Hypergeometric functions over F_p and relations to elliptic curves and modular forms

By Jenny G. Fuselier. 2008

79 citations recorded by the source corpus.

For primes p ≡ 1 (mod 12), we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728 / t and values of a particular 2 F 1 -hypergeometric function over F p . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's T -function in terms of hypergeometric functions.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/861f95adb5c171b42c81c3dd135adb5079a869ab

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In Algebra and Number Theory

1. C. Lennon
2. Dermot McCarthy
3. Zhi-Wei Sun
4. Wadim Zudilin

In Geometry and Topology

1. Dermot McCarthy
2. C. Lennon

In Mathematical Physics

1. C. Lennon
2. Dermot McCarthy

Citations in the corpus

• The trace of Frobenius of elliptic curves and the p-adic gamma function (2012, background): “Hypergeometric functions over finite fields have been applied to many areas but most interestingly perhaps has been their relationships to modular forms [Ahlgren and Ono 2000; Evans 2010; Fuselier 2010; Frechette et al. 2004; McCarthy 2012b; Mortenson 2005; Ono 1998; Papanikolas 2006] and their use ”
• On a supercongruence conjecture of Rodriguez-Villegas (2012, methodology): “ Combining Theorem 1.3 with d= 5 and Theorem 1.4 yields the result. 4. Remark Using (2.7) to count points on certain algebraic varieties is by no means new. This author first observed the technique in [6]. In the proof above, we have applied this technique in the case p≡ 1 (mod 5). We could have also”
• Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves (2010, methodology): “NIUS 3 the Frobenius map on Ecan be expressed as a(E(Fq)) = −q· T q−1 4 a3 27 · 2F 1 Tq−1 12 T 5(q−1) 12 Tq−1 2 − 27b2 4a3 ! q . Proof. The method of this proof follows similarly to that given in [2]. Let |E(Fq)| denote the number of projective points of Ein Fq. If we let P(x,y) = x3 +ax+b−y2, the”
• Hypergeometric Functions over Finite Fields (2015, unknown): “According to the work of the first author [36], when t ∈ Fp and t 6= 1, i.”