arXiv subjects
Publications and source records attributed to Gustavo Rama.
We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the relative class number two problem in all cases where the base field has constant field not equal to $\mathbb{F}_2$.
We examine the capacity of the complementarity spectrum to distinguish non-isomorphic digraphs. We focus on the seven families with exactly three complementarity eigenvalues. Our findings reveal that in some, but not all families, any two non-isomorphic members have different complementarity spectrum. Complementarity eigenvalues outperform traditional eigenvalues in the task of identifying graphs. Indeed, the question of whether graphs are uniquely determined by their complementarity spectrum remains unresolved, highlighting the significance of this tool in graph theory. Moreover, since the complementarity spectrum of a digraph was characterized as the set of spectral radii of the induced strongly connected subdigraphs, the results of this study provides useful structural information for important families of digraphs.
In this paper we present results of several experiments in which we model the repulsion of low-lying zeros of L-functions using random matrix theory. Previous work has typically focused on the twists of L-functions associated to elliptic curves and on families that can be modeled by unitary and orthogonal matrices. We consider families of L-function of modular forms of weight greater than 2 and we consider families that can be modeled by symplectic matrices. Additionally, we explore a model for low-lying zeros of twists that incorporates a discretization arising from the Kohnen--Zagier theorem. Overall, our numeric evidence supports the expectation that the repulsion of zeros decreases as the conductor of the twist increases. Surprisingly, though, it appears that using the discretization that arises from the Kohnen--Zagier theorem does not model the data better than if the discretization is not used for forms of weight 4 or above.
We compute tables of paramodular forms of degree two and cohomological weight via a correspondence with orthogonal modular forms on quinary lattices.
We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev.
For a finite dimensional algebra $A$ with $0 < ϕdim (A) = m < \infty$ we prove that there always exist modules $M$ and $N$ such that $ϕ(M) = m-1$ and $ϕ(N) = 1$. On the other hand, we see an example of an algebra that not every value between $1$ and its $ϕ$-dimension is reached by the $ϕ$ function. We call that values gaps and we prove that the algebras with gaps verifies the finitistic conjecture.
Given a truncated path algebra $A=\frac{\Bbbk Q}{J^k}$ we prove that $\fidim A = \fidim A^{\op}$. We also compute the $ϕ$-dimension of $A$ in function of the $ϕ$-dimension of $\frac{\Bbbk Q}{J^2}$ when $Q$ has no sources nor sinks. This allows us to bound the $ϕ$-dimension for truncated path algebras. Finally, we characterize $A$ when its $ϕ$-dimension is equal to $1$.