scholarly journals A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform

2015 ◽  
Vol 11 (08) ◽  
pp. 2431-2450 ◽  
Author(s):  
Dermot McCarthy ◽  
Matthew A. Papanikolas
Keyword(s):  
Finite Field ◽  
Hypergeometric Function ◽  
Modular Forms ◽  
Hypergeometric Functions ◽  
Siegel Modular Forms ◽  
Hecke Operator ◽  
Special Value ◽  
Forms Of Higher Degree

Although links between values of finite field hypergeometric functions and eigenvalues of elliptic modular forms are well known, we establish in this paper that there are also connections to eigenvalues of Siegel modular forms of higher degree. Specifically, we relate the eigenvalue of the Hecke operator of index p of a Siegel eigenform of degree 2 and level 8 to a special value of a 4F3-hypergeometric function.

scholarly journals Spanning the isogeny class of a power of an elliptic curve

2021 ◽  
Vol 91 (333) ◽  
pp. 401-449
Author(s):  
Markus Kirschmer ◽  
Fabien Narbonne ◽  
Christophe Ritzenthaler ◽  
Damien Robert
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Elliptic Curve ◽  
Finite Fields ◽  
Modular Forms ◽  
Null Point ◽  
Siegel Modular Forms ◽  
Content Type ◽  
Careful Choice ◽  
Isomorphism Classes

Let E E be an ordinary elliptic curve over a finite field and g g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E g E^g . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E 3 E^3 and of the Igusa modular form in dimension 4 4 . We illustrate our algorithms with examples of curves with many rational points over finite fields.

scholarly journals Hypergeometric functions and relations to Dwork hypersurfaces

2017 ◽  
Vol 13 (02) ◽  
pp. 439-485 ◽  
Author(s):  
Heidi Goodson
Keyword(s):  
Finite Field ◽  
Hypergeometric Function ◽  
Finite Fields ◽  
Hypergeometric Functions ◽  
K3 Surfaces ◽  
Point Count ◽  
Higher Dimensional ◽  
The Family ◽  
Trace Of Frobenius ◽  
The Relationship

We give an expression for number of points for the family of Dwork K3 surfaces over finite fields of order [Formula: see text] in terms of Greene’s finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy’s [Formula: see text]-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.

scholarly journals COUNTING POINTS ON DWORK HYPERSURFACES AND -ADIC HYPERGEOMETRIC FUNCTIONS

2016 ◽  
Vol 94 (2) ◽  
pp. 208-216 ◽  
Author(s):  
RUPAM BARMAN ◽  
HASANUR RAHMAN ◽  
NEELAM SAIKIA
Keyword(s):  
Finite Field ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Counting Points

We express the number of points on the Dwork hypersurface $X_{\unicode[STIX]{x1D706}}^{d}:x_{1}^{d}+x_{2}^{d}+\cdots +x_{d}^{d}=d\unicode[STIX]{x1D706}x_{1}x_{2}\cdots x_{d}$ over a finite field of order $q\not \equiv 1\,(\text{mod}\,d)$ in terms of McCarthy’s $p$-adic hypergeometric function for any odd prime $d$.

scholarly journals Higher pullbacks of modular forms on orthogonal groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Brandon Williams
Keyword(s):  
Modular Forms ◽  
Differential Operators ◽  
Jacobi Form ◽  
Siegel Modular Forms ◽  
Orthogonal Groups

Abstract We apply differential operators to modular forms on orthogonal groups O ⁢ ( 2 , ℓ ) {\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature ( 2 , 2 ) {(2,2)} and ( 2 , 3 ) {(2,3)} , for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

Recursion formulas for multivariable hypergeometric functions

2015 ◽  
Vol 08 (04) ◽  
pp. 1550082 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Radiation Field ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Recursion Formulas ◽  
Appl Math ◽  
Appell Functions

Recently, Opps, Saad and Srivastava [Recursion formulas for Appell’s hypergeometric function [Formula: see text] with some applications to radiation field problems, Appl. Math. Comput. 207 (2009) 545–558] presented the recursion formulas for Appell’s function [Formula: see text] and also gave its applications to radiation field problems. Then Wang [Recursion formulas for Appell functions, Integral Transforms Spec. Funct. 23(6) (2012) 421–433] obtained the recursion formulas for Appell functions [Formula: see text] and [Formula: see text]. In our investigation here, we derive the recursion formulas for 14 three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions.

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