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Academic literature on the topic 'Quaternionic modular forms'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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Journal articles on the topic "Quaternionic modular forms"

1

TERRACINI, LEA. "A WEIGHT INDEPENDENCE RESULT FOR QUATERNIONIC HECKE ALGEBRAS." International Journal of Number Theory 09, no. 08 (2013): 1895–922. http://dx.doi.org/10.1142/s1793042113500656.

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Let p be a prime and B be a quaternion algebra indefinite over Q and ramified at p. We consider the space of quaternionic modular forms of weight k and level p∞, endowed with the action of Hecke operators. By using cohomological methods, we show that the p-adic topological Hecke algebra does not depend on the weight k. This result provides a quaternionic version of a theorem proved by Hida for classical modular forms; we discuss the relationship of our result to Hida's theorem in terms of Jacquet–Langlands correspondence.

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BRASCA, RICCARDO. "QUATERNIONIC MODULAR FORMS OF ANY WEIGHT." International Journal of Number Theory 10, no. 01 (2014): 31–53. http://dx.doi.org/10.1142/s1793042113500796.

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In this work we give a geometric definition, as sections of line bundles, of p-adic analytic families of overconvergent modular forms attached to an indefinite quaternion algebra over ℚ. As a consequence of this, we obtain the existence of an eigencurve in this context. Our theory includes the interpretation of a modular form as a rule on test objects. We introduce the Hecke operators U and T l, both in families and for a single weight. We show that the U -operator acts compactly on the space of overconvergent modular forms. We finally construct the eigencurve, a rigid analytic variety whose points correspond to systems of eigenvalues associated to overconvergent eigenforms of finite slope with respect to the U -operator.

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3

Howe, Sean. "The spectral p-adic Jacquet–Langlands correspondence and a question of Serre." Compositio Mathematica 158, no. 2 (2022): 245–86. http://dx.doi.org/10.1112/s0010437x22007308.

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We show that the completed Hecke algebra of $p$ -adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$ -adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$ . This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and $p$ -adic quaternionic eigenforms.

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Freitag, Eberhard, and Riccardo Salvati Salvati Manni. "Vector-valued Hermitian and quaternionic modular forms." Kyoto Journal of Mathematics 55, no. 4 (2015): 819–36. http://dx.doi.org/10.1215/21562261-3157757.

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5

Sharma, Anuradha, and Amit K. Sharma. "Byte weight enumerators and modular forms of genus r." Journal of Algebra and Its Applications 14, no. 06 (2015): 1550080. http://dx.doi.org/10.1142/s0219498815500802.

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For a positive integer m, let R be either the ring ℤ2m of integers modulo 2m or the quaternionic ring Σ2m = ℤ2m + αℤ2m + βℤ2m + γℤ2m with α = 1 + î, β = 1 + ĵ and [Formula: see text], where [Formula: see text] are elements of the ring ℍ of real quaternions satisfying [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we obtain Jacobi forms (or Siegel modular forms) of genus r from byte weight enumerators (or symmetrized byte weight enumerators) in genus r of Type I and Type II codes over R. Furthermore, we derive a functional equation for partial Epstein zeta functions, which are summands of classical Epstein zeta functions associated with quadratic forms.

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6

Arenas, A. "On Hilbert and quaternionic cusp forms." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 1 (2006): 1–6. http://dx.doi.org/10.1017/s0308210500004406.

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The aim of this paper is to determine in a natural manner the subspace of the space of Hilbert modular newforms of level n which correspond to eigenforms of an appropriate quaternion algebra, in the sense of having the same eigenvalues with respect to the corresponding Hecke operators. This study may be seen as a particular case of the Jacquet–Langlands correspondence.

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7

Baba, Srinath, and Håkan Granath. "Quaternionic modular forms and exceptional sets of hypergeometric functions." International Journal of Number Theory 11, no. 02 (2015): 631–43. http://dx.doi.org/10.1142/s1793042115500347.

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We determine the exceptional sets of hypergeometric functions corresponding to the (2, 4, 6) triangle group by relating them to values of certain quaternionic modular forms at CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.

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8

Dou, Ze-Li. "On adelic automorphic forms with respect to a quadratic extension." Bulletin of the Australian Mathematical Society 62, no. 1 (2000): 29–43. http://dx.doi.org/10.1017/s000497270001844x.

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Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

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9

Salazar, Daniel Barrera, та Shan Gao. "Overconvergent Eichler–Shimura isomorphisms for quaternionic modular forms over ℚ". International Journal of Number Theory 13, № 10 (2017): 2687–715. http://dx.doi.org/10.1142/s1793042117501494.

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In this work, we construct overconvergent Eichler–Shimura isomorphisms over Shimura curves over [Formula: see text]. More precisely, for a prime [Formula: see text] and a wide open disk [Formula: see text] in the weight space, we construct a Hecke–Galois-equivariant morphism from the space of families of overconvergent modular symbols over [Formula: see text] to the space of families of overconvergent modular forms over [Formula: see text]. In addition, for all but finitely many weights [Formula: see text], this morphism provides a description of the finite slope part of the space of overconvergent modular symbols of weight [Formula: see text] in terms of the finite slope part of the space of overconvergent modular forms of weight [Formula: see text]. Moreover, for classical weights these overconvergent isomorphisms are compatible with the classical Eichler–Shimura isomorphism.

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10

Baba, Srinath, and Håkan Granath. "Differential equations and expansions for quaternionic modular forms in the discriminant 6 case." LMS Journal of Computation and Mathematics 15 (December 1, 2012): 385–99. http://dx.doi.org/10.1112/s146115701200112x.

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AbstractWe study the differential structure of the ring of modular forms for the unit group of the quaternion algebra over ℚ of discriminant 6. Using these results we give an explicit formula for Taylor expansions of the modular forms at the elliptic points. Using appropriate normalizations we show that the Taylor coefficients at the elliptic points of the generators of the ring of modular forms are all rational and 6-integral. This gives a rational structure on the ring of modular forms. We give a recursive formula for computing the Taylor coefficients of modular forms at elliptic points and, as an application, give an algorithm for computing modular polynomials.

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Dissertations / Theses on the topic "Quaternionic modular forms"

1

Butenuth, Ralf [Verfasser], and Gebhard [Akademischer Betreuer] Böckle. "Quaternionic Drinfeld modular forms / Ralf Butenuth ; Betreuer: Gebhard Böckle." Heidelberg : Universitätsbibliothek Heidelberg, 2012. http://d-nb.info/1177039494/34.

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2

Gehre, Dominic Steffen [Verfasser]. "Quaternionic modular forms of degree two over Q(-3,-1) / Dominic Steffen Gehre." Aachen : Hochschulbibliothek der Rheinisch-Westfälischen Technischen Hochschule Aachen, 2013. http://d-nb.info/1031115951/34.

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3

BRASCA, RICCARDO. "P-ADIC MODULAR FORMS OF NON-INTEGRAL WEIGHT OVER SHIMURA CURVES." Doctoral thesis, Università degli Studi di Milano, 2012. http://hdl.handle.net/2434/172626.

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In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of k-th invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid-analytic space, that parametrizes the weights. Finally, we define Hecke operators. We focus on the U operator, showing that it is completely continuous on the space of overconvergent modular forms.

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4

Daghigh, Hassan. "Modular forms, quaternion algebras, and special values of L-functions." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape11/PQDD_0006/NQ44398.pdf.

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5

Daghigh, Hassan. "Modular forms, quaternion algebras, and special values of L-functions." Thesis, McGill University, 1997. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=34938.

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Let f be a cusp form of weight 2 and level N. Let K be an imaginary quadratic field of discriminant, --D, and A an ideal class of K. We obtain precise formulas for the special values of the L-functions associated to the Rankin convolution of f and a theta series associated to the ideal class A , in terms of the Petersson scalar product of f with the theta series associated to an Eichler order in a positive definite quaternion algebra. Our work is an extension of the work done by Gross [7]. The central tools used in this thesis are Rankin's method and a reformulation of Gross of work of Waldspurger concerning central critical values.

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Socrates, Jude Thaddeus U. "The quaternionic bridge between elliptic curves and Hilbert modular forms." Thesis, 1993. https://thesis.library.caltech.edu/7376/2/Socrates_jtu_1993.pdf.

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<p>The main result of this thesis is a matching between an elliptic curve E over F = Q(√509) which has good reduction everywhere, and a normalized holomorphic Hilbert modular eigenform f for F of weight 2 and full level. The curve E is not F-isogenous to its Galois conjugate E^σ and does not possess potential complex multiplication. The eigenform f has rational eigenvalues, does not come from the base change of an elliptic modular form, and does not satisfy f = f ⊗ ε for any quadratic character ε of F associated to a degree 2 imaginary extension of F. We show that a_ρ(E) = a_ρ(f) for a large set Ʃ of σ invariant primes in F. This provides the first known non-trivial example of the conjectural Langlands correspondence (see Section 1.1) in the everywhere unramified case.</p> <p>The method we use exploits the isomorphism between the spaces of holomorphic Hilbert modular cusp forms and quaternionic cusp forms. The construction of f involves explicity constructing a maximal order O in the quaternion algebra B/F which ramified precisely at the finite primes. We determine the type number T_1 of B as well as the class number H_1 for O, which equals T_1 in our case of interest. We found that for Q(√509), T_1 = H_1 = 24. One sees that the space of weight 2 full level cusp forms for F has dimension 23.</p> <p>The main tools are θ-series attached to ideals and Brandt matrices B(ξ) for an order in B for quadratic fields Q (√m) with class number 1 and whose fundamental unit u has nor -1. (Q(√509) is such a field.) The θ-series gives a way to obtain representatives of left O-ideal classes and hence representatives of maximal orders of different type. The Hecke action on quaternionic cusp forms is given by the modified Brandt matrices B'(ξ), hence a set of simultaneous eigenvectors for these matrices corresponds to the normalized eigenforms for F.</p> <p>Applying these algorithms to Q(√509), we prove that there are exactly three normalized eigenforms which have rational eigenvalues for all the Hecke operators. We show that for one of these eigenforms f, a_ρ(f) ≠ a_ρ(f^σ) for certain primes ρ, proving that f does not come form base change. We also note that there is another elliptic curve E'/Q(√509) which is isogenous to its Galois conjugate and hence not isogenous to either E or E^σ. We show that a_ρ(E') = a_ρ(f') ∀ρ ε Ʃ, where f' is the third normalized eigenform that we found above. This is compatible with the expectation that all three non-isogenous elliptic curves correspond to normalized eigenforms with rational eigenvalues.</p>

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7

Hopkins, Kimberly Michele. "Periods of modular forms and central values of L-functions." Thesis, 2010. http://hdl.handle.net/2152/ETD-UT-2010-05-1423.

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This thesis is comprised of three problems in number theory. The introduction is Chapter 1. The first problem is to partially generalize the main theorem of Gross, Kohnen and Zagier to higher weight modular forms. In Chapter 2, we present two conjectures which do this and some partial results towards their proofs as well as numerical examples. This work provides a new method to compute coefficients of weight k+1/2 modular forms for k>1 and to compute the square roots of central values of L-functions of weight 2k>2 modular forms. Chapter 3 presents four different interpretations of the main construction in Chapter 2. In particular we prove our conjectures are consistent with those of Beilinson and Bloch. The second problem in this thesis is to find an arithmetic formula for the central value of a certain Hecke L-series in the spirit of Waldspurger's results. This is done in Chapter 4 by using a correspondence between special points in Siegel space and maximal orders in quaternion algebras. The third problem is to find a lower bound for the cardinality of the principal genus group of binary quadratic forms of a fixed discriminant. Chapter 5 is joint work with Jeffrey Stopple and gives two such bounds.<br>text

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Books on the topic "Quaternionic modular forms"

1

Modular forms on half-spaces of quaternions. Springer-Verlag, 1985.

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2

Krieg, Aloys. Modular Forms on Half-Spaces of Quaternions. Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/bfb0075946.

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3

Hijikata, Hiroaki. The basis problem for modular forms on [Gamma]o(N). American Mathematical Society, 1989.

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4

Modular Forms On Halfspaces Of Quaternions. Springer, 1985.

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5

Krieg, Aloys. Modular Forms on Half-Spaces of Quaternions. Springer London, Limited, 2006.

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Book chapters on the topic "Quaternionic modular forms"

1

Serre, Jean-Pierre. "Two Letters on Quaternions and Modular Forms (modp)." In Springer Collected Works in Mathematics. Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-642-41978-2_37.

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2

Shimura, Goro. "On certain zeta functions attached to two Hilbert modular forms: II. The case of automorphic forms on a quaternion algebra." In Collected Papers. Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4612-2060-2_9.

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