<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.

John Horton Conway was a British mathematician in the twentieth century. He made notable achievements in fields such as algebra, number theory, and knot theory. He was a renowned professor at Cambridge University and later Princeton. His contributions to algebra include his discovery of the Conway group, a group in twenty-four dimensions, and the Conway Constellation. He contributed to number theory with his development of the surreal numbers. His Game of Life earned him long-lasting fame. He contributed to knot theory with his developments of the Conway polynomial, Conway sphere, and Conway notation.

<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>

We provide explicit equations for moduli spaces of Drinfeld shtukas over the projective line with Γ(N), Γ_1(N) and Γ_0(N) level structures, where N is an effective divisor on P^1 . If the degree of N is big enough, these moduli spaces are relative surfaces. We study how the moduli space of shtukas over P^1 with Γ_0(N) level structure, Sht^{2,tr}(Γ_0(N)), can be used to provide a notion of motivic modularity for elliptic curves defined over function fields. Elliptic curves over function fields are known to be modular in the sense of admitting a parametrization from a Drinfeld modular curve, provided that they have split multiplicative reduction at one place. We conjecture a different notion of modularity that should cover the curves excluded by the reduction hypothesis. We use our explicit equations for Sht^{2,tr}(Γ_0(N)) to verify our modularity conjecture in the cases where N = 2(0) + (1) + (∞) and N = 3(0) + (∞).

No description available.

Thesis advisor: Solomon Friedberg / We study the generalized theta lifting between the double covers of split special orthogonal groups, which uses the non-minimal theta representations constructed by Bump, Friedberg and Ginzburg. We focus on the theta liftings of non-generic representations and make a conjecture that gives an upper bound of the first non-zero occurrence of the liftings, depending only on the unipotent orbit. We prove both global and local results that support the conjecture. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

No description available.

<p>Let S be a polynomial ring K[x1,...,xn] over a field K and let F be a non-negatively graded free module over S generated by m basis elements. In this thesis, we study four kinds of upper bounds: degree bounds for Gröbner bases of submodules of F, bounds for arithmetic degrees of S-ideals, regularity bounds for radicals of S-ideals, and Stillman bounds. </p> <p><br></p> <p>Let M be a submodule of F generated by elements with degrees bounded above by D and dim(F/M)=r. We prove that if M is graded, the degree of the reduced Gröbner basis of M for any term order is bounded above by 2[1/2((Dm)^{n-r}m+D)]^{2^{r-1}}. If M is not graded, the bound is 2[1/2((Dm)^{(n-r)^2}m+D)]^{2^{r}}. This is a generalization of bounds for ideals in a polynomial ring due to Dubé (1990) and Mayr-Ritscher (2013).</p> <p><br></p> <p>Our next results are concerned with a homogeneous ideal I in S generated by forms of degree at most d with dim(S/I)=r. In Chapter 4, we show how to derive from a result of Hoa (2008) an upper bound for the regularity of sqrt{I}, which denotes the radical of I. More specifically we show that reg(sqrt{I})<= d^{(n-1)2^{r-1}}. In Chapter 5, we show that the i-th arithmetic degree of I is bounded above by 2*d^{2^{n-i-1}}. This is done by proving upper bounds for arithmetic degrees of strongly stable ideals and ideals of Borel type.</p> <p><br></p> <p>In the last chapter, we explain our progress in attempting to make Stillman bounds explicit. Ananyan and Hochster (2020) were the first to show the existence of Stillman bounds. Together with G. Caviglia, we observe that a possible way of making their results explicit is to find an effective bound for an invariant called D(k,d) and supplement it into their proof. Although we are able to obtain this bound D(k,d) and realize Stillman bounds via an algorithm, it turns out that the computational complexity of Ananyan and Hochster's inductive proof would make the bounds too large to be meaningful. We explain the bad behavior of these Stillman bounds by giving estimates up to degree 3.</p>

<p>Let p be a prime number and F be a finite extension of Q_p. We established an algorithm to compute the semisimplification of the reduction of some irreducible two dimensional crystalline representations with two parameter {h,a_p} when v_p(a_p) is large enough. We improve the known results when p|h. We also extend the algorithm to the two dimensional semistable and non-crystalline representation. We compute the semi-simplification of the reduction when v_p(L) large enough and p=2. These results solve the difficulties with the case p=2. The strategies are based on the study of the Kisin modules over O_F and Breuil modules over S_F. By the theory of Breuil and Theorem of Colmez-Fontaine, these modules are closely related to semistable representations.</p>