We prove that real topological Hochschild homology $\mathrm {THR}$ for schemes with involution satisfies base change and descent for the ${\mathbb {Z}/2}$-isovariant étale topology. As an application, we provide computations for the projective line (with and without involution) and the higher-dimensional projective spaces.

We present some results related to Zilber’s Exponential-Algebraic Closedness Conjecture, showing that various systems of equations involving algebraic operations and certain analytic functions admit solutions in the complex numbers. These results are inspired by Zilber’s theorems on raising to powers.

We show that algebraic varieties which split as a product of a linear subspace of an additive group and an algebraic subvariety of a multiplicative group intersect the graph of the exponential function, provided that they satisfy Zilber’s freeness and rotundity conditions, using techniques from tropical geometry.

We then move on to prove a similar theorem, establishing that varieties which split as a product of a linear subspace and a subvariety of an abelian variety A intersect the graph of the exponential map of A (again under the analogues of the freeness and rotundity conditions). The proof uses homology and cohomology of manifolds.

Finally, we show that the graph of the modular j-function intersects varieties which satisfy freeness and broadness and split as a product of a Möbius subvariety of a power of the upper-half plane and a complex algebraic variety, using Ratner’s orbit closure theorem to study the images under j of Möbius varieties.

Abstract prepared by Francesco Paolo Gallinaro

E-mail: francesco.gallinaro@mathematik.uni-freiburg.de

URL: https://etheses.whiterose.ac.uk/31077/

Let p be a prime. In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in cyclic extensions of degree p. We also study growth of the p-primary Selmer group and the Shafarevich–Tate group in cyclic degree-p extensions and improve upon previously known results in this direction.

We formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the grand Riemann hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan–Hardy–Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type ${G^{n , 0}_{0 , n}}$.

For a finite extension F of ${\mathbf Q}_p$, Drinfeld defined a tower of coverings of (the Drinfeld half-plane). For $F = {\mathbf Q}_p$, we describe a decomposition of the p-adic geometric étale cohomology of this tower analogous to Emerton’s decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finiteness theorem for the arithmetic étale cohomology modulo p whose proof uses Scholze’s functor, global ingredients, and a computation of nearby cycles which makes it possible to prove that this cohomology has finite presentation. This last result holds for all F; for $F\neq {\mathbf Q}_p$, it implies that the representations of $\mathrm{GL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case $F = {\mathbf Q}_p$.

We show that the first-order logical theory of the binary overlap-free words (and, more generally, the $\alpha $-free words for rational $\alpha $, $2 < \alpha \leq 7/3$), is decidable. As a consequence, many results previously obtained about this class through tedious case-based proofs can now be proved “automatically,” using a decision procedure, and new claims can be proved or disproved simply by restating them as logical formulas.

In this article, we provide an explicit upper bound for $h_K \mathcal {R}_K d_K^{-1/2}$ which depends on an effective constant in the error term of the Ideal Theorem.

Let f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{\lambda_f(n)\}_n$ be its sequence of normalised Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers $n \in \mathcal{N}_K$ yield a sign change for the sequence $\{\lambda_f(n)\}_{n \in \mathcal{N}_K}$. More precisely, for a positive proportion of $n \in \mathcal{N}_K \cap [1,X]$ we have $\lambda_f(n)\lambda_f(n') < 0$, where n′ is the first element of $\mathcal{N}_K$ greater than n for which $\lambda_f(n') \neq 0$.

For example, for $K = \mathbb{Q}(i)$ and $\mathcal{N}_K = \{m^2+n^2 \;:\; m,n \in \mathbb{Z}\}$ the set of sums of two squares, we obtain $\gg_f X/\sqrt{\log X}$ such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author.

In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed $a \neq 0$ there are $\gg_{f,\varepsilon} X^{1/2-\varepsilon}$ sign changes for $\lambda_f$ along the sequence of integers of the form $a + m^2 + n^2 \leq X$.

We give a new proof of Faltings's $p$-adic Eichler–Shimura decomposition of the modular curves via Bernstein–Gelfand–Gelfand (BGG) methods and the Hodge–Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was used in the original proof. Then we construct overconvergent Eichler–Shimura maps for the modular curves providing ‘the second half’ of the overconvergent Eichler–Shimura map of Andreatta, Iovita and Stevens. We use higher Coleman theory on the modular curve developed by Boxer and Pilloni to show that the small-slope part of the Eichler–Shimura maps interpolates the classical $p$-adic Eichler–Shimura decompositions. Finally, we prove that overconvergent Eichler–Shimura maps are compatible with Poincaré and Serre pairings.

We study fluctuations of the error term for the number of integer lattice points lying inside a three-dimensional Cygan–Korányi ball of large radius. We prove that the error term, suitably normalized, has a limiting value distribution which is absolutely continuous, and we provide estimates for the decay rate of the corresponding density on the real line. In addition, we establish the existence of all moments for the normalized error term, and we prove that these are given by the moments of the corresponding density.

Let $\ell $ and p be (not necessarily distinct) prime numbers and F be a global function field of characteristic $\ell $ with field of constants $\kappa $. Assume that there exists a prime $P_\infty $ of F which has degree $1$ and let $\mathcal {O}_F$ be the subring of F consisting of functions with no poles away from $P_\infty $. Let $f(X)$ be a polynomial in X with coefficients in $\kappa $. We study solutions to Diophantine equations of the form $Y^{n}=f(X)$ which lie in $\mathcal {O}_F$ and, in particular, show that if m and $f(X)$ satisfy additional conditions, then there are no nonconstant solutions. The results apply to the study of solutions to $Y^{n}=f(X)$ in certain rings of integers in $\mathbb {Z}_{p}$-extensions of F known as constant $\mathbb {Z}_p$-extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \ldots , T_r]$, where K is any field of characteristic $\ell $, showing that the only solutions must lie in K. We apply our methods to study solutions of Diophantine equations of the form $Y^n=\sum _{i=1}^d (X+ir)^m$, where $m,n, d\geq 2$ are integers.

In this paper, we mainly prove the following conjectures of Z.-W. Sun (J. Number Theory 133 (2013), 2914–2928): let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in \mathbb {Z}$ and $x\equiv 1\ ({\rm {mod}}\ 3)$, then

\[ x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k}{2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^2), \]

$p\equiv 1\pmod 3$

\[ \sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^3), \]

$f_n=\sum _{k=0}^n\binom {n}k^3$

$n$

Given $A\subseteq GL_2(\mathbb {F}_q)$, we prove that there exist disjoint subsets $B, C\subseteq A$ such that $A = B \sqcup C$ and their additive and multiplicative energies satisfying

$$\begin{align*}\max\{\,E_{+}(B),\, E_{\times}(C)\,\}\ll \frac{|A|^3}{M(|A|)}, \end{align*}$$

where

$$ \begin{align*} M(|A|) = \min\Bigg\{\,\frac{q^{4/3}}{|A|^{1/3}(\log|A|)^{2/3}},\, \frac{|A|^{4/5}}{q^{13/5}(\log|A|)^{27/10}}\,\Bigg\}. \end{align*} $$

$A, B, C\subseteq GL_2(\mathbb {F}_q)$

$$\begin{align*}|AB+C|, ~|(A+B)C|\gg q^4,\end{align*}$$

$|A||B||C|\gg q^{10 + 1/2}$

$Forum Math.$

The integrality of the numbers $A_{n,m}={(2n)!(2m)!}/{n!m!(n+m)!}$ was observed by Catalan as early as 1874 and Gessel named $A_{n,m}$ the super Catalan numbers. The positivity of the q-super Catalan numbers (q-analogue of the super Catalan numbers) was investigated by Warnaar and Zudilin [‘A q-rious positivity’, Aequationes Math. 81 (2011), 177–183]. We prove the divisibility of sums of q-super Catalan numbers, which establishes a q-analogue of Apagodu’s congruence involving super Catalan numbers.

We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell–Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients $X_0^+(N)$ of prime level $N$, the curve $X_{S_4}(13)$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve $X_{\scriptstyle \mathrm { ns}} ^+ (17)$.

Let $\varphi $ be Euler’s function and fix an integer $k\ge 0$. We show that for every initial value $x_1\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for all initial values $x_1,x_2\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=\varphi (x_{n+1})+\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that k is even.

We study the asymptotic behavior of the family of holomorphic correspondences $\lbrace \mathcal {F}_a\rbrace _{a\in \mathcal {K}}$, given by

$$ \begin{align*}\bigg(\frac{az+1}{z+1}\bigg)^2+\bigg(\frac{az+1}{z+1}\bigg)\bigg(\frac{aw-1}{w-1}\bigg)+\bigg(\frac{aw-1}{w-1}\bigg)^2=3.\end{align*} $$

$\mathcal {F}_a$

$\operatorname {PSL}_2(\mathbb {Z})$

$a\in \mathcal {K}$

$\mathcal {F}_a$

$\mathcal {F}_a$

We study plane curves over finite fields whose tangent lines at smooth $\mathbb {F}_q$-points together cover all the points of $\mathbb {P}^2(\mathbb {F}_q)$.

We prove the Zariski dense orbit conjecture in positive characteristic for regular self-maps of split semiabelian varieties.

We state a conjecture that relates the derived category of smooth representations of a $p$-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of ${\rm GL}_n$ by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and of Emerton and Helm.