This document summarizes a talk on Darmon points given at the 2013 CMS Winter Meeting. It provides background on Birch and Swinnerton-Dyer conjectures and how Heegner points were originally used to study them. It then discusses Henri Darmon's insight to generalize the construction to number fields without complex multiplication. The talk aims to present a unified perspective for constructing Darmon points in both archimedean and non-archimedean settings that includes all previous constructions as special cases. It concludes by providing an example of constructing an archimedean Darmon point attached to a cubic number field and elliptic curve over that field.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Continuation calculus at Term Rewriting Seminarbgeron
Continuation calculus (CC) is an alternative to lambda calculus, where the order of evaluation is determined by programs themselves. Owing to its simplicity, continuations are no unusual terms. This makes it natural to model programs with nonlocal control flow, as with exceptions and call-by-name functions.
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Continuation calculus at Term Rewriting Seminarbgeron
Continuation calculus (CC) is an alternative to lambda calculus, where the order of evaluation is determined by programs themselves. Owing to its simplicity, continuations are no unusual terms. This makes it natural to model programs with nonlocal control flow, as with exceptions and call-by-name functions.
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
My PhD talk "Application of H-matrices for computing partial inverse"Alexander Litvinenko
Sometimes you need not the whole solution of a partial differential equation, but only a part (e.g. in boundary layer). How to compute not the whole inverse matrix, but only a part or it (which can nevertheless provide you the solution in a subdomain)?
The computation of automorphic forms for a group Gamma is
a major problem in number theory. The only known way to approach the higher rank cases is by computing the action of Hecke operators on the cohomology.
Henceforth, we consider the explicit computation of the cohomology by using cellular complexes. We then explain how the rational elements can be made to act on the complex when it originate from perfect forms. We illustrate the results obtained for the symplectic Sp4(Z) group.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
In a polytope P the faces F (from vertex to facets) define the combinatorics of P. In particular a flag F0, F1, ...., F(n-1) with Fi being a face of dimension i and Fi a subset of F(i+1).
From such a flag system and a subset S of {1,....,n} we can define a new flag system named the Wythoff construction. We consider the l1-embedding of the obtained graphs. We also expose an application of the Wythoff construction to the computation of homology of the Mathieu group M24.
Darmon points for fields of mixed signaturemmasdeu
Fifteen years ago Henri Darmon introduced a construction of p-adic points on elliptic curves. These points were conjectured to be algebraic and to behave much like Heegner points, although so far a proof remains inaccessible. Other constructions emerged in the subsequent years, thanks to work of himself and many others. All of these constructions are local (either non-archimedean like the original one, or archimedean), and so far none of these are proven to yield algebraic points, although there is extensive numerical evidence.
In this talk I will present joint work with Xavier Guitart and Haluk Sengun, in which we propose a framework that includes all the above constructions as particular cases, and which allows us to extend the construction of local points to elliptic curves defined over arbitrary number fields. As a by-product, we provide an explicit (though conjectural) construction of the (isogeny class of the) elliptic curve attached to an automorphic form for GL2.
I will explain our construction and present lots of new numerical evidence.
Non-archimedean construction of elliptic curves and rational pointsmmasdeu
In this talk I will describe a non-archimedean conjectural construction of the Tate lattice
of an elliptic curve E defined over an arbitrary-signature number field F. I will also provide analytic constructions
of algebraic points on such curves, which generalize the so-called Stark--Heegner or Darmon points. One important
feature of all these constructions is their explicitness, which allows for the numerical verification of the conjectures.
This is joint work with Xavier Guitart and Mehmet H. Sengun.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
1. A unified perspective for Darmon points
2013 CMS Winter Meeting, Ottawa
Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3
1Institut f¨ur Experimentelle Mathematik
2Columbia University
3University of Warwick
December 7, 2013
Marc Masdeu A unified perspective for Darmon points December 7, 2013 0 / 17
2. Birch and Swinnerton-Dyer
Brian Birch
Peter Swinnerton-Dyer
Marc Masdeu A unified perspective for Darmon points December 7, 2013 0 / 17
3. The BSD conjecture
Let F be a number field.
Let E/F be an elliptic curve of conductor N = NE.
Let K/F be a quadratic extension of F.
Assume (for simplicity) that N is square-free, coprime to disc(K/F).
Hasse-Weil L-function of the base change of E to K ( (s) >> 0)
L(E/K, s) =
p|N
1 − ap|p|−s −1
×
p N
1
ap(E) = 1 + |p| − #E(Fp).
− ap|p|−s
+ |p|1−2s −1
.
Modularity conjecture (which we assume) =⇒
Analytic continuation of L(E/K, s) to C.
Functional equation relating s ↔ 2 − s.
Coarse version of BSD conjecture
ords=1 L(E/K, s) = rkZ E(K).
So L(E/K, 1) = 0
BSD
=⇒ ∃PK ∈ E(K) of infinite order.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 1 / 17
4. The main tool for BSD: Heegner points
Kurt Heegner
Exist for F totally real and K/F totally complex (CM extension).
We recall the definition of Heegner points in the simplest setting:
F = Q (and K/Q quadratic imaginary), and
Heegner hypothesis: | N =⇒ split in K.
This ensures that ords=1 L(E/K, s) is odd (so ≥ 1).
Marc Masdeu A unified perspective for Darmon points December 7, 2013 2 / 17
5. Heegner Points (K/Q imaginary quadratic)
Attach to E a holomorphic 1-form on H = {z ∈ C : (z) > 0}.
ΦE = fE(z)dz =
n≥1
ane2πinz
dz ∈ H0
(Γ0(N)
Γ0(N) = { a b
c d ∈ SL2(Z): N | c}
, Ω1
H).
Given τ ∈ K ∩ H set Jτ =
τ
∞
ΦE ∈ C.
Consider the lattice ΛE = γ ΦE | γ ∈ H1 Γ0(N)H, Z .
There exists an isogeny η: C/ΛE → E(C).
Set Pτ = η(Jτ ) ∈ E(C).
Fact: Pτ ∈ E(Hτ ), where Hτ /K is a class field attached to τ.
Gross–Zagier: PK = TrHτ /K(Pτ ) is nontorsion iff L (E/K, 1) = 0.
Why did this work?
1 The Riemann surface Γ0(N)H has an algebraic model X0(N)/Q.
2 There is a morphism φ defined over Q, mapping Jac(X0(N)) → E.
3 The “CM point” (τ) − (∞) gets mapped to φ((τ) − (∞)) ∈ E(Hτ ).
Marc Masdeu A unified perspective for Darmon points December 7, 2013 3 / 17
6. Darmon’s Insight
Drop hypothesis of K/F being CM.
Simplest case: F = Q, K real quadratic.
However:
There are no points on the modular curve X0(N) attached to K.
In general there is no morphism φ: Jac(X0(N)) → E.
If F not totally real, even the curve X0(N) is missing!
Nevertheless, one can construct local points in such cases. . .
. . . and hope that they are global.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 4 / 17
7. Goals
In this talk I will:
1 Explain what is a Darmon point.
2 Sketch a general construction.
“The fun of the subject seems to me to be in the examples.
B. Gross, in a letter to B. Birch, 1982
”3 Illustrate with an fun example.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 5 / 17
8. Basic notation
Let v | ∞F be an infinite place of F.
If v is real, then:
1 It may extend to two real places of K (splits), or
2 It may extend to one complex place of K (ramifies).
If v is complex, then it extends to two complex places of K (splits).
n = #{v | ∞F : v splits in K}.
K/F is CM ⇐⇒ n = 0.
If n = 1 we call K/F quasi-CM.
S(E, K) = v | N∞F : v not split in K , s = #S(E, K).
Sign of functional equation for L(E/K, s) should be (−1)#S(E,K).
From now on, we assume that s is odd.
Fix a place ν ∈ S(E, K).
1 If ν = p is finite we are in the non-archimedean case.
2 If ν is infinite we are in the archimedean case.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 6 / 17
10. Non-archimedean History
Warning
These constructions are also known as Stark-Heegner points.
H. Darmon (1999): F = Q, quasi-CM, s = 1.
Darmon-Green (2001): special cases, used Riemann products.
Darmon-Pollack (2002): same cases, overconvergent methods.
Guitart-M. (2012): all cases, overconvergent methods.
M. Trifkovic (2006): F imag. quadratic ( =⇒ quasi-CM)), s = 1.
Trifkovic (2006): F euclidean, E of prime conductor.
Guitart-M. (2013): F arbitrary, E arbitrary.
M. Greenberg (2008): F totally real, arbitrary ramification, s ≥ 1.
Guitart-M. (2013): F = Q, quasi-CM case, s ≥ 1.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 7 / 17
12. Archimedean History
Warning
Initially called Almost Totally Real (ATR) points.
But this name only makes sense in the original setting of Darmon.
H. Darmon (2000): F totally real, s = 1.
Darmon-Logan (2007): F quadratic norm-euclidean, NE trivial.
Guitart-M. (2011): F quadratic and arbitrary, NE trivial.
Guitart-M. (2012): F quadratic and arbitrary, NE arbitrary.
J. Gartner (2010): F totally real, s ≥ 1.
?
Marc Masdeu A unified perspective for Darmon points December 7, 2013 8 / 17
13. Our construction
Xavier Guitart M. Mehmet H.
Sengun
Available for arbitrary base number fields F (mixed signature).
Comes in both archimedean and non-archimedean flavors.
All of the previous constructions become particular cases.
We can provide genuinely new numerical evidence.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 9 / 17
14. Path integrals: archimedean setting
H = (P1(C) P1(R))+ has a complex-analytic structure.
SL2(R) acts on Hν through fractional linear transformations:
a b
c d · z =
az + b
cz + d
, z ∈ H.
Consider a holomorphic 1-form ω ∈ Ω1
H.
Given two points P and Q in H, define:
Q
P
ω = usual path integral.
The resulting pairing
Div0
H × Ω1
H → C
is compatible with the action of SL2(R) on H:
γQ
γP
ω =
Q
P
γ∗
ω.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 10 / 17
15. Path integrals: non-archimedean setting
Hp = P1(Kp) P1(Fp) has a rigid-analytic structure.
SL2(Fp) acts on Hp through fractional linear transformations:
a b
c d · z =
az + b
cz + d
, z ∈ Hp.
Consider a rigid-analytic 1-form ω ∈ Ω1
Hp
.
Given two points P and Q in Hp(Kp), define:
Q
P
ω = Coleman integral.
Again, the resulting pairing:
Div0
Hp × Ω1
Hp
→ Kp,
is compatible with the action of SL2(Fp) on Hp:
γQ
γP
ω =
Q
P
γ∗
ω.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 11 / 17
16. The group Γ
Let B/F = quaternion algebra with Ram(B) = S(E, K) {ν}.
B = M2(F) (split case) ⇐⇒ s = 1.
Otherwise, we are in the quaternionic case.
E and K determine a certain {ν}-arithmetic subgroup Γ ⊂ SL2(Fv):
Let m = l|N, split in K l.
Let R be an Eichler order of level m inside B.
Fix an embedding ιν : R → M2(ZF,ν).
Γ = ιν R[1/ν]×
1 ⊂ SL2(Fν).
Marc Masdeu A unified perspective for Darmon points December 7, 2013 12 / 17
17. (Co)-homology classes
We attach to E a unique cohomology class
ΦE ∈ Hn
Γ, Ω1
Hν
.
We attach to each embedding ψ: K → B a homology class
Θψ ∈ Hn Γ, Div0
Hν .
Well defined up to the image of Hn+1(Γ, Z)
δ
→ Hn(Γ, Div0
Hν).
Cap-product and integration on the coefficients yield an element:
Jψ = Θψ, ΦE ∈ Kν.
Jψ is well-defined up to a lattice
L = δ(θ), ΦE : θ ∈ Hn+1(Γ, Z) .
Marc Masdeu A unified perspective for Darmon points December 7, 2013 13 / 17
18. Conjectures
Jψ = Θψ, ΦE ∈ Kν/L.
Conjecture 1 (Oda, Yoshida, Greenberg, Guitart-M-Sengun)
There is an isogeny β : Kν/L → E(Kν).
Proven in some non-arch. cases (Greenberg, Rotger–Longo–Vigni).
Completely open in the archimedean case.
Darmon point attached to (E, ψ: K → B)
Pψ = β(Jψ) ∈ E(Kν).
Conjecture 2 (Darmon, Greenberg, Trifkovic, Gartner, G-M-S)
1 The local point Pψ is global, and belongs to E(Kab).
2 Pψ is nontorsion if and only if L (E/K, 1) = 0.
Predicts also the exact number field over which Pψ is defined.
Includes a Shimura reciprocity law like that of Heegner points.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 14 / 17
20. Archimedean cubic Darmon point (I)
Let F = Q(r) with r3 − r2 + 1 = 0.
F has discriminant −23, and is of signature (1, 1).
Consider the elliptic curve E/F given by the equation:
E/F : y2
+ (r − 1) xy + r2
− r y = x3
+ −r2
− 1 x2
+ r2
x.
E has prime conductor NE = r2 + 4 of norm 89.
K = F(w), with w2 + (r + 1) w + 2r2 − 3r + 3 = 0.
K has class number 1, thus we expect the point to be defined over K.
The computer tells us that rkZ E(K) = 1
S(E, K) = {σ}, where σ: F → R is the real embedding of F.
Therefore the quaternion algebra B is just M2(F).
The arithmetic group to consider is
Γ = Γ0(NE) ⊂ SL2(OF ).
Γ acts naturally on the symmetric space H
Hyperbolic 3-space
× H3:
H × H3 = {(z, x, y): z ∈ H, x ∈ C, y ∈ R>0}.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 15 / 17
21. Archimedean cubic Darmon point (II)
E ; ωE, an automorphic form with Fourier-Bessel expansion:
ωE(z, x, y) =
α∈δ−1OF
α0>0
a(δα)(E)e−2πi(α0 ¯z+α1x+α2 ¯x)
yH (α1y) ·
−dx∧d¯z
dy∧d¯z
d¯x∧d¯z
H(t) = −
i
2
eiθ
K1(4πρ), K0(4πρ),
i
2
e−iθ
K1(4πρ) t = ρeiθ
.
K0 and K1 are hyperbolic Bessel functions of the second kind:
K0(x) =
∞
0
e−x cosh(t)
dt, K1(x) =
∞
0
e−x cosh(t)
cosh(t)dt.
ωE is a 2-form on ΓH × H3.
The cocycle ΦE is defined as (γ ∈ Γ):
ΦE(γ) =
γ·O
O
ωE(z, x, y) ∈ Ω1
H with O = (0, 1) ∈ H3.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 16 / 17
22. Archimedean cubic Darmon point (III)
Consider the embedding ψ: K → M2(F) given by:
w →
−2r2 + 3r r − 3
r2 + 4 2r2 − 4r − 1
Let γψ = ψ(u), where u is a fundamental norm-one unit of OK.
γψ fixes τψ = −0.7181328459824 + 0.55312763561813i ∈ H.
Construct Θψ = [γψ ⊗τψ] ∈ H1(Γ, Div H).
Θψ is equivalent to a cycle γi ⊗(si − ri) taking values in Div0
H.
Jψ =
i
si
ri
ΦE(γi) =
i
γi·O
O
si
ri
ωE(z, x, y).
We obtain, summing over all ideals (α) of norm up to 400, 000:
Jψ = 0.0005281284234 + 0.0013607546066i ; Pψ ∈ E(C).
Numerically (up to 32 decimal digits) we obtain:
Pψ
?
= −10 × r − 1, w − r2
+ 2r ∈ E(K).
Marc Masdeu A unified perspective for Darmon points December 7, 2013 17 / 17
23. Thank you !
Bibliography, code and slides at:
http://www.math.columbia.edu/∼masdeu/
Marc Masdeu A unified perspective for Darmon points December 7, 2013 17 / 17
24. What to do next?
1 Available code
2 Non-archimedean example
3 Details for Cohomology
4 Details for Homology
5 Bibliography
Marc Masdeu A unified perspective for Darmon points December 7, 2013 17 / 17
25. Available Code
SAGE code for non-archimedean Darmon points when n = 1.
https://github.com/mmasdeu/darmonpoints
Compute with “quaternionic modular symbols”.
Need presentation for units of orders in B (J. Voight, A. Page).
Overconvergent method for arbitrary B.
Obtain a method to find algebraic points.
SAGE code for archimedean Darmon points (in restricted cases).
https://github.com/mmasdeu/atrpoints
Only for the split (B = M2(F)) cases, and:
1 F real quadratic, and K/F ATR (Hilbert modular forms)
2 F cubic (1, 1), and K/F totally complex (cubic automorphic forms).
Marc Masdeu A unified perspective for Darmon points December 7, 2013 1 / 6
26. Non-archimedean cubic example (I)
F = Q(r), with r3 − r2 − r + 2 = 0 and discriminant −59.
Consider the elliptic curve E/F given by the equation:
E/F : y2
+ (−r − 1) xy + (−r − 1) y = x3
− rx2
+ (−r − 1) x.
E has conductor NE = r2 + 2 = p17q2, where
p17 = −r2
+ 2r + 1 , q2 = (r) .
Consider K = F(α), where α =
√
−3r2 + 9r − 6.
The quaternion algebra B/F has discriminant D = q2:
B = F i, j, k , i2
= −1, j2
= r, ij = −ji = k.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 2 / 6
27. Non-archimedean cubic example (II)
The maximal order of K is generated by wK, a root of the polynomial
x2
+ (r + 1)x +
7r2 − r + 10
16
.
One can embed OK in the Eichler order of level p17 by:
wK → (−r2
+ r)i + (−r + 2)j + rk.
We obtain γψ = 6r2−7
2 + 2r+3
2 i + 2r2+3r
2 j + 5r2−7
2 k and:
τψ = (12g+8)+(7g+13)17+(12g+10)172
+(2g+9)173
+(4g+2)174
+· · ·
After integrating we obtain:
Jψ = 16+9·17+15·172
+16·173
+12·174
+2·175
+· · ·+5·1720
+O(1721
),
which corresponds to:
Pψ = −
3
2
× 72 × r − 1,
α + r2 + r
2
∈ E(K).
Marc Masdeu A unified perspective for Darmon points December 7, 2013 3 / 6
28. Cohomology
Recall that S(E, K) and ν determine:
Γ = ιν R[1/ν]×
1 ⊂ SL2(Fν).
e.g. S(E, K) = {p} and ν = p give Γ = SL2 OF [1
p ] .
Choose “signs at infinity” s1, . . . , sn ∈ {±1}.
Theorem (Darmon, Greenberg, Trifkovic, Gartner, G.–M.–S.)
There exists a unique (up to sign) class
ΦE ∈ Hn
Γ, Ω1
Hν
such that:
1 TlΦE = alΦE for all l N.
2 UqΦE = aqΦE for all q | N.
3 Wσi ΦE = siΦE for all embeddings σi : F → R which split in K.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 4 / 6
29. Homology
Let ψ: O → R be an embedding of an order O of K.
Which is optimal: ψ(O) = R ∩ ψ(K).
Consider the group O×
1 = {u ∈ O× : NmK/F (u) = 1}.
rank(O×
1 ) = rank(O×
) − rank(OF ) = n.
Let u1, . . . , un ∈ O×
1 be a basis for the non-torsion units.
; ∆ψ = ψ(u1) · · · ψ(un) ∈ Hn(Γ, Z).
K acts on Hv through ιν ◦ ψ.
Let τψ be the (unique) fixed point of K×
on Hv.
Have the exact sequence
Hn+1(Γ, Z)
δ // Hn(Γ, Div0
Hν) // Hn(Γ, Div Hν)
deg
// Hn(Γ, Z)
Θψ
? // [∆ψ ⊗τψ] // [∆ψ]
Fact: [∆ψ] is torsion.
Can pull back a multiple of [∆ψ ⊗τψ] to Θψ ∈ Hn(Γ, Div0
Hν).
Well defined up to δ(Hn+1(Γ, Z)).
Marc Masdeu A unified perspective for Darmon points December 7, 2013 5 / 6
30. Bibliography
H. Darmon and A. Logan. Periods of Hilbert modular forms and rational points on elliptic curves.
Int. Math. Res. Not. (2003), no. 40, 2153–2180.
H. Darmon and P. Green. Elliptic curves and class fields of real quadratic fields: Algorithms and evidence.
Exp. Math., 11, No. 1, 37-55, 2002.
H. Darmon and R. Pollack. Efficient calculation of Stark-Heegner points via overconvergent modular symbols.
Israel J. Math., 153:319–354, 2006.
J. G¨artner. Darmon points and quaternionic Shimura varieties.
Canad. J. Math. 64 (2012), no. 6.
X. Guitart and M. Masdeu. Elementary matrix Decomposition and the computation of Darmon points with higher conductor.
Math. Comp. (arXiv.org, 1209.4614), 2013.
X. Guitart and M. Masdeu. Computation of ATR Darmon points on non-geometrically modular elliptic curves.
Exp. Math., 2012.
X. Guitart and M. Masdeu. Computation of quaternionic p-adic Darmon points.
(arXiv.org, 1307.2556), 2013.
M. Greenberg. Stark-Heegner points and the cohomology of quaternionic Shimura varieties.
Duke Math. J., 147(3):541–575, 2009.
D. Pollack and R. Pollack. A construction of rigid analytic cohomology classes for congruence subgroups of SL3(Z).
Canad. J. Math., 61(3):674–690, 2009.
M. Trifkovic. Stark-Heegner points on elliptic curves defined over imaginary quadratic fields.
Duke Math. J., 135, No. 3, 415-453, 2006.
Marc Masdeu A unified perspective for Darmon points December 7, 2013 6 / 6