<p dir="ltr">We investigate several diophantine problems in many variables through analytic method.</p>

<pre>This dissertation concerns two questions involving the injectivity of specialization homomorphisms for elliptic surfaces. We primarily focus on elliptic surfaces over the projective line defined over the rational numbers. The specialization theorem of Silverman proven in 1983 says that, for a fixed surface, all but finitely many specialization homomorphisms are injective. Given a subgroup of the group of rational sections with explicit generators, we thus ask the following.</pre><pre>Given some rational number, how can we effectively determine whether or not the associated specialization map is injective?</pre><pre>What is the set of rational numbers such that the corresponding specialization maps are injective?</pre><pre>The classical specialization theorem of Neron proves that there is a set S which differs from a Hilbert subset of the rational numbers by finitely many elements such that for each number in S the associated specialization map is injective. We expand this into an effective procedure that determines if some rational number is in S, yielding a partial answer to question 1. Computing the Hilbert set provides a partial answer to question 2, and we carry this out for some examples. We additionally expand an effective criterion of Gusic and Tadic to include elliptic surfaces with a rational 2-torsion curve.<br></pre>

<p>See abstract. </p>

<p>In this thesis, for symplectic and split odd special orthogonal groups over a p-adic field, we provide an extensive account of the intersection of local Arthur packets. More specifically, following Atobe's reformulation of Moeglin's construction of local Arthur packets, we give precise algorithms to determine whether a given representation lies in any local Arthur packet. Furthermore, if the representation does lie in a local Arthur packet, we give a systematic approach to determine all the other local Arthur packets which also contain the representation. These algorithms are based on certain operators which act on certain construction data coming from Atobe's parametrization of the local Arthur packets.</p>

<p>This thesis is focused on determinantal rings in 2 different contexts. In Chapter 3 the homological properties of powers of determinantal ideals are studied. In particular the focus is on local cohomology of determinantal thickenings and we explicitly describe the $R$-module structure of some of these local cohomology modules. In Chapter 4 we introduce \textit{generalized diagonal} matrices, a class of sparse matrices which contain diagonal and upper triangular matrices. We study the ideals of minors of such matrices and describe their properties such as height, multiplicity, and Cohen-Macaulayness. </p>

<p>See abstract. </p>

<p>This thesis has three major topics. The first is on generalized multiplicities. The second is on height bounds for ideals of minors of matrices with a given rank. The last topic is on the ideal of minors of generic generalized diagonal matrices.</p> <p>In the first part of this thesis, we discuss various generalizations of Hilbert-Samuel multiplicity. These include the Buchsbaum-Rim multiplicity, mixed multiplicities, $j$-multiplicity, and $\varepsilon$-multiplicity. For $(R,m)$ a Noetherian local ring of dimension $d$ and $I$ a $m$-primary ideal in $R$, Lech showed the following bound for the Hilbert-Samuel multiplicity of $I$, $e(I) \leq d!\lambda(R/I)e(m)$. Huneke, Smirnov, and Validashti improved the bound to $e(mI) \leq d!\lambda(R/I)e(m)$. We generalize the improved bound to the Buchsbaum-Rim multiplicity and to mixed multiplicities. </p> <p>For the second part of the thesis we discuss bounds on heights of ideals of minors of matrices. A classical bound for these heights was shown by Eagon and Northcott. Bruns' bound is an improvement on the Eagon-Northcott bound taking into consideration the rank of the matrix. We prove an analogous bound to Bruns' bound for alternating matrices. We then discuss an open problem by Eisenbud, Huneke, and Ulrich that asks for height bounds for symmetric matrices given their rank. We show a few reduction steps and prove some small cases of this problem. </p> <p>Finally, for the last topic we explore properties of the ideal of minors of generic generalized diagonal matrices. Generalized diagonal matrices are matrices with two ladders of zeros in the bottom left and top right corners. We compute their initial ideals and give a description of the facets of their Stanley-Reisner complex. Using this description, we characterize when these ideals are Cohen-Macaulay. In the special case where the ladders of zeros are triangles, we compute the height and multiplicity</p>

<div><div><div><p>It is well-known that the sum of reciprocals of twin primes converges or is a finite sum.</p><p>In the same spirit, Samuel Wagstaff proved in 2021 that the sum of reciprocals of primes p</p><p>such that ap + b is prime also converges or is a finite sum for any a, b where gcd(a, b) = 1</p><p>and 2 | ab. Wagstaff gave upper and lower bounds in the case that ab is a power of 2. Here,</p><p>we expand on his work and allow any a, b satisfying gcd(a, b) = 1 and 2 | ab. Let Πa,b be the</p><p>product of p−1 over the odd primes p dividing ab. We show that the upper bound of these p−2</p><p>sums is Πa,b times the upper bound found by Wagstaff and provide evidence as to why we cannot hope to do better than this. We also give several examples for specific pairs (a, b).</p><p><br></p><p>Next, we turn our attention to elliptic Carmichael numbers. In 1987, Dan Gordon defined the notion of an elliptic Carmichael number as a composite integer n which satisfies a Fermat- like criterion on elliptic curves with complex multiplication. More recently, in 2018, Thomas Wright showed that there are infinitely such numbers. We build off the work of Wright to prove that there are infinitely many elliptic Carmichael numbers of the form a (mod M) for a certain M, using an improved lower bound due to Carl Pomerance. We then apply this result to comment on the infinitude of strong pseudoprimes and strong Lucas pseudoprimes.</p><p><br></p><p>Finally, we consider the problem of classifying for which k does one have Φk(x) | Φn(x)−1, where Φn(x) is the nth cyclotomic polynomial. We provide a motivating example as to how this can be applied to primality proving. Then, we complete the case k = 8 and give a partial characterization for the case k = 16. This leads us to conjecture necessary and sufficient conditions for when Φk(x) | Φn(x) − 1 whenever k is a power of 2.</p></div></div></div>

In this paper, we prove the equality of the local arithmetic and analytic epsilon- and L-factors attached to the twisted exterior and symmetric square representations of GL(N). We will construct the twisted symmetric square local analytic gamma- and L-factor of GL(N) by applying Langlands-Shahidi method to odd GSpin groups. Then we reduce the problem to the stablity of local coefficients, and eventually prove the analytic stabitliy in this case by some analysis on the asymptotic behavior of certain partial Bessel functions.

<p>For a prime p>2 and a smooth proper p-adic formal scheme X over O_K where K is a p-adic field of absolute ramification degree e, we study a series of conditions (Cr_s), s>=0 that partially control the G_K-action on the image of the associated Breuil-Kisin prismatic cohomology RΓ_Δ(X/S) inside the A_inf-prismatic cohomology RΓ_Δ(X_Ainf/A_inf). The condition (Cr₀) is a criterion for a Breuil-Kisin-Fargues G_K-module to induce a crystalline representation used by Gee and Liu, and thus leads to a proof of crystallinity of Hⁱ_et(X_CK, Q_p) that avoids the crystalline comparison. The higher conditions (Cr_s) are used in an adaptation of a ramification bounds strategy of Caruso and Liu. As a result, we establish ramification bounds for the mod p representations Hⁱ_et(X_CK, Z/pZ) for arbitrary e and i, which extend or improve existing bounds in various situations.</p>