$q$-Analogues of some supercongruences related to Van Hamme's (E.2) and (F.2) supercongruences

Liton Karmakar; Arijit Jana

Home/Archive/arXiv

arXiv · 2610.11786

$q$-Analogues of some supercongruences related to Van Hamme's (E.2) and (F.2) supercongruences

Liton Karmakar·Arijit Jana

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we present a one-parameter extension of some supercongruences related to Van Hamme's $\mathrm{(E.2)}$ and $\mathrm{(F.2)}$ supercongruences. To deduce these results, we first establish the corresponding $q$-supercongruences modulo the third power of a cyclotomic polynomial. Our proofs rely on the \emph{creative microscoping} method devised by Guo and Zudilin~\cite{guo2019q}, together with Jackson's ${}_{6}ϕ_{5}$ summation formula.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Liton Karmakar, Arijit Jana. 2026-10-08. $q$-Analogues of some supercongruences related to Van Hamme's (E.2) and (F.2) supercongruences. https://arxiv.org/abs/2610.11786

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Solvable extensions of number fields ramified at only one prime are Ostrowski

In this note, we show that, under a certain condition, solvable extensions of number fields ramifed at only one prime are Ostrowski. As a corollary, we deduce a generalization of Hilbert Theorem 94 to cyclic extensions ramifed at one prime.

math.NT↗

arXiv2026

The integer group determinants for GA(1,p) and related semidirect products

We consider the integer group determinants for groups that are semidirect products of $\mathbb Z_p$ and $\mathbb Z_n$ with $p$ prime and $n\mid p-1$. We give a complete description of the integer group determinants for the general affine groups of degree one GA(1,$p$) when $p=5,7,11$ and $23$, and for $\mathbb Z_7\rtimes \mathbb Z_3,$ $\mathbb Z_{11}\rtimes \mathbb Z_5$ and $\mathbb Z_{13}\rtimes \mathbb Z_6,$ showing that the obvious divisibility and congruence conditions arising from the form of the group determinant when $n=p-1$ or $\frac{1}{2}(p-1)$, can be sufficient as well as necessary for these types of groups (although in the latter case we must work with norms of integers in a quadratic field). For $p=13$ this also happens for the remaining groups of this type, $\mathbb Z_{13}\rtimes_5 \mathbb Z_4$ and $\mathbb Z_{13}\rtimes \mathbb Z_3$, (working in an appropriate cubic and quartic field).

math.NT↗

arXiv2026

The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny

We determine the probability that a random Weierstrass equation with coefficients in the $p$-adic integers defines an elliptic curve with a non-trivial $3$-torsion point, or with a degree $3$ isogeny, defined over the field of $p$-adic numbers. We determine these densities by calculating the corresponding $p$-adic volume integrals and analyzing certain modular curves. Additionally, we explore the case of $\ell$-torsion for $\ell>3$ prime.

math.NT↗