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Shimura curves and bounds for the $abc$ conjecture

By Hector Pasten

Also appears in collection : Diophantine geometry / Géométrie diophantienne

I will explain some new connections between the $abc$ conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the $abc$ conjecture, which lies beyond the existing techniques in this context. The proof involves a number of tools such as Shimura curves, CM points, analytic number theory, and Arakelov geometry. It also requires some intermediate results of independent interest, such as bounds for the Manin constant beyond the semi-stable case. If time permits, I will also explain some results towards Szpiro's conjecture over totally real number fields which are compatible with the discriminant term appearing in Vojta's conjecture for algebraic points of bounded degree.

Information about the video

Date of recording 23/05/2018
Date of publication 29/05/2018
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19408103
Cite this video Pasten, Hector (23/05/2018). Shimura curves and bounds for the $abc$ conjecture. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19408103
URL https://dx.doi.org/10.24350/CIRM.V.19408103

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35:33

published on August 4, 2016

Optimized Schwarz waveform relaxation methods: theory and applications

By Pascal Omnes

46:12

published on August 3, 2016

Parametrized model order reduction for component-to-system synthesis

By Anthony Patera

01:05:19

published on August 8, 2016

Data-driven wildfire behavior modelling: focus on front level-set data assimilation

By Mélanie Rochoux

44:00

published on August 8, 2016

A discussion about the impact of scalability constraint on the design of futur dynamical kernels of NWP models

By Fabrice Voitus

39:44

published on August 8, 2016

Modeling and data assimilation in cardiac electrophysiology

By Annabelle Collin

57:16

published on August 10, 2016

High performance climate modelling : mimetic finite differences, and beyond?

By Thomas Dubos

57:49

published on August 10, 2016

Direct and inverse biomechanical modeling of the heart

By Dominique Chapelle

54:02

published on September 20, 2016

Crystalline cohomology, period maps, and applications to K3 surfaces

By Christian Liedtke

56:03

published on September 20, 2016

The non-archimedean SYZ fibration and Igusa zeta functions - Part 2

By Johannes Nicaise

53:48

published on September 20, 2016

The non-archimedean SYZ fibration and Igusa zeta functions - Part 3

By Johannes Nicaise

47:11

published on September 22, 2016

The unitary extension principle on LCA groups

By Ole Christensen

50:19

published on September 27, 2016

Some recent insights into computing with positive definite kernels

By Greg Fasshauer

51:15

published on September 22, 2016

Algebraic multigrid and subdivision

By Maria Charina

26:29

published on September 27, 2016

Spherical splines

By Hartmut Prautzsch

54:30

published on October 11, 2016

Forcing theory for transverse trajectories of surface homeomorphisms - Part 1

By Patrice Le Calvez

53:25

published on October 11, 2016

Forcing theory for transverse trajectories of surface homeomorphisms - Part 2

By Patrice Le Calvez

54:00

published on October 11, 2016

Forcing theory for transverse trajectories of surface homeomorphisms - Part 3

By Fabio Tal

48:42

published on October 11, 2016

Forcing theory for transverse trajectories of surface homeomorphisms - Part 4

By Fabio Tal

47:37

published on November 10, 2016

Transferring information across image and shape collections

By Niloy Mitra

43:32

published on November 16, 2016

Quasisemisimple classes

By Jean Michel

01:13:58

published on December 14, 2016

Operators in ergodic theory - Lecture 1: Operators dynamics versus base space dynamics

By Markus Haase

01:01:43

published on December 14, 2016

Operators in ergodic theory - Lecture 2: Dilations and joinings

By Markus Haase

49:47

published on December 14, 2016

Operators in ergodic theory - Lecture 3: Compact semigroups and splitting theorems

By Markus Haase

59:07

published on December 14, 2016

Multiple ergodic theorems: old and new - Lecture 2

By Bryna Kra

01:06:39

published on December 14, 2016

Multiple ergodic theorems: old and new - Lecture 3

By Bryna Kra

01:12:37

published on January 6, 2017

Möbius randomness and dynamics six years later

By Peter Sarnak

01:00:08

published on January 6, 2017

Ergodicity of the Liouville system implies the Chowla conjecture

By Nikos Frantzikinakis

48:04

published on January 6, 2017

The diameter of the symmetric group: ideas and tools

By Harald Helfgott

01:01:12

published on January 6, 2017

Primes with missing digits

By James Maynard

57:14

published on January 6, 2017

The unsolved problems of Halmos

By Benjamin Weiss

50:33

published on March 2, 2017

Generic properties of the geodesic flow in nonpositive curvature

By Yves Coudène

55:54

published on March 3, 2017

Central limit theorems for circle packings

By Mark Pollicott

58:14

published on March 3, 2017

Mixing and rates of mixing for infinite measure flows

By Ian Melbourne

01:01:15

published on March 2, 2017

Fast-Slow partially hyperbolic systems: an example

By Carlangelo Liverani

40:09

published on March 10, 2017

Conditioned determinantal processes are determinantal

By Alexander Shamov

43:02

published on March 8, 2017

A functional limit theorem for the sine-process

By Andrey Dymov

57:46

published on March 23, 2017

Correlation spectrum of Morse-Smale flows

By Gabriel Rivière

59:43

published on April 24, 2017

Skeletons and moduli of Stokes torsors

By Jean-Baptiste Teyssier

01:02:52

published on April 24, 2017

Semi-infinite Hodge structures in noncommutative geometry

By Dmytro Shklyarov

01:25:17

published on April 24, 2017

On the Riemann-Hilbert correspondence for irregular holonomic D-modules

By Andrea D'Agnolo

50:11

published on May 4, 2017

Tilings and non-intersecting paths beyond integrable cases

By Vadim Gorin

01:48:41

published on May 11, 2017

Informatique et physique : quelques interactions

By Gilles Dowek

01:13:28

published on May 11, 2017

La difficile explication du résultat des calculs : des preuves automatiques à l'apprentissage automatique

By Gilles Dowek

02:30:39

published on May 11, 2017

Arithmétique à virgule flottante : plaie ou terrain de jeu ?

By Sylvie Boldo

42:01

published on May 25, 2017

Curvature measures of random sets

By Martina Zähle

45:31

published on May 25, 2017

Random walks on dynamical percolation

By Perla Sousi

42:13

published on June 1, 2017

Angles of Gaussian primes

By Zeév Rudnick

41:55

published on June 1, 2017

Small sumsets in continuous and discrete settings

By Anne de Roton

32:38

published on June 1, 2017

Bounded remainder sets for the discrete and continuous irrational rotation

By Sigrid Grepstad

44:58

published on June 1, 2017

Large character sums

By Youness Lamzouri

43:13

published on August 3, 2017

Splitting algorithm for nested events

By Ludovic Goudenège

53:14

published on August 3, 2017

Projected particle methods for solving McKean Vlasov SDEs

By Denis Belomestny

55:27

published on August 3, 2017

Some asymptotic results about American options and volativity

By Stefano De Marco

47:28

published on August 10, 2017

Mean field type control with congestion

By Mathieu Lauriere

49:24

published on August 28, 2017

Forward and backward simulation of Euler scheme

By Emmanuel Gobet

01:09:54

published on September 8, 2017

Minicourse shape spaces and geometric statistics

By Alain Trouvé , Xavier Pennec

01:31:17

published on September 8, 2017

Geometry of quantum entanglement

By Karol Zyczkowski

54:56

published on September 14, 2017

Coloring graphs with forbidden induced paths

By Maria Chudnovsky

59:25

published on September 14, 2017

Combinatorial and algorithmic properties of Robinsonian matrices

By Monique Laurent

53:47

published on September 14, 2017

Random hyperbolic graphs

By Marcos Kiwi

01:06:10

published on September 14, 2017

Restricted types of matchings

By Dieter Rautenbach

01:03:20

published on September 22, 2017

Real curves and a Klein TQFT

By Penka Georgieva

01:06:05

published on September 21, 2017

Sums of three squares and Noether-Lefschetz loci

By Olivier Benoist

01:04:57

published on September 22, 2017

Complex torus, its good compactifications and the ring of conditions

By Askold Khovanskii

$Homotopy theory of strict $\omega$-categories and its connections with homology of monoids - Lecture 2$
01:31:10

published on September 28, 2017

Homotopy theory of strict $\omega$-categories and its connections with homology of monoids - Lecture 2

By François Métayer

01:01:16

published on September 28, 2017

Shuffles of trees

By Eric Hoffbeck

56:01

published on September 28, 2017

Type theories and polynomial monads

By Steve Awodey

$Homotopy theory of strict $\omega$-categories and its connections with homology of monoids - Lecture 3$
01:22:19

published on September 28, 2017

Homotopy theory of strict $\omega$-categories and its connections with homology of monoids - Lecture 3

By François Métayer

50:58

published on October 6, 2017

Ill-posedness for Leray solutions of the ipodissipative Navier-Stokes equations

By Camillo de Lellis

56:20

published on October 6, 2017

30 years of $T(b)$ theorems

By Pascal Auscher

55:29

published on October 6, 2017

Localization of eigenfunctions via an effective potential

By David Jerison

59:07

published on October 6, 2017

Parametrizing with Guy

By Tatiana Toro

57:14

published on October 12, 2017

From forcing models to realizability models

By Laura Fontanella

50:22

published on October 12, 2017

The hierarchy of second-order set theories between GBC and KM and beyond

By Joel David Hamkins

$Monochromatic sumsets for colourings of $\mathbb{R}$$
31:42

published on October 12, 2017

Monochromatic sumsets for colourings of $\mathbb{R}$

By Daniel T. Soukup

51:32

published on October 12, 2017

Prikry type forcing and combinatorial properties

By Dima Sinapova

54:45

published on October 26, 2017

Interlacements and the uniform spanning forest

By Tom Hutchcroft

51:45

published on October 26, 2017

Weighted distances in scale free random graphs

By Julia Komjathy

48:31

published on October 26, 2017

Extinction time for the weaker of two competing stochastic SIS logistic epidemics

By Malwina Luczak

19:41

published on November 2, 2017

The quantum Vlasov equation

By Norbert Mauser

37:25

published on November 16, 2017

Viability and arbitrage under Knightian uncertainty

By Matteo Burzoni

36:53

published on November 16, 2017

Market delay and $G$-expectations

By Yan Dolinsky

27:35

published on November 21, 2017

Rough volatility from an affine point of view

By Christa Cuchiero

27:21

published on November 21, 2017

Affine Volterra processes and models for rough volatility

By Martin Larsson

35:36

published on November 16, 2017

Reconstruction by optimal transport: applications in cosmology and finance

By Grégoire Loeper

58:21

published on December 7, 2017

Algebraic sums and products of univoque bases

By Karma Dajani

01:00:42

published on December 7, 2017

A perspective on the The Fibonacci trace map

By David Damanik

55:32

published on December 7, 2017

Yet another characterization of the Pisot conjecture

By Paul Mercat

01:02:42

published on December 11, 2017

Dimension groups and recurrence for tree subshifts

By Valérie Berthé

32:38

published on January 10, 2018

Eigenvalues and variance components

By Iain Johnstone

29:22

published on December 21, 2017

Estimation of functions depending on a parameter observed in Gaussian noise

By Ildar Ibragimov

01:00:53

published on January 10, 2018

Some memories and facts about the work of Oleg Lepski: beyond a 'discourse on method'

By Marc Hoffmann

54:19

published on January 31, 2018

Complex Monge-Ampere equations with prescribed singularities

By Eleonora Di Nezza

50:31

published on January 31, 2018

Kähler-Ricci solitons on crepant resolutions of finite quotients of $C^n$

By Heather Macbeth

49:09

published on January 31, 2018

Extremal Poincaré type metrics and stability of pairs on Hirzebruch surfaces

By Lars Martin Sektnan

01:02:22

published on February 7, 2018

Finite type invariants of knots in homology 3-spheres

By Delphine Moussard

01:06:16

published on February 7, 2018

Equidistribution of square-tiled surfaces, meanders, and Masur-Veech volumes

By Anton Zorich

01:03:10

published on February 18, 2018

Trisections diagrams and surgery operations on embedded surfaces

By David Gay

01:02:31

published on February 18, 2018

Covering spaces and spanning trees

By David Cimasoni

$Quantum $\mathfrak{sl}_n$ knot cohomology and the slice genus$
50:00

published on February 19, 2018

Quantum $\mathfrak{sl}_n$ knot cohomology and the slice genus

By Andrew Lobb

58:15

published on March 7, 2018

Groups with Bowditch boundary a 2-sphere

By Bena Tshishiku

01:05:09

published on March 7, 2018

The Farrell-Jones conjecture for free-by-cyclic groups

By Mladen Bestvina

57:08

published on March 7, 2018

Homomorphisms to 3-manifold groups and other families

By Daniel Groves

47:35

published on March 14, 2018

Ideals in $L(L_p)$

By William B. Johnson

47:10

published on March 14, 2018

On a difference between two methods of low-distortion embeddings of finite metric spaces into non-superreflexive Banach spaces

By Beata Randrianantoanina

44:49

published on March 14, 2018

$L^2$ spectral gap and group actions on Banach spaces

By Mikael de la Salle

47:52

published on March 15, 2018

Finite quotients in coarse geometry

By Anastasia Khukhro

54:04

published on March 28, 2018

The coherent Satake category

By Harold Williams

51:09

published on March 29, 2018

From $Q$-systems to quantum affine algebras and beyond

By Rinat Kedem

51:51

published on March 28, 2018

Newton-Okounkov bodies for Grassmannians

By Lauren Williams

56:25

published on March 28, 2018

Preprojective algebras and Cluster categories

By Osamu Iyama

45:15

published on April 5, 2018

Symmetry and computation: a few of my favourite things

By Hans Z. Munthe-Kaas

51:33

published on April 5, 2018

Hybrid finite element methods preserving local symmetries and conservation laws

By Ari Stern

57:54

published on April 12, 2018

Deformation theory of twistor spaces of K3 surfaces

By Ana-Maria Brecan

53:55

published on April 12, 2018

Improved bounds for the Kobayashi conjecture on generic hyperbolicity

By Jean-Pierre Demailly

50:18

published on April 12, 2018

Nonabelian Hodge correspondences for klt varieties and quasi-etale uniformisation

By Stefan Kebekus

50:08

published on April 12, 2018

Signature morphisms from the Cremona group

By Susanna Zimmermann

$The weak-$A_\infty$ condition for harmonic measure$
58:21

published on April 25, 2018

The weak-$A_\infty$ condition for harmonic measure

By Xavier Tolsa

01:17:43

published on May 10, 2018

Méthodes probabilistes

By Marthe Bonamy

01:16:20

published on May 15, 2018

SMT solving

By David Monniaux

$Global solutions to elliptic and parabolic $\Phi^4$ models in Euclidean space$
44:58

published on May 22, 2018

Global solutions to elliptic and parabolic $\Phi^4$ models in Euclidean space

By Martina Hofmanova

49:39

published on May 22, 2018

Bessel-like SPDEs

By Lorenzo Zambotti

55:36

published on May 29, 2018

Recent progress in the classification of torsion subgroups of elliptic curves

By Alvaro Lozano-Robledo

55:55

published on May 30, 2018

A new Northcott property for Faltings height

By Lucia Mocz

01:00:35

published on May 29, 2018

Shimura curves and bounds for the $abc$ conjecture

By Hector Pasten

52:52

published on May 31, 2018

A supercongruence and hypergeometric motives

By Frits Beukers

54:38

published on May 31, 2018

Around Jouanolou-type theorems

By Rahim Moosa

53:45

published on May 31, 2018

Gamma functions, monodromy and Apéry constants

By Masha Vlasenko

58:41

published on June 7, 2018

Geometric recursion

By Jorgen Ellegaard Andersen

56:15

published on June 7, 2018

Exact density matrix for quantum group invariant sector of XXZ

By Fedor Smirnov

54:38

published on June 7, 2018

An introduction to the BV-BFV formalism

By Alberto S. Cattaneo

55:53

published on June 7, 2018

Capelli eigenvalue problem for Lie superalgebras and supersymmetric polynomials

By Vera Serganova

01:00:14

published on June 21, 2018

On the motive of the stack of vector bundles on a curve

By Victoria Hoskins

58:08

published on June 21, 2018

On Hitchin's hyperkähler metric on moduli spaces of Higgs bundles

By Andrew Neitzke

57:39

published on June 21, 2018

The ends of the Hitchin moduli space

By Laura Fredrickson

01:01:23

published on June 21, 2018

The large scale geometry of the Higgs bundle moduli space

By Jan Swoboda

01:23:49

published on June 28, 2018

Establishment in a new habitat under the infinitesimal model

By Alison M. Etheridge , Nicholas H. Barton

01:10:43

published on July 10, 2018

A Jacobian criterion for smoothness of algebraic diamonds

By Laurent Fargues

01:03:48

published on July 12, 2018

Modeling the chemotherapy-induced selection of drug-resistant traits during tumor growth

By Doron Levy

52:47

published on August 1, 2018

Macroscopic limit from a structured population model to the Kirkpatrick-Barton model

By Gaël Raoul

33:21

published on August 3, 2018

Mathematical model of cortical neurogenesis

By Marie Postel

28:04

published on August 3, 2018

Modelling of complex dynamical networks: from agent-based to continuum models

By Diane Peurichard

46:01

published on December 6, 2018

Bayesian capture-recapture in social justice research

By David Corliss

33:03

published on December 6, 2018

Big data for health: a Bayesian spatio-temporal analysis for predicting cardiac risk in Ticino and optimal defibrillators positioning

By Antonietta Mira

49:18

published on April 18, 2019

The partial compactification of the universal centralizer

By Ana Balibanu

01:00:32

published on April 25, 2019

BFN Springer theory

By Joel Kamnitzer

58:38

published on April 18, 2019

$T$-structures from categorical actions

By Clemens Koppensteiner

58:17

published on April 18, 2019

Coulomb branches of quiver gauge theories with symmetrizers

By Hiraku Nakajima

59:44

published on April 18, 2019

Gelfand-Tsetlin theory and Coulomb branches

By Ben Webster

59:16

published on April 18, 2019

Character stacks and $(q-)$geometric representation theory

By Sam Gunningham

52:34

published on April 18, 2019

Schubert calculus and self-dual puzzles

By Iva Halacheva

01:03:31

published on April 18, 2019

Skeins, clusters, and character sheaves

By David Jordan

01:02:09

published on April 18, 2019

Quantum character varieties at roots of unity

By Pavel Safronov

59:48

published on April 19, 2019

Fusion rings from quantum groups and DAHA actions

By Catharina Stroppel

01:00:06

published on April 25, 2019

Partial resolutions/deformations of diagonal invariants and representations of finite reductive groups

By Cédric Bonnafé

58:05

published on April 18, 2019

Symplectic singularities and nilpotent orbits

By Yoshinori Namikawa

01:05:54

published on April 24, 2019

Higher representations of $gl(1|1)^+$

By Raphaël Rouquier

59:52

published on April 21, 2019

Langlands duality and quantum field theory

By Philsang Yoo

01:04:54

published on April 24, 2019

Irreducible equivariant perverse coherent sheaves on affine Grassmannians of type $A$ and dual canonical bases

By Michael Finkelberg

01:03:04

published on April 23, 2019

Mirabolic Satake equivalence

By Victor Ginzburg

01:02:49

published on April 24, 2019

A geometric $R$-matrix for the Hilbert scheme of points on a general surface

By Noah Arbesfeld

56:24

published on April 24, 2019

A sheaf theoretic approach towards calculating simple characters of algebraic groups

By Peter Fiebig

01:05:20

published on April 24, 2019

Affine Springer fibers and G1T modules

By Eric Vasserot

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