| 1. |
- Baba, Srinath, et al.
(författare)
-
Quaternionic modular forms and exceptional sets of hypergeometric functions
- 2015
-
Ingår i: International Journal of Number Theory. - Singapure : World Scientific Publishing. - 1793-0421. ; 11:2, s. 631-643
-
Tidskriftsartikel (refereegranskat)abstract
- 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 formsat CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.
|
|
| 2. |
- Brzezinski, Juliusz, 1939, et al.
(författare)
-
Smooth lattices over quadratic integers
- 2008
-
Ingår i: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 1432-1823 .- 0025-5874. ; 258:1, s. 161-184
-
Tidskriftsartikel (refereegranskat)abstract
- We construct lattices with quadratic structure over the integers in quadratic number fields having the property that the rank of the quadratic structure is constant and equal to the rank of the lattice in all reductions modulo maximal ideals. We characterize the case in which such lattices are free. The construction gives a representative of the genus of such lattices as an orthogonal sum of "standard" pieces of ranks 1-4 and covers the case of the discriminant of the real quadratic number field congruent to 1 modulo 8 for which a general construction was not known. © 2007 Springer-Verlag.
|
|
| 3. |
- Granath, Håkan, 1969-, et al.
(författare)
-
Differential equations and expansions for quaternionic modular forms in the discriminant 6 case
- 2012
-
Ingår i: LMS Journal of Computation and Mathematics. - : Cambridge University Press. - 1461-1570. ; 15, s. 385-399
-
Tidskriftsartikel (refereegranskat)abstract
- We 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.
|
|
| 4. |
- Granath, Håkan, 1969-
(författare)
-
On inequalities and asymptotic expansions for the Landau constants
- 2012
-
Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 386:2, s. 738-743
-
Tidskriftsartikel (refereegranskat)abstract
- We demonstrate how arbitrarily sharp asymptotic expansions for the Landau constants can be derived, and we prove some related inequalities. The main tool used is Brounckerʼs continued fraction formula. We also show that the implied series expansion of the Landau constants is in fact divergent.
|
|
| 5. |
|
|
| 6. |
- Granath, Håkan, 1969-, et al.
(författare)
-
Orthogonal Systems of Modular Forms and Supersingular Polynomials
- 2011
-
Ingår i: International Journal of Number Theory. - : World Scientific. - 1793-0421. ; 7:1, s. 249-259
-
Tidskriftsartikel (refereegranskat)abstract
- We extend a construction of Kaneko and Zagier to obtain modular forms which, modulo a prime, vanish at the supersingular points. These modular forms arise simultaneously as solutions of certain second order differential equations, and as an orthogonal basis for an inner product on the space of modular forms.
|
|
| 7. |
- Granath, Håkan, 1969-
(författare)
-
The discriminant 10 Shimura curve and its associated Heun functions
- 2016
-
Ingår i: Bulletin of the London Mathematical Society. - : Oxford University Press. - 0024-6093 .- 1469-2120. ; 48, s. 957-967
-
Tidskriftsartikel (refereegranskat)abstract
- The Shimura curve of discriminant 10 is uniformized by a subgroup of an arithmetic $(2,2,2,3)$ quadrilateral group. We derive the differential structure of the ring of modular forms for the Shimura curve and relate the ring generators to explicit Heun functions for the quadrilateral group. We also show that the Picard–Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms. These results are used to completely describe the exceptional sets of the Heun functions, and we show how to find examples like \[ Hl\left(\frac{27}{2},\frac{7}{36}; \frac{1}{12},\frac{7}{12},\frac{2}{3},\frac{1}{2}; -\frac{96}{25}\right)=\frac{2^{1/2}5^{2/3}}{3^{4/3}}. \]
|
|
