hindry, Marc | CIRM H

Multi angle

Algebraic and Complex Geometry;Number Theory

The classical Brauer-Siegel theorem can be seen as one of the first instances of description of asymptotical arithmetic: it states that, for a family of number fields $K_i$, under mild conditions (e.g. bounded degree), the product of the regulator by the class number behaves asymptotically like the square root of the discriminant.
This can be reformulated as saying that the Brauer-Siegel ratio log($hR$)/ log$\sqrt{D}$ has limit 1.
Even if some of the fundamental problems like the existence or non-existence of Siegel zeroes remains
unsolved, several generalisations and analog have been developed: Tsfasman-Vladuts, Kunyavskii-Tsfasman, Lebacque-Zykin, Hindry-Pacheco and lately Griffon. These analogues deal with number fields for which the limit is different from 1 or with elliptic curves and abelian varieties either for a fixed variety and varying field or over a fixed field with a family of varieties.
The classical Brauer-Siegel theorem can be seen as one of the first instances of description of asymptotical arithmetic: it states that, for a family of number fields $K_i$, under mild conditions (e.g. bounded degree), the product of the regulator by the class number behaves asymptotically like the square root of the discriminant.
This can be reformulated as saying that the Brauer-Siegel ratio log($hR$)/ log$\sqrt{D}$ has limit 1.
Even if some ...

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## Maps between curves and diophantine obstructions Voloch, José Felipe | CIRM H

Multi angle

Algebraic and Complex Geometry;Number Theory

Given two algebraic curves $X$, $Y$ over a finite field we might want to know if there is a rational map from $Y$ to $X$. This has been looked at from a number of perspectives and we will look at it from the point of view of diophantine geometry by viewing the set of maps as $X(K)$ where $K$ is the function field of $Y$. We will review some of the known obstructions to the existence of rational points on curves over global fields, apply them to this situation and present some results and conjectures that arise. Given two algebraic curves $X$, $Y$ over a finite field we might want to know if there is a rational map from $Y$ to $X$. This has been looked at from a number of perspectives and we will look at it from the point of view of diophantine geometry by viewing the set of maps as $X(K)$ where $K$ is the function field of $Y$. We will review some of the known obstructions to the existence of rational points on curves over global fields, apply them to ...

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## Local densities compute isogeny classes Achter, Jeffrey | CIRM H

Multi angle

Algebraic and Complex Geometry;Number Theory

Consider an ordinary isogeny class of elliptic curves over a finite, prime field. Inspired by a random matrix heuristic (which is so strong it's false), Gekeler defines a local factor for each rational prime. Using the analytic class number formula, he shows that the associated infinite product computes the size of the isogeny class.
I'll explain a transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality. Moreover, the new perspective allows a natural generalization to higher-dimensional abelian varieties. This is joint work with Julia Gordon and S. Ali Altug.
Consider an ordinary isogeny class of elliptic curves over a finite, prime field. Inspired by a random matrix heuristic (which is so strong it's false), Gekeler defines a local factor for each rational prime. Using the analytic class number formula, he shows that the associated infinite product computes the size of the isogeny class.
I'll explain a transparent proof of this formula; it turns out that this product actually computes an adelic ...

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## Rational points on curves over finite fields Serre, Jean-Pierre ; Howe, Everett ; Oesterlé, Joseph ; Ritzenthaler, Christophe ; Bassa, Alp ; Lorenzo Garcia, Elisa ; Schoof, René | Société Mathématique de France 2020

Ouvrage

- x; 187 p.
ISBN 978-2-85629-923-4

Documents mathématiques , 0018

Localisation : Collection 1er étage

champ fini # courbe # point rationnel # théorie des corps de classes # résultat asymptotique # genre 2 # genre 3 # limite # formule explicite

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