<p dir="ltr">In this thesis, we prove the rationality results for the ratio of the critical values of certain <i>L</i>-functions, which appear in the constant term of Eisenstein series associated with the exceptional group <i>G</i>_<em>2</em> over a totally imaginary field. Our methodology builds upon the works of Harder and Raghuram, who established rationality results for special values of Rankin-Selberg <i>L</i>-functions for<i> </i><i>GL</i>_<em>n</em><i>× GL</i>_<em>n'</em> by studying the rank-one Eisenstein cohomology of the ambient group <i>GL</i>_n+n' over a totally real field, as well as its generalization by Raghuram [35] for the case over a totally imaginary field.</p><p dir="ltr">The <i>L</i>-functions in this thesis were constructed using the Langlands-Shahidi method for <i>G</i>_<em>2</em> over a totally imaginary field, attached to maximal parabolic subgroups. This is the first instance of applying the Harder-Raghuram method to an exceptional group, and the first case involving more than one function appearing in the constant term. Our results demonstrate the relationship between the rationality of different <i>L</i>-functions appearing in the constant term, allowing one to prove the rationality of one <i>L</i>-function based on the known rationality result of another <i>L</i>-functions.</p>

<p dir="ltr">In this article, we establish results concerning the nilpotent representation theory of groups. In particular, we utilize a theorem of Stallings to provide a general method that constructs pairs of groups that have isomorphic universal nilpotent quotients. We then prove by counterexample that absolute Galois groups of number fields are not determined by their universal nilpotent quotients. We also show that this is the case for residually nilpotent Kleinian groups and in fact, there exist non-isomorphic pairs that have arbitrarily large nilpotent genus. We additionally provide examples of non-isomorphic curves whose geometric fundamental groups have isomorphic universal nilpotent quotients and the isomorphisms are compatible with the outer Galois actions. </p>

There has been in the past ten to fifteen years a surge of activity concerning the cohomology of semi-direct product groups of the form $\mathbb{Z}^{n}\rtimes$G with G finite. A problem first stated by Adem-Ge-Pan-Petrosyan asks for suitable conditions for the Lyndon-Hochschild-Serre Spectral Sequence associated to this group extension to collapse at second page of the Lyndon-Hochschild-Serre spectral sequence. In this thesis we use facts from integer representation theory to reduce this problem to only considering representatives from each genus of representations, and establish techniques for constructing new examples in which the spectral sequence collapses.

<div> <div> <div> <p>Let K be a discrete valuation field with perfect residue field, we study the functor from weakly admissible filtered (φ,N,G_K)-modules over K to the isogeny category of Breuil- Kisin-Fargues G_K-modules. This functor is the composition of a functor defined by Fargues-Fontaine from weakly admissible filtered (φ,N,G_K)-modules to G_K-equivariant modifications of vector bundles over the Fargues-Fontaine curve X_FF , with the functor of Fargues-Scholze that between the category of admissible modifications of vector bundles over X_FF and the isogeny category of Breuil-Kisin-Fargues modules. We characterize the essential image of this functor and give two applications of our result. First, we give a new way of viewing the p-adic monodromy theorem of p-adic Galois representations. Also we show our theory provides a universal theory that enable us to compare many integral p-adic Hodge theories at the A_inf level. </p> </div> </div> </div>

Square Form Factoring is an <i>O</i>(<i>N</i>^1/4) factoring algorithm developed by D. Shanks using certain properties of quadratic forms. Central to the original algorithm is an iterative search for a square form. We propose a new subexponential-time algorithm called SQUFOF2, based on ideas of D. Shanks and R. de Vogelaire, which replaces the iterative search with a sieve, similar to the Quadratic Sieve.