Research
Research
Research
Research
44+ open-access research outputs.
By extending the notion of spin of prime ideals, we show that a short character sum conjecture implies that the set of primes raising the level of a certain even Galois representation has density 2/3,…
In this work, we show that given a finite p-group G, a number field K having a trivial p-class group Cl K , and a finite set of primes S of K, there exists a finite extension F/K such that the S-split…
Using the WZ method to prove supercongruences critically depends on an inspired WZ pair choice. This paper demonstrates a procedure for finding WZ pair candidates to prove a given supercongruence. Whe…
We describe a class of Lie superalgebras in characteristic $3$, containing the Elduque-Cunha superalgebras $\mathfrak{g}(3,3), \mathfrak{g}(6,6)$ and the Elduque superalgebra $\mathfrak{el}(5,3)$, usi…
Let $p\geq 5$ be a prime number. In this paper, we construct Galois representations associated with modular forms for which the dimension of the $p$-torsion in the Bloch-Kato Selmer group can be made …
By the FKG inequality for FK-percolation, the probability of the alternating two-arm event is smaller than the product of the probabilities of having a primal arm and a dual arm, respectively. In this…
This is an expository paper intended to introduce the polynomial time lattice basis reduction algorithm first described by Arjen Lenstra, Hendrik Lenstra, and L\'aszl\'o Lov\'asz in 1982. We begin by …
In this paper we generalize the forcing ramification argument of Khare-Ramakrishna to the setting of the lifting result by Fakhruddin-Khare-Patrikis. In particular, we show that in the relative deform…
The `odd cover number' of a complete graph is the smallest size of a family of complete bipartite graphs that covers each edge an odd number of times. For $n$ odd, Buchanan, Clifton, Culver, Nie, O'Ne…
In this paper we obtain explicit formulas for the traces of Hecke operators on spaces of cusp forms in certain instances related to arithmetic triangle groups. These expressions are in terms of hyperg…
In the early 2000s, Ramakrishna asked the question: For the elliptic curve $$ E: y^2 = x^3 - x, $$ what is the density of primes $p$ for which the Fourier coefficient $a_p(E)$ is a cube modulo $p$? As…
If an $n$-uniform hypergraph can be 2-colored, then it is said to have property B. Erd\H{o}s (1963) was the first to give lower and upper bounds for the minimal size $m(n)$ of an $n$-uniform hypergrap…
Cohen and Taylor introduced Plesken Lie algebra as certain Lie algebra constructed using finite groups. Arjun and Romeo described the linear representation of these Lie algebras induced from group rep…
We prove a new lower bound for the almost 20 year old problem of determining the smallest possible size of an essential cover of the $n$-dimensional hypercube $\{\pm 1\}^n$, i.e. the smallest possible…
In this paper, we mainly establish two supercongruences involving truncated hypergeometric series by using some hypergeometric transformation formulas. The first supercongruence confirms a recent conj…
We study the problem of extracting randomness from somewhere-random sources, and related combinatorial phenomena: partition analogues of Shearer's lemma on projections. A somewhere-random source is …
Using a ${}_7F_6$ hypergeometric transformation formula, we prove two supercongruences. In particular, one of these supercongruences confirms a recent conjecture of Guo, Liu and Schlosser, and gives a…
An essential cover of the vertices of the $n$-cube $\{0,1\}^n$ by hyperplanes is a minimal covering where no hyperplane is redundant and every variable appears in the equation of at least one hyperpla…
Given a finite simple graph $G$, an odd cover of $G$ is a collection of complete bipartite graphs, or bicliques, in which each edge of $G$ appears in an odd number of bicliques and each non-edge of $G…
The $n$-queens problem is to determine $\mathcal{Q}(n)$, the number of ways to place $n$ mutually non-threatening queens on an $n \times n$ board. We show that there exists a constant $\alpha = 1.942 …
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