Thesis advisor: Mark Reeder / This thesis contributes to the construction of supercuspidal representations in small residual characteristics. Let G be a connected, quasi-split, semisimple reductive algebraic group defined and quasi-split over a non-archimedean local field k and splitting over a tamely, totally ramified extension of k. To each parahoric subgroup of G(k), Moy and Prasad have attached a natural filtration by compact open subgroups, the first of which is called the pro-unipotent radical of the parahoric subgroup. The first main result of this thesis is to characterize shallow characters of a pro-unipotent radical, those being complex characters that vanish on the smallest Moy-Prasad subgroup containing all commutators of linearly-dependent affine k-root groups. Through low-rank examples, we illustrate how this characterization can be used to explicitly construct all shallow characters. Next, we provide a natural sufficient condition under which a shallow character compactly induces as a direct sum of supercuspidal representations of G(k). Through examples, however, we show that this sufficient condition need not be necessary, all while constructing new supercuspidal representations of Sp_4(k) when p = 2 and the split form of G_2 over k when p = 3. This work extends the construction of the simple supercuspidal representations given by Gross and Reeder and the epipelagic supercuspidal representations given by Reeder and Yu. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis advisor: Dubi Kelmer / In complex analysis, analytic continuation is a common and important tool to study the properties of complex functions. In this paper, we will introduce and define the global zeta function with an associated function f, where f is a polynomial with n variables with integer coefficients. With the usage of p-adic integration, we can conclude that the global zeta function is analytically continued to all s ∈ C when f is the sum of squares with n variables. / Thesis (BS) — Boston College, 2022. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Departmental Honors. / Discipline: Mathematics.

We discuss two versions of the Hecke algebra of a locally profinite group G, one that is complex valued and one that is p-adic valued. We reproduce several results which are well known for the complex valued Hecke algebra for the p-adic valued Hecke algebra. Specifically we show the equivalence of smooth representations of G and smooth modules of the Hecke algebra of G. We specialize to the group G=GLn(F) for F an extension of Qp, and show that the spherical Hecke algebra of G is finitely generated, and exhibit its generators. This is a standard fact for the complex valued Hecke algebra that we reproduce for the p-adic valued case. We then show that the spherical Hecke algebra of SLnF is isomorphic to a subalgebra of the spherical Hecke algebra of GLnF. Then a character of the spherical Hecke algebra ofGLn(F) can also be viewed as a character of the spherical Hecke algebra of SLn(F). Therefore such a character has two induced modules, one for the Hecke algebra of GLn(F) and another for the Hecke algebra of SLn(F). Theorem 3.4.3 and corollary 3.4.4give a condition under which the coinduced and induced modules of such a character areisomorphic as vector spaces.

Let $\mathbb{G}$ be a connected reductive group defined over a p-adic field F which splits over a tamely ramified extension of F, and let G = $\mathbb{G}(F)$. We also assume that the residual characteristic of F does not divide the order of the Weyl group of $\mathbb{G}$. Following J.K. Yu's construction, the irreducible supercuspidal representation constructed from the G-datum $\Psi$ is denoted $\pi_G(\Psi)$. The datum $\Psi$ contains an irreducible depth-zero supercuspidal representation, which we refer to as the depth-zero part of the datum. Under our hypotheses, the J.K. Yu Construction is exhaustive. Given a connected reductive F-subgroup $\mathbb{H}$ that contains the derived subgroup of $\mathbb{G}$, we study the restriction $\pi_G(\Psi)|_H$ and obtain a description of its decomposition into irreducible components along with their multiplicities. We achieve this by first describing a natural restriction process from which we construct H-data from the G-datum $\Psi$. We then show that the obtained H-data, and conjugates thereof, construct the components of $\pi_G(\Psi)|_H$, thus providing a very precise description of the restriction. Analogously, we also describe an extension process that allows to construct G-data from an H-datum $\Psi_H$. Using Frobenius Reciprocity, we obtain a description for the components of $\Ind_H^G\pi_H(\Psi_H)$. From the obtained description of $\pi_G(\Psi)|_H$, we prove that the multiplicity in $\pi_G(\Psi)|_H$ is entirely determined by the multiplicity in the restriction of the depth-zero piece of the datum. Furthermore, we use Clifford theory to obtain a formula for the multiplicity of each component in $\pi_G(\Psi)|_H$. As a particular case, we take a look at the regular depth-zero supercuspidal representations and obtain a condition for a multiplicity free restriction. Finally, we show that our methods can also be used to define a restriction of Kim-Yu types, allowing to study the restriction of irreducible representations which are not supercuspidal.

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<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>

在這篇論文裡, 我們探討體中之賦值的一般理論. 最主要的, 我們證明了多個賦值等價條件. 進而我們探討p進位數體中的分析, 得到ㄧ些新的現象與例子. / In this thesis, we study some general theory of valuations on a field. Especially, we obtain several equivalent conditions on the equivalence of two valuations on a field, some of them are new in the literature. Moreover, we study the $p$-adic analysis on the p-adic number fields and obtain some new phenomena and examples.

In this thesis, we report three preprints [Li17a] [Li17b] and [HL17] the author wrote (the last one was written jointly with D. Hansen) during his pursuing of PhD at Columbia. We study smooth proper rigid varieties which admit formal models whose special fibers are projective. The main theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that non-archimedean Hopf varieties do not have a projective reduction. The proof of our main theorem uses the theory of moduli of semistable coherent sheaves. Combine known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory, We then prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over p-adic fields.

Let R(G) denote the category of smooth representations of a p-adic group. Bernstein has constructed an indexing set B(G) such that R(G) decomposes into a direct sum over s ∈ B(G) of full subcategories Rs(G) known as Bernstein subcategories. Bushnell and Kutzko have developed a method to study the representations contained in a given subcategory. One attempts to associate to that subcategory a smooth irreducible representation (τ,W) of a compact open subgroup J < G. If the functor V ↦ HomJ(W,V) is an equivalence of categories from Rs(G) → H(G,τ)mod we call (J,τ) a type. Given a Levi subgroup L < G and a type (JL, τL) for a subcategory of representations on L, Bushnell and Kutzko further show that one can construct a type on G that “lies over” (JL, τL) by constructing an object known as a cover. In particular, a cover implements induction of H(L,τL)-modules in a manner compatible with parabolic induction of L-representations. In this thesis I construct a cover for certain representations of the Siegel Levi subgroup of Sp(2k) over an archimedean local field of characteristic zero. In partic- ular, the representations I consider are twisted by highly ramified characters. This compliments work of Bushnell, Goldberg, and Stevens on covers in the self-dual case. My construction is quite concrete, and I also show that the cover I construct has a useful property known as splitness. In fact, I prove a fairly general theorem characterizing when covers are split.