Henri Darmon - Academia.edu (original) (raw)

Papers by Henri Darmon

arXiv (Cornell University), Jul 4, 2022

Michigan Mathematical Journal

arXiv (Cornell University), Jul 4, 2022

The figure on the front cover, “Rational Curves on a K3 Surface,”

— Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadrati... more — Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field K. They are constructed analytically as local points on elliptic curves with multiplicative reduction at a prime p that remains inert in K, but are conjectured to be rational over ring class fields of K and to satisfy a Shimura reciprocity law describing the action of GK on them. The main conjectures of [Dar] predict that any linear combination of Stark-Heegner points weighted by the values of a ring class character ψ of K should belong to the corresponding piece of the Mordell-Weil group over the associated ring class field, and should be non-trivial when L′(E/K,ψ, 1) 6= 0. Building on the results described in Chapters 2, 3 and 4, this chapter shows that such linear combinations arise from global classes in the idoneous pro-p Selmer group, and are non-trivial when the first derivative of a weightvariable p-adic L-function Lp(f/K,ψ) does not vanish at the point associated to (E/K,ψ). The proof rests on the construction of a three-variable family κ(f ,g,h) of cohomology classes associated to a triple of Hida families and a direct comparison between Stark-Heegner points and the generalised Kato classes arising by specialising κ(f ,g,h) at weights (2, 1, 1) for a suitable choice of Hida families. The explicit formula that emerges from this comparison is of independent interest and supplies theoretical evidence for the elliptic Stark Conjectures of [DLR].

Harmonic weak Maass forms are instances of real analytic modular forms which have recently found ... more Harmonic weak Maass forms are instances of real analytic modular forms which have recently found applications in several areas of mathematics. They provide a framework for Ramanujan’s theory of mock modular forms ([Ono08]), arise naturally in investigating the surjectivity of Borcherds’ singular theta lift ([BF04]), and their Fourier coefficients seem to encode interesting arithmetic information ([BO]). Until now, harmonic weak Maass forms have been studied solely as complex analytic objects. The aim of this thesis is to recast their definition in more conceptual, algebro-geometric terms, and to lay the foundations of a padic theory of harmonic weak Maass forms analogous to the theory of p-adic modular forms formulated by Katz in the classical context. This thesis only discusses harmonic weak Maass forms of weight 0. The treatment of more general integral weights requires no essentially new idea but involves further notational complexities which may obscure the main features of our ...

Advances in Mathematics, 2022

This article is the first in a series devoted to the Euler system arising from p-adic families of... more This article is the first in a series devoted to the Euler system arising from p-adic families of Beilinson-Flach elements in the first K-group of the product of two modular curves. It relates the image of these elements under the p-adic syntomic regulator (as described by Besser (2012)) to the special values at the near-central point of Hida’s p-adic

Rational Points on Modular Elliptic Curves, 2003

Annals of Mathematics, 2001

Notices of the American Mathematical Society, 2017

The rst of these properties illustrates the central role of the j-function in the theory of compl... more The rst of these properties illustrates the central role of the j-function in the theory of complex multiplication, and the Kronecker Jugendtraum in the context of Hilbert’s 12th problem. The second and third properties correspond to the algebraic and analytic parts of the paper of Gross–Zagier [GZ85], which was the gateway to their landmark results in Gross–Zagier [GZ86] and Gross–Kohnen–Zagier [GKZ87]. An explicit formula for the factorisation of the norm was given after reinterpreting the exponent of a prime q as a certain arithmetic intersection number of two points on the (0-dimensional) Shimura variety associated to a de nite quaternion algebra Bq∞ rami ed at q and∞. The third property is exploited to give an analytic proof of the same explicit formula, and is of particular importance for our seminar. In this seminar, we will study real quadratic analogues of singular moduli. A basic obstacle to using the j-invariant is the fact that its domain is the Poincaré upper half plane...

Publications Mathématiques de Besançon, 2012

arXiv (Cornell University), Jul 4, 2022

Michigan Mathematical Journal

arXiv (Cornell University), Jul 4, 2022

The figure on the front cover, “Rational Curves on a K3 Surface,”

— Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadrati... more — Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field K. They are constructed analytically as local points on elliptic curves with multiplicative reduction at a prime p that remains inert in K, but are conjectured to be rational over ring class fields of K and to satisfy a Shimura reciprocity law describing the action of GK on them. The main conjectures of [Dar] predict that any linear combination of Stark-Heegner points weighted by the values of a ring class character ψ of K should belong to the corresponding piece of the Mordell-Weil group over the associated ring class field, and should be non-trivial when L′(E/K,ψ, 1) 6= 0. Building on the results described in Chapters 2, 3 and 4, this chapter shows that such linear combinations arise from global classes in the idoneous pro-p Selmer group, and are non-trivial when the first derivative of a weightvariable p-adic L-function Lp(f/K,ψ) does not vanish at the point associated to (E/K,ψ). The proof rests on the construction of a three-variable family κ(f ,g,h) of cohomology classes associated to a triple of Hida families and a direct comparison between Stark-Heegner points and the generalised Kato classes arising by specialising κ(f ,g,h) at weights (2, 1, 1) for a suitable choice of Hida families. The explicit formula that emerges from this comparison is of independent interest and supplies theoretical evidence for the elliptic Stark Conjectures of [DLR].

Harmonic weak Maass forms are instances of real analytic modular forms which have recently found ... more Harmonic weak Maass forms are instances of real analytic modular forms which have recently found applications in several areas of mathematics. They provide a framework for Ramanujan’s theory of mock modular forms ([Ono08]), arise naturally in investigating the surjectivity of Borcherds’ singular theta lift ([BF04]), and their Fourier coefficients seem to encode interesting arithmetic information ([BO]). Until now, harmonic weak Maass forms have been studied solely as complex analytic objects. The aim of this thesis is to recast their definition in more conceptual, algebro-geometric terms, and to lay the foundations of a padic theory of harmonic weak Maass forms analogous to the theory of p-adic modular forms formulated by Katz in the classical context. This thesis only discusses harmonic weak Maass forms of weight 0. The treatment of more general integral weights requires no essentially new idea but involves further notational complexities which may obscure the main features of our ...

Advances in Mathematics, 2022

This article is the first in a series devoted to the Euler system arising from p-adic families of... more This article is the first in a series devoted to the Euler system arising from p-adic families of Beilinson-Flach elements in the first K-group of the product of two modular curves. It relates the image of these elements under the p-adic syntomic regulator (as described by Besser (2012)) to the special values at the near-central point of Hida’s p-adic

Rational Points on Modular Elliptic Curves, 2003

Annals of Mathematics, 2001

Notices of the American Mathematical Society, 2017

The rst of these properties illustrates the central role of the j-function in the theory of compl... more The rst of these properties illustrates the central role of the j-function in the theory of complex multiplication, and the Kronecker Jugendtraum in the context of Hilbert’s 12th problem. The second and third properties correspond to the algebraic and analytic parts of the paper of Gross–Zagier [GZ85], which was the gateway to their landmark results in Gross–Zagier [GZ86] and Gross–Kohnen–Zagier [GKZ87]. An explicit formula for the factorisation of the norm was given after reinterpreting the exponent of a prime q as a certain arithmetic intersection number of two points on the (0-dimensional) Shimura variety associated to a de nite quaternion algebra Bq∞ rami ed at q and∞. The third property is exploited to give an analytic proof of the same explicit formula, and is of particular importance for our seminar. In this seminar, we will study real quadratic analogues of singular moduli. A basic obstacle to using the j-invariant is the fact that its domain is the Poincaré upper half plane...

Publications Mathématiques de Besançon, 2012