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Topic: Hilbert modular form

	Modular form -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		Modular form theory is a special case of the more general theory of (additional info and facts about automorphic form) automorphic forms, and therefore can now be seen as just the most concrete part of a rich theory of (additional info and facts about discrete group) discrete groups.
		Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.
		Hilbert modular forms are functions in n variables, each a complex number in the upper half plane, satisfying a modular relation for 2×2 matrices with entries in a (additional info and facts about totally real number field) totally real number field.
	www.absoluteastronomy.com /encyclopedia/m/mo/modular_form.htm (2048 words)

	Modular form (Site not responding. Last check: 2007-10-09)
		A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition.
		The crucialconceptual link between modular forms and number theory are furnished by the theory of Hecke operators, which also gives the link between the theory of modular forms and representation theory.
		Hilbert modular forms are functions in n variables, each a complexnumber in the upper half plane, satisfying a modular relation for 2×2 matrices with entries in a totally real number field.
	www.therfcc.org /modular-form-79309.html (1123 words)

	Hilbert Modular Forms And p-Adic Hodge Theory (ResearchIndex)
		Hilbert Modular Forms And p-Adic Hodge Theory (ResearchIndex)
		15 Automorphic forms on GL (context) - Jacquet, Langlands - 1970
		1 Modular forms and #-adic representations (context) - Langlands - 1973
	citeseer.ist.psu.edu /471771.html (557 words)

	ipedia.com: Modular form Article (Site not responding. Last check: 2007-10-09)
		The theory of modular forms therefore belongs to complex analysis but the main importance of the theory lies in its connections with number theory.
		Modular functions can be thought of as functions on the moduli space of isomorphism classes of complex elliptic curves.
		The crucial conceptual link between modular forms and number theory are furnished by the theory of Hecke operators, which also gives the link between the theory of modular forms and representation theory.
	www.ipedia.com /modular_form.html (1209 words)

	Working Seminar Spring 2004
		A modular form of weight 1 and level 283 will be constructed which has coefficients related to the number of solutions of x^4-x-1 modulo p.
		The number of zeros of x^4-x-1 modulo p do not form a pattern related to quadratic number fields or congruence conditions on the prime p, but nevertheless is related to a modular form of weight 1.
		Modular forms of prime level p are congruent modulo p to modular forms of full level.
	www.math.rutgers.edu /~tunnell/working.html (826 words)

	Automorphic form -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		The Hilbert modular forms (Hilbert-Blumenthal, as one should say) were proposed not long after that, though a full theory was long in coming.
		The Siegel modular forms, for which G is a (additional info and facts about symplectic group) symplectic group, arose naturally from considering (additional info and facts about moduli space) moduli spaces and (additional info and facts about theta function) theta functions.
		It does not actually completely include the automorphic form idea introduced above, in that the (additional info and facts about adele) adele approach is a way of dealing with the whole family of (additional info and facts about congruence subgroup) congruence subgroups at once.
	www.absoluteastronomy.com /encyclopedia/A/Au/Automorphic_form.htm (845 words)

	The Computational Theory of Mind
		The most influential strategy for formalization was that of Hilbert, who treated formalized reasoning as a "symbol game", in which the rules of derivation were expressed in terms of the syntactic (or perhaps better, non-semantic) properties of the symbols employed.
		Hilbert himself carried out such a project with respect to geometry, and Whitehead and Russell extended such a method to arithmetic.
		This is complicated, however, by Fodor's [1984] distinction between "modular" and "global" mental processes, and his judgement (in Fodor [2000]) that it is only the former that are likely to be computational in the classic sense.
	plato.stanford.edu /entries/computational-mind (6686 words)

	Modular form (Site not responding. Last check: 2007-10-09)
		Hilbert modular forms are functions in n variables, each a complex number in the upper half plane, satisfying a modular relation for 2×2 matrices with entries in a totally real number field.
		For an elementary introduction to the theory of modular forms, see Chapter VII of Jean-Pierre Serre: A Course in Arithmetic.
		For an introduction to modular forms from the point of view of representation theory, one might consult Stephen Gelbart: Automorphic forms on adele groups.
	www.portaljuice.com /modular_form.html (1121 words)

	London Number Theory Seminar: Previous Talks
		Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography.
		The Manin constant of a (modular) elliptic curve was introduced by Manin in a paper in the mid-70s, in which he also conjectured that it is always 1.
		Second-order modular forms are functions that have recently appeared in several contexts: Eisenstein series formed with modular symbols, converse theorems of L-functions, percolation theory etc. They satisfy a functional equation that extends naturally that of the usual modular forms and their study is important for the topics that have motivated their introduction.
	www.mth.kcl.ac.uk /~alj1s/kcl/numbtheo_past.html (5723 words)

	Automorphic form (Site not responding. Last check: 2007-10-09)
		In mathematics, the general notion of automorphic form isthe extension to analytic functions, perhaps of several complex variables, of the theory of modular forms.
		The Hilbert modular forms (Hilbert-Blumenthal, as one should say) were proposed not long after that,though a full theory was long in coming.
		The Siegel modular forms, for which G is a symplectic group, arose naturally from considering moduli spaces and theta functions.
	www.therfcc.org /automorphic-form-79316.html (671 words)

	NMBRTHRY Archives -- October 2000 (#12)
		I started learning the theory of modular forms when I was a graduate student in a very concrete way, by reading Serre's Course in Arithmetic and noting that he gave some very concrete formulae for level 1 modular forms, e.g.
		I decided recently to do the same thing for Hilbert modular forms because although I know some of the high-brow story, my "practical" hold on the theory is almost non-existent.
		The only Hilbert modular forms I know explicitly though are eigenforms, so in order to get somewhere with this method I will have to multiply two forms together.
	listserv.nodak.edu /cgi-bin/wa.exe?A2=ind0010&L=nmbrthry&F=&S=&P=1274 (595 words)

	[No title]
		The existence of a Hilbert cusp form with the correct behavior is deduced from the theorem of Jacquet and Langlands.
		As explained earlier on in the paper, the expected modularity of the two-dimensional representation $\sigma$ of $Gal(\bar F/F)$ is equivalent to the existence of a holomorphic Hilbert modular form $\mathfrak f$ of weight ${\bf k} = (2,4)$.
		Roughly speaking, the ``basis problem'' asks whether all elements of of a certain space of modular forms may be expressed as linear combinations of theta-series attached to the norm form of a definite quaternion algebra (\cf~\cite{HPS}).
	www.math.ias.edu /preprints/katia/0009134.tex (13579 words)

	A Mathematical Lie
		This fact was already clear in the 19th century - the moduli of elliptic curves are expressible in terms of modular forms of the parameter tau in the upper half plane.
		This is from "Topics in Elliptic Curves and Modular Forms," by J. William Hoffman, Sept. 28, 2001, p.
		Eisenstein [1823-1852] that modular forms on a congruence subgroup Gamma of SL
	log24.com /log03/1130.htm (1789 words)

	week197
		Then, we say a modular function is a "modular form" if it doesn't blow up as you march up the upper half-plane to the point at infinity.
		If this modular form has weight 0, the partition function is an honest-to-goodness function on the moduli space of elliptic curves: for any elliptic curve the partition function is an actual number.
		In fact, the ring of modular forms is generated by one of weight 4 and one of weight 6: these are both "Eisenstein series", which are well-understood, so we just need someone to cook up conformal field theories having these as partition functions.
	www.math.ucr.edu /home/baez/week197.html (5732 words)

	UNC Charlotte Mathematics Department - What We Know About Fermat's Last Theorem
		It is a conjecture of Mazur that R = T, and it would follow from this that every lift of rho_p which ``looks modular'' (in particular the one we are interested in) is attached to a modular form.
		One of these is related to the Selmer group of the symmetric square of the given modular lifting of rho_p, and the other is related (by work of Hida) to an L -value.
		Let X denote the modular curve whose points correspond to pairs (A, C) where A is an elliptic curve and C is a subgroup of A isomorphic to the group scheme E[5].
	www.math.uncc.edu /flt.php (3199 words)

	GALOIS THEORY and ARITHMETIC, Bonn, 1-4 June 2004
		This conjecture was proved in this form for $n\le2$ by de Jong, and a proof for general $n$ and $\ell\neq2$ was given by Gaitsgory in a recent preprint using methods from the geometric Langlands program.
		The talk is about an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two.
		We also determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and study Faltings heights of arithmetic Hirzebruch-Zagier divisors.
	www.math.uni-bonn.de /people/gata/abstracts.html (1348 words)

	math lessons - Theta function
		The most common form of theta function is that occurring in the theory of elliptic functions.
		With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions (sometimes called quasi-periodicity, though this is not related to the use of that term for dynamical systems).
		This theta function is a modular form of weight n/2 (on an appropriately defined subgroup) of the modular group.
	www.mathdaily.com /lessons/Theta-function (1094 words)

	Eknath Ghate's Papers
		Congruences between base-change and non-base-change Hilbert modular forms (Cohomology of arithmetic groups, Automorphic forms and $L$-functions, Narosa, 2001) is an expository account of a conjecture of Doi, Hida and Ishii.
		In Endomorphism algebras of motives attached to elliptic modular forms (Ann.
		On the local behaviour of ordinary modular Galois representations (Progress in Mathematics 224, Birkhauser-Verlag (2004), 105-124) we investigate a question of Greenberg concerning the spitting behaviour of the restriction to a decomposition group at a prime p of the p-adic Galois representation attached to a p-ordinary elliptic modular cusp form.
	www.math.tifr.res.in /~eghate/math.html (1044 words)

	[No title]
		A theorem of Skinner and Wiles shows that nearly ordinary lifts of modular Galois representations are modular.
		There's a slight wrinkle: the weight 1 Hilbert modular form f attached to E[3] by Langlands-Tunnell might not be ordinary, and thus one cannot necessarily find a higher-weight modular lift of E[3] whose restriction to decomposition groups at 3 we can control.
		When E is a quadratic field of discriminant 5 or 8, we produce the desired pairs of Hilbert modular forms explicitly and show how they can be used to compute the group of Néron components of a Hilbert-Blumenthal abelian variety with real multiplication by E..dvi version.pdf version
	www.math.princeton.edu /~ellenber/papers.html (1955 words)

	Eknath Ghate's Papers
		An introduction to congruences between modular forms (Current trends in number theory, Hindustan Book Agency (2002), 39-58) introduces some of the basic concepts in the theory of congruences between modular forms.
		The program asks for input a Galois conjugacy class of an elliptic modular cusp form of arbitrary weight and level and real nebentypus (the nebentypus is specified by typing in a string of 1's and 2's).
		Ordinary forms and their local Galois representations is an exposition of some related issues (this paper will appear in the proceedings of a conference held in Hyderabad in 2003; some slides of the talk are
	www.math.tifr.res.in /%7Eeghate/math.html (1044 words)

	U of R Number Theory seminar (Site not responding. Last check: 2007-10-09)
		Tate's conjecture from arithmetic algebraic geometry is a p-adic analog of the Hodge conjecture.
		I will describe the conjecture for elliptic curves and abelian varieties and mention how modular forms were used by Ribet to solve the conjecture in some special cases, before Faltings gave a general proof.
		Next I will describe the general form of Tate's conjecture for an arbitrary variety, and discuss how modular forms (or automorphic representations) can be used to approach the problem for modular varieties (like modular curves, Hilbert modular surfaces and Picard modular surfaces), and products of modular varieties.
	www.math.rochester.edu /research/algebra_and_number_theory/10.24.03.html (123 words)

	ArithmeTexas 2005: Texas A&M University, April 2-3, 2005
		Periods of automorphic forms are investigated in many different guises, as integrals, as special values of L-functions, as Fourier coefficients of automorphic forms, and so on.
		To any Hilbert symbol equivalence between two number fields one associates a set of prime ideals, called the wild set of the Hilbert symbol equivalence.
		We will focus on the Hilbert symbol self-equivalences of the field of rational numbers (called rational self-equivalences), and present a characterization of the finite sets of primes that are wild sets for rational self-equivalences.
	www.math.tamu.edu /~map/nt/arithmetexas/schedabs.html (1738 words)

	Preprints
		Keywords: Hilbert modular forms, quaternionic automorphic forms, theta functions, theta correspondence, periods of automorphic forms, Fourier coefficients of automorphic forms.
		This follows from the construction of a space of forms associated to $\lap_\mu$ which satisfy an ``extension by zero'' property.
		The results of this paper extend to the case of differential forms on $M$ with values in a flat Hermitian vector bundle.
	www.math.tcu.edu /Preprints/Preprints.html (725 words)

	Continued Fractions and Modular Forms
		This incursion into the realm of elementary and probabilistic number theory of continued fractions, via modular forms, allows us to study the alternating sum of coefficients of a continued fraction, thus solving the longstanding open problem of their limit law.
		To this aim, let us recall a few facts about modular forms and Kloosterman sums, since these objects appeared to be the key to the asymptotic analysis.
		A modular form of weight k is a holomorphic function on H satisfying:
	algo.inria.fr /banderier/Seminar/vardi00.html (1322 words)

	Periods of Hilbert modular forms and (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		Abstract: Let E be a modular elliptic curve over a totally real eld.
		Chapter 8 of [Dar2] formulates a conjecture allowing the construction of canonical algebraic points on E by suitably integrating the associated Hilbert modular form.
		The main goal of the present work is to obtain numerical evidence for this conjecture in the rst case where it asserts something nontrivial, namely, when E has everywhere good reduction over a real quadratic eld.
	citeseer.ist.psu.edu /604001.html (353 words)

	UIUC Dept. of Mathematics Seminar Calendar
		Congruences for the coefficients of weakly holomorphic modular forms, I
		Abstract: Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli.
		We show that this type of phenomena is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form.
	torus.math.uiuc.edu /cal/math/cal?year=2005&month=02&day=15&interval=day (707 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		It follows that f is a modular form of weight (k, , k) (f (is a modular form of weight (k, , k) (f — f (is a modular form of weight (k, , k).
		Bounding the denominator of (K (1 — k): Corollary (of a theorem to be stated): Two modular forms f and h of weights (k,,k) and (k’,,k’) whose q-expansions have p-integral coefficients and are congruent modulo p, satisfy k(k’ mod (p-1).
		There exist g modular forms in characteristic p EMBED Equation.3 , parameterized by the embeddings of K into a suitable p-adically complete field, such that the following holds: The weight of EMBED Equation.3 is (0,...,0, p,0,, 0, -1, 0, , 0).
	www.math.mcgill.ca /goren/Colloquium.doc (1728 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Start with a mod p representation of the Galois group of Q which is known to be modular.
		If you're lucky, this will be known to be modular, because of results of Jerry Tunnell (on base change).
		You use Hilbert irreducibility and the Cebotarev density theorem (in some way that hasn't yet sunk in) to produce a non-cuspidal rational point of X over which the covering remains irreducible.
	www.cs.rpi.edu /~kennyz/doc/humor/fermat.proof (602 words)

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