Generalised Kummer constructions and Weil restrictions

We classify newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. Under the extended Riemann hypothesis for odd real Dirichlet characters, these newforms are finite in number. We produce tables for weights 3 and 4, where finiteness holds unconditionally.

The problem of summation of Fourier series in finite form is formulated in the weak sense, which allows one to consider this problem uniformly both for classically convergent and for divergent series. For series with polynomial Fourier coefficients \(a_n, b_n \in \mathbb{R}[n]\), it is proved that the sum of a Fourier series can be represented as a linear combination of 1, \(\delta(x)\), \(\cot \frac{x}{2}\) and their derivatives. It is shown that this representation can be found in a finite number of steps. For series with rational Fourier coefficients \(a_n, b_n \in \mathbb{R}(n)\), it is shown that the sum of such a series is always a solution of a linear differential equation with constant coefficients whose right-hand side is a linear combination of 1, \(\delta(x)\), \(\cot \frac{x}{2}\) and their derivatives. Thus, the issue of summing a Fourier series with rational coefficients is reduced to the classical problem of the theory of integration in elementary functions.

Let f be a newform of weight k ≥ 3 with Fourier coefficients in a number field K . We show that the universal deformation ring of the mod λ Galois representation associated to f is unobstructed, and thus isomorphic to a power series ring in three variables over the Witt vectors, for all but finitely many primes λ of K . We give an explicit bound on such λ for the 6 known cusp forms of level 1, trivial character, and rational Fourier coefficients. We also prove a somewhat weaker result for weight 2.

Special values of L-functions by a Siegel–Weil–Kudla–Rallis formula

We study the complex structure moduli dependence of the scalar Laplacian eigenmodes for one-parameter families of Calabi-Yau n-folds in ℙn+1. It was previously observed that some eigenmodes get lighter while others get heavier as a function of these moduli, which leads to eigenvalue crossing. We identify the cause for this behavior for the torus. We then show that at points in a sublocus of complex structure moduli space where Laplacian eigenmodes cross, the torus has complex multiplication. We speculate that the generalization to arbitrary Calabi-Yau manifolds could be that level crossing is related to rank one attractor points. To test this, we compute the eigenmodes numerically for the quartic K3 and the quintic threefold, and match crossings to CM and attractor points in these varieties. To quantify the error of our numerical methods, we also study the dependence of the numerical spectrum on the quality of the Calabi-Yau metric approximation, the number of points sampled from the Calabi-Yau variety, the truncation of the eigenbasis, and the distance from degeneration points in complex structure moduli space.

In this paper we discuss four methods of proving of Calabi--Yau threefolds with $h^{12}=1$: existence of elliptic ruled surfaces inside (Hulek-Verrill), correspondence with a product of an elliptic curve and a K3 surface (Livn\'e-Yui), correspondence with a (modular) rigid Calabi-Yau threefold, and existence of an involution splitting the fourdimensional representation into twodimensional subrepresentations. We apply these methods to prove of 17 out of 18 double octic Calabi-Yau threefolds for which numerical evidence of modularity was found in the second Author's thesis. We observe that holds for those elements in a pencil having some additional geometric properties. In the proofs we use representations of the considered Calabi-Yau threefolds as a Kummer fibration associated to a fiber product of rational elliptic fibrations.

The purpose of this note is to prove that under certain hypotheses, the graded ring of integral automorphic forms, with respect to an arithmetic group operating On a tube domain, is generated as a graded algebra over the complex numbers by a finite number of automorphic forms having rational integral Fourier coefficients. The result, which seems to have some potential numbertheoretic interest, was inspired by notes of M. Eichler [8, esp. Satz 49] and by questions raised in correspondence with that author and with I. I. Pyatetskii-Shapiro. One should also note the similarity of ideas here with those used in the proof of Satz D of [9]. We wish to remark, furthermore, that in most cases lengthy computations will be needed to verify the applicability of our theorem here, which should therefore be regarded more as a technical lemma than as a substantial contribution in itself. As in [2], let ~ denote a (hermitian) symmetric tube domain, let F be a discrete, arithmetically defined subgroup of the group of all holomorphic automorphisms of ~ , and assume that ~ has a zerodimensional rational boundary component Fo, which we take to be the zero-dimensional rational boundary component of ~ at infinity as in [2]. Let

In this article we study combinatorial degenerations of minimal surfaces of\nKodaira dimension 0 over local fields, and in particular show that the `type'\nof the degeneration can be read off from the monodromy operator acting on a\nsuitable cohomology group. This can be viewed as an arithmetic analogue of\nresults of Persson and Kulikov on degenerations of complex surfaces, and\nextends various particular cases studied by Matsumoto, Liedtke/Matsumoto and\nHern\\'andez-Mada. We also study `maximally unipotent' degenerations of\nCalabi--Yau threefolds, following Koll\\'ar/Xu, showing in this case that the\ndual intersection graph is a 3-sphere.\n

Calabi–Yau threefolds obtained as fiber products of elliptic and quasi-elliptic rational surfaces

We construct complete Calabi–Yau metrics on non-compact manifolds that are smoothings of an initial complete intersection V0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_0$$\\end{document} that is a Calabi–Yau cone, extending the work of Székelyhidi (Duke Math J 168(14):2651–2700, 2019). The constructed Calabi–Yau manifold has tangent cone at infinity given by C×V0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}\ imes V_0$$\\end{document}. This construction produces Calabi–Yau metrics with fibers having varying complex structures and possibly isolated singularities.

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.

We study the web of dualities relating various enumerative invariants, notably Gromov–Witten invariants and invariants that arise in topological gauge theory. In particular, we study Donaldson–Thomas gauge theory and its reductions to D=4D=4 and D=2D=2 which are relevant to the local theory of surfaces in Calabi–Yau threefolds.

We study the syzygetic structure of projections of del Pezzo surfaces in order to construct singular Calabi-Yau threefolds. By smoothing those threefolds,we obtain new examples of Calabi-Yau threefolds with Picard group of rank 1. We also give an example of type II primitive contraction whose exceptional divisor is the blow-up of the projective plane at a point.

In this paper, we propose a combinatorial approach to study Calabi–Yau threefolds constructed as a resolution of singularities of a double covering of [Formula: see text] branched along an arrangement of eight planes. We use this description to give a complete classification of arrangements of eight planes in [Formula: see text] defining Calabi–Yau threefolds modulo projective transformation with [Formula: see text] and to derive their geometric properties (Kummer surface fibrations, automorphisms, special elements in families).

In this final chapter we look at the map W : CrGr(s+1) in the case of quantum cohomology and, more generally, abstract quantum cohomology. That is, we apply the fundamental construction of Chapter 8 to the situation of Chapters 5 and 6. It should not be surprising that this yields benefits, as the map W is a geometrical manifestation of the quantum cohomology D-module, and we have seen already that W is of great importance in other integrable systems such as the KdV equation and the harmonic map equation. In fact these expectations will be exceeded handsomely: in the case of the quantum cohomology of a Calabi–Yau manifold, W can be interpreted as a variation of Hodge structure (VHS) on a ‘mirror dual’ Calabi–Yau manifold.