On transformation formulas of $p$-adic hypergeometric functions

Technical Summary: Transformation Formulas of p-Adic Hypergeometric Functions

Problem Statement
This paper addresses the relationship between transformation formulas for three distinct classes of $p$-adic hypergeometric functions: Dwork's functions ($F^{Dw}_a(t)$), and two logarithmic-type functions introduced by Asakura and Wang ($F^{(\sigma)}_a(t)$ and $\tilde{F}^{(\sigma)}_a(t)$). Specifically, the authors investigate two conjectured transformation formulas under the inversion symmetry $t \mapsto t^{-1}$:

1. Conjecture 5.5: A transformation formula for Dwork's functions, relating $F^{Dw}_a(t)$ to $F^{Dw}_a(t^{-1})$ via a sign factor and a power of $t$.
2. Conjecture 5.2: A transformation formula relating the logarithmic-type functions $F^{(\sigma)}_a(t)$ and $\tilde{F}^{(\sigma)}_a(t)$, positing that $F^{(\sigma)}_a(t) = -\tilde{F}^{(\sigma)}_a(t^{-1})$ (with appropriate Frobenius adjustments).

While Conjecture 5.5 had been proven for specific cases (e.g., $s=2$ or under conditions on $a$ and $p$ by Nemoto), and Conjecture 5.2 was known to hold modulo $p$ or in specific geometric settings, the logical dependency between these two conjectures remained unestablished. The paper seeks to determine if the validity of the transformation formula for Dwork's functions implies the validity of the transformation formula for the logarithmic-type functions.

Methodology
The authors employ a combination of $p$-adic analysis, congruence relations, and coefficient manipulation within the framework of Witt rings and formal power series.

1. Definitions and Preliminaries: The paper reviews the definitions of the three hypergeometric functions. $F^{Dw}_a(t)$ is defined as the ratio of hypergeometric power series $F_a(t)/F_{a'}(t^p)$. The logarithmic functions $F^{(\sigma)}_a(t)$ and $\tilde{F}^{(\sigma)}_a(t)$ are defined via integrals involving $p$-adic digamma functions ($\psi_p$), Euler constants ($\gamma_p$), and Frobenius endomorphisms.
2. Coefficient Analysis: The core technical work involves analyzing the coefficients of the power series expansions of these functions. The authors define normalized coefficients $\beta_k = B_k/A_k$ and $\tilde{\beta}_k = \tilde{B}_k/A_k$, where $A_k$ are coefficients of the base hypergeometric series, and $B_k, \tilde{B}_k$ are coefficients of the numerators of the logarithmic functions.
3. Key Lemma (Lemma 4.2): A central result established is a functional relation between the $p$-adic interpolations of these coefficients: $\beta_\lambda + \tilde{\beta}_{-\lambda-a} = 0$ for $\lambda \in \mathbb{Z}_p$. This relation is proven using properties of the $p$-adic Gamma function and specific congruences involving the Pochhammer symbols.
4. Congruence Reduction: The proof of the main implication relies on Lemma 5.9, which demonstrates that if Conjecture 5.5 holds, then specific sums of products of coefficients of the base series satisfy a congruence relation modulo $p^{d+1}$.
5. Inductive Argument: Theorem 5.10 utilizes the coefficient relation from Lemma 4.2 and the congruence derived from Conjecture 5.5 to show that the transformation formula for the logarithmic functions (Conjecture 5.2) holds modulo $p^n$ for all $n$, thereby establishing the identity in the relevant ring of convergent functions.

Key Contributions and Results

• Main Theorem (Theorem 5.10): The primary result is the proof that Conjecture 5.5 implies Conjecture 5.2. Specifically, if the transformation formula for Dwork's $p$-adic hypergeometric functions holds, then the transformation formula relating $F^{(\sigma)}_a(t)$ and $\tilde{F}^{(\sigma)}_a(t)$ necessarily follows.
• Correction of Conjecture 5.5: The authors identify and correct an error in the original formulation of Conjecture 5.5 found in prior literature ([9, Conjecture 4.18]). They establish that for $p=2$, a sign ambiguity exists depending on the parity of $s$ and the Dwork prime $a'$, which was previously overlooked.
• New Cases for Conjecture 5.2: By combining their main theorem with existing results on Conjecture 5.5 (specifically Theorem 5.8 by Nemoto and Example 3 for $s=1$), the authors deduce that Conjecture 5.2 is true in the following cases:
  • When $s=1$ (Corollary 5.11).
  • When $s \ge 2$, $a \in N^{-1}\mathbb{Z}$, $0 < a < 1$, and $p \nmid N$ for some $N \ge 2$ (Corollary 5.12).

Significance
The paper claims significance primarily as a theoretical bridge connecting two distinct areas of $p$-adic hypergeometric theory. By establishing that the transformation formula for Dwork's functions (which have geometric applications related to unit roots of elliptic curves) implies the transformation formula for the logarithmic-type functions (which are linked to syntomic regulators and the $p$-adic Beilinson conjecture), the work unifies these frameworks.

The authors state that this implication allows for the deduction of new cases for Conjecture 5.2 without requiring new geometric proofs, leveraging the recent generalizations of Conjecture 5.5 by Nemoto. The paper does not propose new experimental applications or future directions beyond the scope of proving these specific transformation formulas and their logical interdependence.