We construct an anticyclotomic Euler system for the Rankin–Selberg convolutions of two modular forms, using p-adic families of generalised Gross–Kudla–Schoen diagonal cycles. As applications of this construction, we prove new results on the Bloch–Kato conjecture in analytic ranks zero and one, and a divisibility towards an Iwasawa main conjecture.

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/

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 $F = {\mathbf Q}_p$.

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 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 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.

Romyar Sharifi has constructed a map $\varpi _M$ from the first homology of the modular curve $X_1(M)$ to the K-group $K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M+\zeta _M^{-1}, \frac {1}{M}]) \otimes \operatorname {\mathrm {\mathbf {Z}}}[1/2]$, where $\zeta _M$ is a primitive Mth root of unity. Sharifi conjectured that $\varpi _M$ is annihilated by a certain Eisenstein ideal. Fukaya and Kato proved this conjecture after tensoring with $\operatorname {\mathrm {\mathbf {Z}}}_p$ for a prime $p\geq 5$ dividing M. More recently, Sharifi and Venkatesh proved the conjecture for Hecke operators away from M. In this note, we prove two main results. First, we give a relation between $\varpi _M$ and $\varpi _{M'}$ when $M' \mid M$. Our method relies on the techniques developed by Sharifi and Venkatesh. We then use this result in combination with results of Fukaya and Kato in order to get the Eisenstein property of $\varpi _M$ for Hecke operators of index dividing M.

In [20], Rohrlich proved a modular analog of Jensen’s formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert $ of a ${\mathrm {PSL}}(2,{\mathbb {Z}})$ modular form f in terms of the Dedekind Delta function evaluated at the divisor of f. In [2], the authors re-interpreted the Rohrlich–Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert $ and extended the result to compute a regularized inner product of $\log \Vert f \Vert $ with what amounts to powers of the Hauptmodul of $\mathrm {PSL}(2,{\mathbb {Z}})$. In the present article, we revisit the Rohrlich–Jensen formula and prove that in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur–Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass–Selberg relation. In this form, we develop a Rohrlich–Jensen formula associated with any Fuchsian group $\Gamma $ of the first kind with one cusp by employing a type of Kronecker limit formula associated with the resolvent kernel. We present two examples of our main result: First, when $\Gamma $ is the full modular group ${\mathrm {PSL}}(2,{\mathbb {Z}})$, thus reproving the theorems from [2]; and second when $\Gamma $ is an Atkin–Lehner group $\Gamma _{0}(N)^+$, where explicit computations of inner products are given for certain levels N when the quotient space $\overline {\Gamma _{0}(N)^+}\backslash \mathbb {H}$ has genus zero, one, and two.

We study the discriminants of the minimal polynomials $\mathcal {P}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive $n\equiv 11\pmod {24}$. We show that $\Delta (\mathcal {P}_n)$ divides $\Delta (H_n)$, where $H_n$ is the ring class polynomial, with quotient a perfect square and determine the sign of $\Delta (\mathcal {P}_n)$ based on the ideal class group structure of the order of discriminant $-n$. We also show that the discriminant of the number field generated by $j({(-1+\sqrt {-n})}/{2})$, where j is the j-invariant, divides $\Delta (\mathcal {P}_n)$. Moreover, using Ye’s computation of $\log|\Delta(H_n)|$ [‘Revisiting the Gross–Zagier discriminant formula’, Math. Nachr. 293 (2020), 1801–1826], we show that 3 never divides $\Delta(H_n)$, and thus $\Delta(\mathcal{P}_n)$, for all squarefree $n\equiv11\pmod{24}$.

Let p and $\ell $ be primes such that $p> 3$ and $p \mid \ell -1$ and k be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight k and level $\Gamma _0(\ell )$ at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the $\mathbb {Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for $k=2$ using our methods. In addition, we prove some $R=\mathbb {T}$ theorems under certain hypotheses.

In this paper, we prove the algebraicity of some L-values attached to quaternionic modular forms. We follow the rather well-established path of the doubling method. Our main contribution is that we include the case where the corresponding symmetric space is of non-tube type. We make various aspects very explicit, such as the doubling embedding, coset decomposition, and the definition of algebraicity of modular forms via CM-points.

Let F be a CM number field. We generalise existing automorphy lifting theorems for regular residually irreducible p-adic Galois representations over F by relaxing the big image assumption on the residual representation.

An explicit formula forthe mean value of $\vert L(1,\chi )\vert ^2$ is known, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Lately, the authors obtained that the mean value of $\vert L(1,\chi )\vert ^2$ is asymptotic to $\pi ^2/6$, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\ \ \pmod {2d}$ which are trivial on a subgroup H of odd order d of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$, provided that $d\ll \frac {\log p}{\log \log p}$. Bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Here, for a given integer $d_0>1$, we consider the same questions for the nonprimitive odd Dirichlet characters $\chi '$ modulo $d_0p$ induced by the odd primitive characters $\chi $ modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi ')\vert ^2$ is asymptotic to $\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order $d\ll \frac {\log p}{\log \log p}$. As a consequence, we improve the previous bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$. Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.

In this paper, we prove one divisibility of the Iwasawa–Greenberg main conjecture for the Rankin–Selberg product of a weight two cusp form and an ordinary complex multiplication form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm {GU}(3,1)$, generalizing an earlier result of the third-named author to allow nonordinary cusp forms. The main result is a key input in the third-named author’s proof of Kobayashi’s $\pm $-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semiordinary Hida theory along an appropriate smaller weight space and a study of the semiordinary Eisenstein family.

In [14], Jacquet–Piatetskii-Shapiro–Shalika defined a family of compact open subgroups of p-adic general linear groups indexed by nonnegative integers and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of nonnegative integers. For the proof, we introduce the Rankin–Selberg integrals for Speh representations.

We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a $p$-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms.

A superelliptic curve over a discrete valuation ring $\mathscr{O}$ of residual characteristic p is a curve given by an equation $\mathscr{C}\;:\; y^n=\,f(x)$, with $\textrm{Disc}(\,f)\neq 0$. The purpose of this article is to describe the Galois representation attached to such a curve under the hypothesis that f(x) has all its roots in the fraction field of $\mathscr{O}$ and that $p \nmid n$. Our results are inspired on the algorithm given in Bouw and WewersGlasg (Math. J. 59(1) (2017), 77–108.) but our description is given in terms of a cluster picture as defined in Dokchitser et al. (Algebraic curves and their applications, Contemporary Mathematics, vol. 724 (American Mathematical Society, Providence, RI, 2019), 73–135.).

We prove Fermat’s Last Theorem over $\mathbb {Q}(\sqrt {5})$ and $\mathbb {Q}(\sqrt {17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer–Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of $\mathbb {Q}(\sqrt {5})$ is a generalization to a real quadratic base field of the one used by Chen and Siksek. For the case of $\mathbb {Q}(\sqrt {17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett, Chen, Dahmen, and Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.

We settle a part of the conjecture by Bandini and Valentino [‘On the structure and slopes of Drinfeld cusp forms’, Exp. Math. 31(2) (2022), 637–651] for $S_{k,l}(\Gamma _0(T))$ when $\mathrm {dim}\ S_{k,l}(\mathrm {GL}_2(A))\leq 2$. We frame and check the conjecture for primes $\mathfrak {p}$ and higher levels $\mathfrak {p}\mathfrak {m}$, and show that a part of the conjecture for level $\mathfrak {p} \mathfrak {m}$ does not hold if $\mathfrak {m}\ne A$ and $(k,l)=(2,1)$.