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number theory, a cusp form is a particular kind of modular form with zero constant coefficient in its Fourier series expansion. A cusp form is distinguished
Cusp_form
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
growth conditions at the cusps of a fundamental domain of Γ {\displaystyle \Gamma } . In contrast to modular forms, Maass forms need not be holomorphic
Maass_wave_form
Topics referred to by the same term
systems Cusp form, in modular form theory Cusp neighborhood, a set of points near a cusp Cuspidal representation, a generalization of cusp forms in the
Cusp
Analytic function on the upper half-plane with a certain behavior under the modular group
{\displaystyle {\text{im}}(z)\to \infty } . In addition, a modular form is called a cusp form if it satisfies the following growth condition: Cuspidal condition:
Modular_form
Major type of automorphic form in mathematics
Schottky form. The space of cusp forms of weight 10 has dimension 1, the space of cusp forms of weight 12 has dimension 2, the space of cusp forms of weight
Siegel_modular_form
The philosophy of cusp forms was a slogan of Harish-Chandra, expressing his idea of a kind of reverse engineering of automorphic form theory, from the
Parabolic_induction
Type of generalization of periodic functions in Euclidean space
this problem, leaving the cusp form or discrete part to investigate. From the point of view of number theory, the cusp forms had been recognised, since
Automorphic_form
Shoreline formations made up of various grades of sediment in an arc pattern
Beach cusps are shoreline formations made up of various grades of sediment in an arc pattern. The horns are made up of coarser material and the embayment
Beach_cusps
proved this formula, when k = Q {\displaystyle k=\mathbb {Q} } and f is a cusp form; Günter Harder made the same discovery at the same time in an unpublished
Waldspurger_formula
Flap of tissue that prevent backflow of blood around the heart
cavities. The invaginated margins form the rudiments of the lateral cusps of the AV valves. The middle and septal cusps develop from the downward extension
Heart_valve
Natural number
{\displaystyle 12\pi } . Twelve is the smallest weight for which a cusp form exists. This cusp form is the discriminant Δ ( q ) {\displaystyle \Delta (q)} whose
12_(number)
Point on a curve where motion must move backwards
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is
Cusp_(singularity)
Rare dental anomaly resulting in teeth having more than one cusp
Talon cusp is a rare dental anomaly resulting in an extra cusp or cusp-like projection on an anterior tooth, located on the inside surface of the affected
Talon_cusp
Part of the theory of modular forms
Atkin–Lehner theory is based on the concept of a newform, which is a cusp form 'new' at a given level N, where the levels are the nested congruence subgroups:
Atkin–Lehner_theory
Natural number
{\displaystyle \Delta (\tau )=\eta (\tau )^{24}=q\prod _{n>0}(1-q^{n})^{24},} a cusp form of weight 12 for S L 2 ( Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb
24_(number)
Unsolved problem in mathematics
generalized it for coefficients of modular forms. In the version for modular forms, the conjecture says that for any cusp form of weight k {\displaystyle k} with
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Complex-differentiable part of a Maass wave function
k modular forms, so these are all examples of harmonic weak Maass forms. (A Maass form is a weak Maass form that decreases rapidly at cusps.) So a harmonic
Mock_modular_form
Feature of human teeth
The cusp of Carabelli, Carabelli's tubercle, or tuberculum anomale of Georg Carabelli is a small additional cusp at the mesiopalatal line angle of maxillary
Cusp_of_Carabelli
Pointed, projecting, or elevated anatomical feature
A cusp is a pointed, projecting, or elevated feature. In animals, it is usually used to refer to raised points on the crowns of teeth. The concept is also
Cusp_(anatomy)
Neighborhood of a singularity of cusp type
In mathematics, a cusp neighborhood is defined as a set of points near a cusp singularity. The cusp neighborhood for a hyperbolic Riemann surface can
Cusp_neighborhood
Relates rational elliptic curves to modular forms
coefficients of f. The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform (an eigenvector of
Modularity_theorem
a cusp form. Or a more general factor of automorphy as discussed in Kollár 1995, §5.2. Kollár, János (1995), Shafarevich maps and automorphic forms, M
Poincaré series (modular form)
Poincaré_series_(modular_form)
Linear operator acting on modular forms
{\textstyle S_{k}} of cusp forms of weight k {\displaystyle k} is preserved by the Hecke operators. If a (non-zero) cusp form f {\displaystyle f} is
Hecke_operator
Conjectures connecting number theory and geometry
The Langlands program is built on existing ideas: the philosophy of cusp forms formulated a few years earlier by Harish-Chandra and Gelfand, the work
Langlands_program
Mathematical concept
{\displaystyle f} . The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform (an eigenvector of
Modular_elliptic_curve
Mathematical formula in harmonic analysis
integral operations on f than the Fourier transform. Let ƒ be a Maass cusp form for the modular group PSL(2,Z) and a(n) its Fourier coefficients. Let
Voronoi_formula
Siegel modular form
Klingen Eisenstein series is a Siegel modular form of weight k and degree g depending on another Siegel cusp form f of weight k and degree r<g, given by a
Klingen_Eisenstein_series
Cohort born during the late 1970s to early 1980s
also called Xillenials, are a social cohort encompassing people born on the cusp of or during the later years of Generation X and the early years of the Millennials
Xennials
term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations
Cuspidal_representation
data of the surface. The zeros are at the following points: For every cusp form with eigenvalue s 0 ( 1 − s 0 ) {\displaystyle s_{0}(1-s_{0})} there exists
Selberg_zeta_function
Area of mathematics
the fold A3 - the cusp A4 - the swallowtail A5 - the butterfly Ak - a representative of an infinite sequence of one variable forms V = x k + 1 + ⋯ {\displaystyle
Catastrophe_theory
Semilunar valve of the heart
lies between the right ventricle and the pulmonary artery, and has three cusps. It is one of the four valves of the heart and one of the two semilunar
Pulmonary_valve
Long, pointed tooth in mammals
or cusp, which projects beyond the level of the other teeth. The root is single, but longer and thicker than that of the incisors, conical in form, compressed
Canine_tooth
Indian mathematician (1887–1920)
generating function as the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence
Srinivasa_Ramanujan
In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form J of degree 4 and weight 8, introduced by Friedrich Schottky (1888, 1903)
Schottky_form
Function studied by Ramanujan
Because the modular discriminant Δ ( z ) {\textstyle \Delta (z)} is a cusp form of weight 12, it gives rise to an L {\displaystyle L} -function, called
Ramanujan_tau_function
Valve in the human heart between the left ventricle and the aorta
has three cusps however there is some discrepancy in their naming. They may be called the left coronary, right coronary and non-coronary cusp. Some sources
Aortic_valve
Local zeta function Weil conjectures Modular form modular group Congruence subgroup Hecke operator Cusp form Eisenstein series Modular curve Ramanujan–Petersson
List_of_number_theory_topics
Type of vector space
classical elliptic modular form theory, the Hecke operators Tn with n coprime to the level acting on the space of cusp forms of a given weight are self-adjoint
Hecke_algebra
Type of small omnivorous rodent
eyes and ears; and differently formed molars (high-crowned with angular cusps instead of low-crowned with rounded cusps). They are sometimes known as meadow
Vole
Large tooth at the back of the mouth
the result of different enamel thickness. Cusps are occasionally joined to form ridges and expanded to form crests. Cingula are often incomplete ridges
Molar_(tooth)
Mathematical ideal related to a modular curve
operators acting on all modular forms for Γ0(N) (not just the cusp forms). The ring T of Hecke operators on the cusp forms is a quotient of T*, so Spec(T)
Eisenstein_ideal
kernel is the space of Siegel cusp forms of degree d. Klingen, Helmut (2003), Introductory Lectures on Siegel Modular Forms, Cambridge University Press
Siegel_operator
Turbulent layer of water that washes ashore after a wave breaks
patterns of erosion. The cusp embayments form at the eroding points and cusp horns occur at the edge wave nodes. The beach cusp spacing can be predicted
Swash
Mathematical function
at the cusps. If the eigenvalue of f {\displaystyle f} under the Laplacian is zero, then f {\displaystyle f} is called a harmonic weak Maass form, or briefly
Harmonic_Maass_form
{\displaystyle k} and S k {\displaystyle \mathbb {S} _{k}} the space of cusp forms. The mapping ⟨ ⋅ , ⋅ ⟩ : M k × S k → C {\displaystyle \langle \cdot ,\cdot
Petersson_inner_product
Mathematical theorem
of Hecke operators on spaces of cusp forms and, in particular cases, dimension formulas for spaces of modular forms. More generally, the trace formula
Selberg_trace_formula
Complex-valued arithmetic function
_{1}}(z)\in M_{k}(MN^{2},\chi \chi _{1}^{2})} . If f {\displaystyle f} is a cusp form so is f χ 1 . {\displaystyle f_{\chi _{1}}.} See theta series of a Dirichlet
Dirichlet_character
Mathematical function
be sufficient to construct a basis for a vector space of modular forms and cusp forms. A useful theorem for limiting the number of modular eta quotients
Dedekind_eta_function
Species of small mammal from the middle Jurassic of Madagascar
They consist of two groups of cusps: a trigonid of three cusps at the front and a talonid with a main cusp, a smaller cusp, and a crest at the back. Features
Ambondro_mahabo
a Siegel modular form of degree 1 and weight 2k, and a Siegel cusp form of degree r and weight k + n + r and constructs a Siegel form of degree 2n + r
Miyawaki_lift
Mathematical function
form is similar to a holomorphic modular form, except that it is allowed to have poles at cusps. Examples include modular functions and modular forms
Weakly holomorphic modular form
Weakly_holomorphic_modular_form
Device for plasma confinement
The biconic cusp, also known as the picket fence reactor, was one of the earliest suggestions for plasma confinement in a fusion reactor. It consists of
Biconic_cusp
f corresponding to the Hecke operator Ti. In the case when f is not a cusp form, the eigenvalues can be given explicitly. An eigenform which is cuspidal
Eigenform
Canadian mathematician
showing that all automorphic forms arise in terms of cusp forms and the residues of Eisenstein series induced from cusp forms on smaller subgroups. As a
Robert_Langlands
(link) Ikeda, Tamotsu (2001), "On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n", Annals of Mathematics, Second Series, 154 (3):
Ikeda_lift
Human tooth
three cusps but can have two as well. The three cusp variety has one large cusp on the buccal with two smaller lingual cusps. The mesiolingual cusp is twice
Mandibular_second_premolar
Field of anatomy dedicated to human teeth
molars have two buccal cusps and two lingual cusps. A fifth cusp that may form on the maxillary first molar is known as the cusp of Carabelli. Mandibular
Dental_anatomy
symbols. There is a pairing between cusp forms f of weight 2 and modular symbols given by integrating the cusp form, or rather fdτ, along the path corresponding
Modular_symbol
Second astrological sign of the zodiac
alongside Virgo and Capricorn, it forms the Earth triplicity. Astronomical symbols Chinese zodiac Circle of stars Cusp (astrology) Elements of the zodiac
Taurus_(astrology)
Cohomology theory
is an isomorphism between an Eichler cohomology group and a space of cusp forms. There are several variations of the Eichler–Shimura isomorphism, because
Eichler–Shimura_isomorphism
Genus of armored ankylosaurine dinosaurs
7 mm (0.28 in). The teeth have in total twelve to fourteen cusps on their edges. The cusp forming the tip of the tooth is off-set to behind. Around the neck
Zuul
Process of extending a representation of a subgroup to the parent group
representations of its parabolic subgroups. This leads, via the philosophy of cusp forms, to the Langlands program. Restricted representation Nonlinear realization
Induced_representation
Eleventh astrological sign of the zodiac
timing rituals (Muhurta). Astronomical symbols Chinese zodiac Circle of stars Cusp (astrology) Elements of the zodiac Astronomical Applications Department 2011
Aquarius_(astrology)
Awarded every year by the American Mathematical Society
\operatorname {GL} _{2}(\mathbb {Z} _{p}[[X]])} attached to ordinary cusp forms". Inventiones Mathematicae. 85 (3): 545–613. doi:10.1007/BF01390329. Zbl 0612
Leroy_P._Steele_Prize
Australian-American mathematician
multiplicities of unitary representations of cohomological type in spaces of cusp forms". Annals of Mathematics. Second Series. 170 (3): 1437–1446. arXiv:0704
Frank_Calegari
Cohort born during the early 1990s to late 1990s
occasionally called Zennials, are a social cohort encompassing people born on the cusp of or during the later years of the Millennial generation and the early years
Zillennials
First astrological sign of the zodiac
portal Stars portal Astronomical symbols Chinese zodiac Circle of stars Cusp (astrology) Elements of the zodiac History of astrology Fire (classical element)
Aries_(astrology)
Mathematical concept
conjectures for cusp forms on general linear groups. The Ramanujan conjectures, in turn, are central to the study of automorphic forms, as they predict
Arthur's_conjectures
Branch of mathematics that studies abstract algebraic structures
topics List of harmonic analysis topics Numerical analysis Philosophy of cusp forms Representation (mathematics) Representation theorem Universal algebra
Representation_theory
Unsolved problem in physics
The cuspy halo problem (also known as the core-cusp problem) is a discrepancy between the inferred dark matter density profiles of low-mass galaxies and
Cuspy_halo_problem
Raga in Hindustani classical music
this raga very distinct from other ragas. Deliberate oscillation on the cusp formed between Ma-Ma# and Ni is commonly heard. In the Gwalior tradition of
Lalit_(raga)
to the eigenvalue of Tn on f. Let f {\displaystyle f} be a holomorphic cusp form with weight ( 2 k + 1 ) / 2 {\displaystyle (2k+1)/2} and character χ {\displaystyle
Shimura_correspondence
complicated but a similar theory can be established via the notion of cusp form. The spectral gap of the surface S {\displaystyle S} is by definition
Arithmetic_Fuchsian_group
Division of the horoscope into 12 sectors
the horoscope in question. They are numbered counter-clockwise from the cusp of the first house. Commonly, houses one through six are below the horizon
House_(astrology)
Cohort born from 1965 to 1980
sometimes called Generation Jones. People born in the Generation X / millennial cusp years of the late 1970s and early to mid-1980s are sometimes called Xennials
Generation_X
South African-born mathematician
number theorists of his generation. His early work on the existence of cusp forms led to the disproof of a conjecture of Atle Selberg. He has obtained the
Peter_Sarnak
American mathematician
Georg; Pixton, Aaron (2018), "Holomorphic anomaly equations and the Igusa cusp form conjecture", Inventiones Mathematicae, 213 (2): 507–587, arXiv:1706.10100
Aaron_Pixton
Architectural element
a cusped arch, polylobed arch, or scalloped arch, is an arch characterized by multiple circular arcs or leaf shapes (called foils, lobes, or cusps) that
Multifoil_arch
example, that a motive can be attached to a newform (i.e. a primitive cusp form), hence their L-functions are motivic. Several conjectures exist concerning
Motivic_L-function
German mathematician
with Rainer Weissauer and Richard Borcherds the existence of a Siegel cusp form of degree 12 and weight 12 using the theta series associated with the
Eberhard_Freitag
Symbol representing the heart
shape consisting of "two similar curves meeting in a point at one end and a cusp at the other," not an anatomically correct representation of a heart. Often
Heart_symbol
Indian-American mathematician (1923–1983)
S.S.D. College, Kanpur celebrates his birthday every year in different forms, which includes lectures from students and professors from various colleges
Harish-Chandra
Formula in analytic number theory
Boston, MA, (1990). Kuznecov, N. V. (1981). "Petersson's Conjecture for Cusp Forms of Weight Zero and Linnik's Conjecture. Sums of Kloosterman Sums". Mathematics
Kuznetsov_trace_formula
Compact astronomical body
singularity from forming. A black star would be gravitationally collapsing slowly enough that quantum effects would keep it just on the cusp of fully collapsing
Black_hole
Type of roulette curve
generating the roulette lies on the rolling circle; the resulting curve has a cusp. Depending on the position of the point generating the curve, it may have
Limaçon
Valve in the heart connecting the left atrium and left ventricle
left atrioventricular valve, is one of the four heart valves. It has two cusps or flaps and lies between the left atrium and the left ventricle of the
Mitral_valve
Zodiac sign over which a planet rules
also used as an implied focus of interpretation for the signs on house cusps in a chart. A planet is considered to be in domal dignity when it is positioned
Domicile_(astrology)
plaster cast. From the above observation, 33° 'cuspal formed' teeth were introduced such that the cuspal inclination would be parallel with the condylar
Complete_denture_occlusion
Extinct order of mammaliaforms
additional lingual cusps may be present: cusp e and cusp df. Cusp e lies in front of cusp g and is roughly lingual to cusp b. Cusp df ("docodont cuspule
Docodonta
pillar-like cusps as for "selenes" crescent-like cusps. The suffix "-conule /-conulid" (upper molar/lower molar) is added to the secondary cusps. For example
Glossary of mammalian dental topography
Glossary_of_mammalian_dental_topography
Astrological chart or diagram
in relation to the zodiacal placement of the Sun on the month of birth, cusp (two days before or after any particular sign, an overlap), or decant (the
Horoscope
Extinct genus of mammal
of cusps (the paracone and the protocone) had no additional ridges. The bases of the two cusps were connected to each other, and a side cusp formed a paraconule
Saloumia
Distribution of stars around a supermassive black hole
Bahcall–Wolf cusp refers to a particular distribution of stars around a massive black hole at the center of a galaxy or globular cluster. If the nucleus
Bahcall–Wolf_cusp
French mathematician (born 1946)
Volume 303, 1986, pp. 85, 161 Kloosterman sums and Fourier coefficients of cusp forms. In: Inventiones Mathematicae. Volume 70, 1982, pp. 219–288 A complete
Jean-Marc_Deshouillers
German mathematician
devoted most of his research to the theory of modular forms, creating the general theory of cusp forms (holomorphic, for GL(2)), as it is now understood in
Erich_Hecke
A829––A832, ISSN 0151-0509, MR 0263977 Shalika, J. A. (1970), "On the space of cusp forms of a P-adic Chevalley group", Annals of Mathematics, Second Series, 92
Steinberg_representation
Japanese mathematician (1930–2019)
Eichler–Shimura isomorphism between Eichler cohomology groups and spaces of cusp forms which would be used in Pierre Deligne's proof of the Weil conjectures
Goro_Shimura
Point on a curve not given by a smooth embedding of a parameter
A cusp: x 3 − y 2 = 0 , {\displaystyle x^{3}-y^{2}=0,} also called a spinode A tacnode: x 4 − y 2 = 0 {\displaystyle x^{4}-y^{2}=0} A rhamphoid cusp: x
Singular_point_of_a_curve
Curve generated by rolling a circle inside another circle with 4x or (4/3)x the radius
astroid is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside
Astroid
CUSP FORM
CUSP FORM
Biblical
a hill; cup
Biblical
threshold; silver cup
Boy/Male
Biblical
Cup-bearer of the prince.
Boy/Male
Biblical
Ethiopians, blackness.
Girl/Female
Arabic, Muslim
Wine Cup
Biblical
Cushan, Cushi, Ethiopians; blackness
Girl/Female
Biblical
Hill, cup, thing lifted up.
Surname or Lastname
English
English : variant of Kiss.Americanized spelling of German and Jewish Kusch.
Girl/Female
British, English
Happy
Biblical
cup-bearer of the prince
Girl/Female
Gujarati, Indian
Cup
Boy/Male
Greek Latin
Cup bearer to the gods.
Boy/Male
Biblical
Threshold, silver cup.
Girl/Female
Biblical, Dutch, German
A Hill; Cup
Boy/Male
English American
Forest; cup bearer.
Boy/Male
Arthurian Legend
Name of a king.
Male
English
Anglicized form of Hebrew Kuwsh, CUSH means "black," i.e. "Ethiopian." In the bible, this is the name of a land and its people. It is also the name of a Benjamite and the son of Ham and grandson of Noah.
Surname or Lastname
English
English : possibly a variant of Copp.Possibly an Americanized spelling of German Kopp.
Boy/Male
English
Cup bearer.
Girl/Female
Arabic, Muslim
Wine Cup
CUSP FORM
CUSP FORM
Boy/Male
Hindu
Victory, Victorious
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
The Earth
Boy/Male
Latin
Descendant of Acrisius.
Boy/Male
Tamil
Manmada | மாநà¯à®®à®¤à®¾
Any time young
Boy/Male
Arabic, Muslim
A Small Cultivator
Boy/Male
Indian
Graceful, Elegant, Connoisseur
Boy/Male
Hindu
Sreyas
Girl/Female
Hindu
Bright star
Surname or Lastname
English
English : topographic name for someone who lived by a postern gate, from Old French posterne; in some cases it would have been a metonymic occupational name for a gatekeeper.English : habitational name from Poston in Herefordshire or Poston in Shropshire, which is named with an Old English personal name Possa + þorn ‘thorn tree’.
Boy/Male
Tamil
Karthiga | கரà¯à®¤à¯€à®•à®¾
God name
CUSP FORM
CUSP FORM
CUSP FORM
CUSP FORM
CUSP FORM
v. t.
To supply with cups of wine.
v. t.
To furnish with a cusp or cusps.
n.
A sharp and rigid point.
a.
Having a calyx or cup; cup-shaped.
n.
A prominence or point, especially on the crown of a tooth.
p. pr. & vb. n.
of Cup
n.
A cup used for holding an egg, at table.
n.
A small vessel, used commonly to drink from; as, a tin cup, a silver cup, a wine cup; especially, in modern times, the pottery or porcelain vessel, commonly with a handle, used with a saucer in drinking tea, coffee, and the like.
n.
The point or horn of the crescent moon or other crescent-shaped luminary.
n.
A multiple point of a curve at which two or more branches of the curve have a common tangent.
n.
The cusk. See Cusk.
n.
The beginning or first entrance of any house in the calculations of nativities, etc.
n.
One who sneaks from his cups; one who balks his glass.
v. t.
To make concave or in the form of a cup; as, to cup the end of a screw.
p. pr. & vb. n.
of Cusp
n.
A triangular protection from the intrados of an arch, or from an inner curve of tracery.
a.
Cup-shaped.
imp. & p. p.
of Cusp
n.
Anything shaped like a cup; as, the cup of an acorn, or of a flower.