Mathematics › Geometry and Topology

Algebraic Geometry and Number Theory

Works Count: 92380

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This cluster of papers covers a wide range of topics in algebraic geometry and moduli theory, including Gromov-Witten theory, log canonical singularities, motivic cohomology, minimal models, tropical geometry, intersection theory, symplectic geometry, birational geometry, and diophantine equations.

Keywords

Moduli Theory; Gromov-Witten Theory; Log Canonical Singularities; Motivic Cohomology; Minimal Models; Tropical Geometry; Intersection Theory; Symplectic Geometry; Birational Geometry; Diophantine Equations

The Arithmetic of Elliptic Curves

1986-01-01

Joseph H. Silverman

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Cohomology of Number Fields

2008-01-01

Jürgen Neukirch , Alexander Schmidt , Kay Wingberg

Introduction to Cyclotomic Fields

1982-01-01

Lawrence C. Washington

Geometric Invariant Theory

1994-01-01

David Mumford , John E. Fogarty , Frances Kirwan

Advanced Topics in the Arithmetic of Elliptic Curves

1994-01-01

Joseph H. Silverman

Higher-Dimensional Algebraic Geometry

2001-01-01

Olivier Debarre

Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

1993-01-01

Victor V. Batyrev

We consider families ${\cal F}(Δ)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $Δ$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $Δ$ in … We consider families ${\cal F}(Δ)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $Δ$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $Δ$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(Δ)$ defined by a Newton polyhedron $Δ$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron $Δ^*$ in the dual space defines another family ${\cal F}(Δ^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(Δ)$ and ${\cal F}(Δ^*)$.

Mirror Symmetry and Algebraic Geometry

1999-03-02

David Cox , Sheldon Katz

Introduction The quintic threefold Toric geometry Mirror symmetry constructions Hodge theory and Yukawa couplings Moduli spaces Gromov-Witten invariants Quantum cohomology Localization Quantum differential equations The mirror theorem Conclusion Singular varieties … Introduction The quintic threefold Toric geometry Mirror symmetry constructions Hodge theory and Yukawa couplings Moduli spaces Gromov-Witten invariants Quantum cohomology Localization Quantum differential equations The mirror theorem Conclusion Singular varieties Physical theories Bibliography Index.

On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises

2001-05-15

Christophe Breuil , Brian Conrad , Fred Diamond +1 more

We complete the proof that every elliptic curve over the rational numbers is modular. We complete the proof that every elliptic curve over the rational numbers is modular.

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Sur les Groupes Hyperboliques d’après Mikhael Gromov

1990-01-01

Étienne Ghys , Pierre de la Harpe

Moduli spaces of local systems and higher Teichmüller theory

2006-06-01

V. V. Fock , A. B. Goncharov

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Good Reduction of Abelian Varieties

1968-11-01

Jean-Pierre Serre , John Tate

Introduction to the Minimal Model Problem

2018-06-06

Yūjirō Kawamata , Katsumi Matsuda , Kenji Matsuki

Chapter O. Notation and preliminaries § 0-1.Kleiman's criterion for ampleness § 0-2.Definitions of terminal, canonical and (weak) log-terminal singularities § 0-3.Canonical varieties § 0-4.The minimal model conjecture Chapter 1. Vanishing … Chapter O. Notation and preliminaries § 0-1.Kleiman's criterion for ampleness § 0-2.Definitions of terminal, canonical and (weak) log-terminal singularities § 0-3.Canonical varieties § 0-4.The minimal model conjecture Chapter 1. Vanishing theorems § 1-1.Covering Lemma § 1-2.Vanishing theorem of Kawamata and Viehweg § 1-3.Vanishing theorem of Elkik and Fujita Chapter 2. Non-Vanishing Theorem § 2-1.Non-Vanishing Theorem Chapter 3. Base Point Free Theorem § 3-1.Base Point Free Theorem § 3-2.Contractions of extremal faces § 3-3.Canonical rings of varieties of general type Chapter 4. Cone Theorem § 4-1.Rationality Theorem § 4-2.The proof of the Cone Theorem Chapter 5. Flip Conjecture § 5-1.Types of contractions of extremal rays § 5-2.Flips of toric morphisms Chapter 6. Abundance Conjecture § 6-1.Nef and abundant divisors

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Variation of hodge structure: The singularities of the period mapping

1973-09-01

Wilfried Schmid

Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques

1971-12-01

Jean-Pierre Serre

THE GEOMETRY OF TORIC VARIETIES

1978-04-30

V. I. Danilov

Contents Introduction Chapter I. Affine toric varieties § 1. Cones, lattices, and semigroups § 2. The definition of an affine toric variety § 3. Properties of toric varieties § 4. … Contents Introduction Chapter I. Affine toric varieties § 1. Cones, lattices, and semigroups § 2. The definition of an affine toric variety § 3. Properties of toric varieties § 4. Differential forms on toric varieties Chapter II. General toric varieties § 5. Fans and their associated toric varieties § 6. Linear systems § 7. The cohomology of invertible sheaves § 8. Resolution of singularities § 9. The fundamental group Chapter III. Intersection theory § 10. The Chow ring § 11. The Riemann-Roch theorem § 12. Complex cohomology Chapter IV. The analytic theory § 13. Toroidal varieties § 14. Quasi-smooth varieties § 15. Differential forms with logarithmic poles Appendix 1. Depth and local cohomology Appendix 2. The exterior algebra Appendix 3. Differentials References

Faisceaux Algebriques Coherents

1955-03-01

Jean-Pierre Serre

Localization of virtual classes

1999-01-15

T.M. Graber , Rahul Pandharipande

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Endlichkeitss�tze f�r abelsche Variet�ten �ber Zahlk�rpern

1983-10-01

Герд Фалтингс

Complex analytic connections in fibre bundles

1957-01-01

Michael Atiyah

Introduction. In the theory of differentiable fibre bundles, with a Lie group as structure group, the notion of a connection plays an important role. In this paper we shall consider … Introduction. In the theory of differentiable fibre bundles, with a Lie group as structure group, the notion of a connection plays an important role. In this paper we shall consider complex analytic connections in complex analytic fibre bundles. The situation is then radically different from that in the differentiable case. In the differentiable case connections always exist, but may not be integrable; in the complex analytic case connections may not exist at all. In both cases we are led therefore to certain obstructions, an obstruction to the integrability of a connection in the differentiable case, an obstruction to the existence of a connection in the complex analytic case. It is a basic theorem that, if the structure group is compact, the obstruction in the differentiable case (the curvature) generates the characteristic cohomology ring of the bundle (with real coefficients). What we shall show is that, in a large class of important cases, the obstruction in the complex analytic case also generates the characteristic cohomology ring. Using this fact we can then give a purely cohomological definition of the characteristic ring. This has a number of advantages over the differentiable approach: in the first place the definition is a canonical one, not depending on an arbitrary choice of connection; secondly we remain throughout in the complex analytic domain, our characteristic classes being expressed as elements of cohomology groups with coefficients in certain analytic sheaves; finally the procedure can be carried through without change for algebraic fibre bundles. The ideas outlined above are developed in considerable detail, and they are applied in particular to a problem first studied by Weil [17], namely the problem of characterizing those fibre bundles which arise from a representation of the fundamental group. We show how Weil's main result fits into the general picture, and we discuss various aspects of the problem. As no complete exposition of the theory of complex analytic fibre bundles has as yet been published, this paper should start with a basic exposition of this nature. However this would be a major undertaking in itself, and instead we shall simply summarize in ?1 the terminology and results on vector bundles which we require, and for the rest we refer to Grothendieck [8], Serre [12], and Hirzebruch [9].

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Space of Unitary Vector Bundles on a Compact Riemann Surface

1967-03-01

C. S. Seshadri

Let X be a compact Riemann surface of genus g _ 2. A holomorphic vector bundle on X is said to be unitary if it arises from a unitary representation … Let X be a compact Riemann surface of genus g _ 2. A holomorphic vector bundle on X is said to be unitary if it arises from a unitary representation of the fundamental group of X. We prove in this paper that on the space of isomorphic classes of unitary vector bundles on X of a given rank, there is a natural structure of a normal projective variety (Theorem 8.1). We recall that Mumford has proved that, on the space of (isomorphic classes) stable vector bundles on X of a given rank and degree, there is a natural structure of a non-singular quasi-projective variety (cf. [7]); further, it was proved in [9] that a vector bundle on X of degree zero is stable if and only if it is associated to an irreducible unitary representation of the fundamental group of X. Thus our result shows the existence of a canonical compactification (as an algebraic variety) of the space of stable bundles on X of a given rank and degree zero. We shall now give a brief outline of the proof. It consists in a refinement of the proof of Mumford for the existence of a natural structure of a quasiprojective variety on the space of stable bundles of a given rank and degree (loc. cit.). Let us fix a very ample invertible sheaf OX(1) on X; then if m is a positive integer which is sufficiently large, we have H'(V(m)) 0 0 and H0( V(m)) generates V(m) for any Ve Or,, where Or, stands for the category of unitary vector bundles on X of rank r. Then the rank of H0(V(m)) is the same whatever be V e OR9; let this be p. The Hilbert polynomial of V(m), is also the same whatever be V e OR,; let this be P. Let Q = Quot(E/P) be the scheme in the sense of Grothendieck; E being the free coherent sheaf of rank p on X (cf. [4]). Let R be the open subscheme of Q consisting of the points which represent quotients of E which are locally free, and whose sections can be canonically identified with H0(E). Thus one has a family of vector bundles {Fq}qeR on X such that every Fq can be canonically considered as a quotient vector bundle of the trivial bundle E on X of rank p. The linear group G = Aut E acts on Q, and R is a G-invariant subscheme; further given V e Or there is a q e R such that Fq V, and the set of such points q con-

The symplectic nature of fundamental groups of surfaces

1984-11-01

William M. Goldman

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Théorie de hodge, III

1974-01-01

Pierre Deligne

Abelian l-Adic Representations and Elliptic Curves

1997-11-15

Jean-Pierre Serre

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on … This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one

Fundamentals of Diophantine Geometry

1983-01-01

Serge Lang

Modular curves and the eisenstein ideal

1977-12-01

Barry Mazur

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Endomorphisms of abelian varieties over finite fields

1966-04-01

John Tate

The Yang-Mills equations over Riemann surfaces

1983-03-17

Michael Atiyah , Raoul Bott

The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account … The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account is taken of its gauge symmetry. This enables topological conclusions to be drawn about the critical sets and leads eventually to information about the moduli space of algebraic bundles over the Riemann surface. This in turn depends on the interplay between the holomorphic and unitary structures, which is analysed in detail.

INTEGRAL SYMMETRIC BILINEAR FORMS AND SOME OF THEIR APPLICATIONS

1980-02-28

Vadim V. Nikulin

We set up the technique of discriminant-forms, which allows us to transfer many results for unimodular symmetric bilinear forms to the nonunimodular case and is convenient in calculations. Further, these … We set up the technique of discriminant-forms, which allows us to transfer many results for unimodular symmetric bilinear forms to the nonunimodular case and is convenient in calculations. Further, these results are applied to Milnor's quadratic forms for singularities of holomorphic functions and also to algebraic geometry over the reals. Bibliography: 57 titles.

The Geometry of Moduli Spaces of Sheaves

1997-01-01

Daniel Huybrechts , Manfred Lehn

Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. … Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.

La conjecture de Weil. I

1974-12-01

Pierre Deligne

Anti Self-Dual Yang-Mills Connections Over Complex Algebraic Surfaces and Stable Vector Bundles

1985-01-01

Simon Donaldson

Moduli of representations of the fundamental group of a smooth projective variety. II

1994-12-01

Carlos Simpson

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Vector Bundles Over an Elliptic Curve

1957-01-01

Michael Atiyah

Smoothness, semi-stability and alterations

1996-12-01

A. J. de Jong

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Existence of minimal models for varieties of log general type

2009-11-13

Caucher Birkar , Paolo Cascini , Christopher D. Hacon +1 more

We prove that the canonical ring of a smooth projective variety is finitely generated. We prove that the canonical ring of a smooth projective variety is finitely generated.

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Modular Elliptic Curves and Fermat's Last Theorem

1995-05-01

Andrew Wiles

When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the … When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.

Algebraic number theory

1999-11-22

Kazuya Katô , Nobushige Kurokawa , Takeshi Saito

This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background … This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties. This version can now be seen as a precursor of Arakelov theory. Part II covers class field theory, and Part III is devoted to analytic methods, including an exposition of Tate's thesis, the Brauer-Siegel theorem, and Weil's explicit formulas. The second edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten.

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The intrinsic normal cone

1997-03-19

Kai Behrend , Barbara Fantechi

La Conjecture de Weil. II

1980-12-01

Pierre Deligne

Théorie de Hodge, II

1971-12-01

Pierre Deligne

Characteristic Classes. (AM-76)

1974-12-31

John Milnor , James Stasheff

The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a … The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.

Fourier-Mukai Transforms in Algebraic Geometry

2006-04-20

Daniel Huybrechts

Abstract This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of … Abstract This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

The projectivity of the moduli space of stable curves, II: The stacks $M_{g,n}$

1983-12-01

Finn F. Knudsen

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Structure theorem, injectivity theorem and vanishing theorem for the cohomology groups of pseudo-effective line bundles

2025-06-25

Chu Qing , Xiangyu Zhou | Journal of Algebraic Geometry

In this paper, we establish a structure theorem and prove an isomorphism theorem for cohomology groups of pseudo-effective line bundles over holomorphically convex manifolds, which generalizes the results of Takegoshi, … In this paper, we establish a structure theorem and prove an isomorphism theorem for cohomology groups of pseudo-effective line bundles over holomorphically convex manifolds, which generalizes the results of Takegoshi, Demailly-Peternell-Schneider, Meng-Zhou, and Wu. As applications, we first give an answer to a question proposed by Matsumura, and establish an injectivity theorem for purely log terminal pairs generalized to pseudo-effective line bundles with transcendental singularities, and then we obtain a Kollár-Nadel-Ohsawa type vanishing theorem which extends the results of Matsumura, Fujino, Meng-Zhou, and others.

On the $\mathfrak{M}_H(G)$ -property

2025-06-24

Sören Kleine , Ahmed Matar , R. Sujatha | Mathematical Proceedings of the Cambridge Philosophical Society

Abstract Let E be an elliptic curve defined over ${{\mathbb{Q}}}$ which has good ordinary reduction at the prime p . Let K be a number field with at least one … Abstract Let E be an elliptic curve defined over ${{\mathbb{Q}}}$ which has good ordinary reduction at the prime p . Let K be a number field with at least one complex prime which we assume to be totally imaginary if $p=2$ . We prove several equivalent criteria for the validity of the $\mathfrak{M}_H(G)$ -property for ${{\mathbb{Z}}}_p$ -extensions other than the cyclotomic extension inside a fixed ${{\mathbb{Z}}}_p^2$ -extension $K_\infty/K$ . The equivalent conditions involve the growth of $\mu$ -invariants of the Selmer groups over intermediate shifted ${{\mathbb{Z}}}_p$ -extensions in $K_\infty$ , and the boundedness of $\lambda$ -invariants as one runs over ${{\mathbb{Z}}}_p$ -extensions of K inside of $K_\infty$ . Using these criteria we also derive several applications. For example, we can bound the number of ${{\mathbb{Z}}}_p$ -extensions of K inside $K_\infty$ over which the Mordell–Weil rank of E is not bounded, thereby proving special cases of a conjecture of Mazur. Moreover, we show that the validity of the $\mathfrak{M}_H(G)$ -property sometimes can be shifted to a larger base field K ′ .

None

2025-06-23

Hanson Smith | Journal de Théorie des Nombres de Bordeaux

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi></mml:math> be an integer and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> a prime so that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi> <mml:mi>p</mml:mi> </mml:msup><mml:mo>-</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:math> is irreducible. Write <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> to indicate the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-fold composition … Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi></mml:math> be an integer and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> a prime so that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi> <mml:mi>p</mml:mi> </mml:msup><mml:mo>-</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:math> is irreducible. Write <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> to indicate the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-fold composition of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> with itself. We study the monogenicity of number fields defined by roots of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> and give necessary and sufficient conditions for a root of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> to yield a power integral basis for each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>. Further, we generalize these criteria to an arbitrary number field.

Exponential BPS Graphs and D-Brane Counting on Toric Calabi–Yau Threefolds: Part II

2025-06-23

Sibasish Banerjee , Pietro Longhi , Mauricio Romo | Communications in Mathematical Physics

Abstract We study BPS states of 5d $$\mathcal {N}=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> SU (2) Yang-Mills theory on $$S^1\times \mathbb {R}^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> … Abstract We study BPS states of 5d $$\mathcal {N}=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> SU (2) Yang-Mills theory on $$S^1\times \mathbb {R}^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:math> . Geometric engineering relates these to enumerative invariants for the local Hirzebruch surface $$\mathbb {F}_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> . We illustrate computations of Vafa-Witten invariants via exponential networks, verifying fiber-base symmetry of the spectrum at certain points in moduli space, and matching with mirror descriptions based on quivers and exceptional collections. Albeit infinite, parts of the spectrum organize in families described by simple algebraic equations. Varying the radius of the M-theory circle interpolates smoothly with the spectrum of 4d $$\mathcal {N}=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> Seiberg–Witten theory, recovering spectral networks in the limit.

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None

2025-06-23

Richard A. P. Birkett | Annales de la faculté des sciences de Toulouse Mathématiques

The dynamics of a rational surface map <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo>⤏</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math> are easier to analyse when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> is “algebraically stable”. Here we investigate when and how this condition can be achieved … The dynamics of a rational surface map <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo>⤏</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math> are easier to analyse when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> is “algebraically stable”. Here we investigate when and how this condition can be achieved by conjugating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> with a birational change of coordinates. We show that if this can be done with a birational morphism, then there is a minimal such conjugacy. For birational <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> we also show that repeatedly lifting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> to its graph gives a stable conjugacy. Finally, we give an example in which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> can be birationally conjugated to a stable map, but the conjugacy cannot be achieved solely by blowing up.

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2025-06-23

Byoung Du Kim | Journal de Théorie des Nombres de Bordeaux

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>T</mml:mi> <mml:mi>E</mml:mi> </mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>W</mml:mi> <mml:mn>2</mml:mn> </mml:msup></mml:mrow></mml:math> be a rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math> crystalline <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi> <mml:msub><mml:mi>ℚ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:msub></mml:math>-representation of weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math> with non-ordinary reduction where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi></mml:math> is … Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>T</mml:mi> <mml:mi>E</mml:mi> </mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>W</mml:mi> <mml:mn>2</mml:mn> </mml:msup></mml:mrow></mml:math> be a rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math> crystalline <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi> <mml:msub><mml:mi>ℚ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:msub></mml:math>-representation of weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math> with non-ordinary reduction where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi></mml:math> is the ring of integers of some extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℚ</mml:mi> <mml:mi>p</mml:mi> </mml:msub></mml:math>, and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover accent="true"><mml:mi>T</mml:mi> <mml:mo>¯</mml:mo></mml:mover> <mml:mi>E</mml:mi> </mml:msub></mml:math> be its residual representation. Suppose <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>l</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> and fix some big enough <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> which only depends on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi> <mml:mi>E</mml:mi> </mml:msub></mml:math>. We show that the group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi mathvariant="normal">Ext</mml:mi> <mml:mrow><mml:mi>c</mml:mi><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>l</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>T</mml:mi> <mml:mo>¯</mml:mo></mml:mover> <mml:mi>E</mml:mi> </mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>T</mml:mi> <mml:mo>¯</mml:mo></mml:mover> <mml:mi>E</mml:mi> </mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> (Definition 2.30) of extensions with crystalline liftings of weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>l</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math>, which are themselves extensions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi> <mml:msub><mml:mi>ℚ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:msub></mml:math>-representations which are congruent to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>T</mml:mi> <mml:mi>E</mml:mi> </mml:msub><mml:mspace width="4.44443pt"/><mml:mrow><mml:mo>(</mml:mo><mml:mo form="prefix">mod</mml:mo><mml:mspace width="0.277778em"/><mml:msup><mml:mi>p</mml:mi> <mml:mi>N</mml:mi> </mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, is isomorphic to the group of finite flat extensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi mathvariant="normal">Ext</mml:mi> <mml:mrow><mml:mi>f</mml:mi><mml:mi>l</mml:mi></mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>T</mml:mi> <mml:mo>¯</mml:mo></mml:mover> <mml:mi>E</mml:mi> </mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>T</mml:mi> <mml:mo>¯</mml:mo></mml:mover> <mml:mi>E</mml:mi> </mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> ([18, Chapter 1.1]). In addition, we construct a certain functor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝒟</mml:mi></mml:math> of deformations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover accent="true"><mml:mi>T</mml:mi> <mml:mo>¯</mml:mo></mml:mover> <mml:mi>E</mml:mi> </mml:msub></mml:math> with liftings of certain type and weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>l</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math>, satisfying certain congruences with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi> <mml:mi>E</mml:mi> </mml:msub></mml:math>, show <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝒟</mml:mi></mml:math> has a representable hull, and demonstrate some evidence that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>t</mml:mi> <mml:mi>𝒟</mml:mi> </mml:msub><mml:mo>⊂</mml:mo><mml:msubsup><mml:mi mathvariant="normal">Ext</mml:mi> <mml:mrow><mml:mi>c</mml:mi><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>l</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>T</mml:mi> <mml:mo>¯</mml:mo></mml:mover> <mml:mi>E</mml:mi> </mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>T</mml:mi> <mml:mo>¯</mml:mo></mml:mover> <mml:mi>E</mml:mi> </mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>V</mml:mi> <mml:msub><mml:mi mathvariant="bold">T</mml:mi> <mml:mi>𝔪</mml:mi> </mml:msub> </mml:msub><mml:mo>⊗</mml:mo><mml:mi>W</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mi>𝔪</mml:mi> <mml:mi>W</mml:mi> </mml:msub><mml:mo>∈</mml:mo><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi mathvariant="bold">T</mml:mi> <mml:mi>𝔪</mml:mi> </mml:msub><mml:mo>⊗</mml:mo><mml:mi>W</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mi>𝔪</mml:mi> <mml:mi>W</mml:mi> </mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="bold">T</mml:mi></mml:math> is the Hecke algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="bold">T</mml:mi> <mml:mi>l</mml:mi> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>Γ</mml:mi> <mml:mn>1</mml:mn> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>M</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="bold">m</mml:mi></mml:math> is its maximal ideal given by a weight <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>l</mml:mi></mml:math> eigenform of level <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Γ</mml:mi> <mml:mn>1</mml:mn> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>M</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> whose Galois representation is congruent modulo <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>p</mml:mi> <mml:mi>N</mml:mi> </mml:msup></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi> <mml:mi>E</mml:mi> </mml:msub></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>V</mml:mi> <mml:msub><mml:mi mathvariant="bold">T</mml:mi> <mml:mi>𝔪</mml:mi> </mml:msub> </mml:msub></mml:math> is its associated Galois representation.

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2025-06-23

David E. Rohrlich | Journal de Théorie des Nombres de Bordeaux

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> be a finite Galois extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℚ</mml:mi></mml:math> and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> be an irreducible self-dual complex representation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">Gal</mml:mi><mml:mo>(</mml:mo><mml:mi>K</mml:mi><mml:mo>/</mml:mo><mml:mi>ℚ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. For an elliptic curve <mml:math … Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> be a finite Galois extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℚ</mml:mi></mml:math> and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> be an irreducible self-dual complex representation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">Gal</mml:mi><mml:mo>(</mml:mo><mml:mi>K</mml:mi><mml:mo>/</mml:mo><mml:mi>ℚ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. For an elliptic curve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi></mml:math> over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℚ</mml:mi></mml:math> let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>W</mml:mi><mml:mo>(</mml:mo><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> be the root number in the functional equation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. We give an example where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> is of dimension 4 and Schur index 1 but <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>W</mml:mi><mml:mo>(</mml:mo><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi></mml:math> over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℚ</mml:mi></mml:math>. The image of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> has order 32.

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2025-06-23

Erik Holmes | Journal de Théorie des Nombres de Bordeaux

For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> prime and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow> <mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:math>, we show that the shapes of pure prime degree number fields lie on one of two <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℓ</mml:mi></mml:math>-dimensional subspaces of the space … For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> prime and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow> <mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:math>, we show that the shapes of pure prime degree number fields lie on one of two <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℓ</mml:mi></mml:math>-dimensional subspaces of the space of shapes, and which of the two subspaces is dictated by whether or not <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> ramifies wildly. When the fields are ordered by absolute discriminant we show that the shapes are equidistributed, in a regularized sense, on these subspaces. We also show that the shape is a complete invariant within the family of pure prime degree fields. This extends the results of Harron, in [15], who studied shapes in the case of pure cubic number fields. Furthermore we translate the statements of pure prime degree number fields to statements about Frobenius number fields with a fixed resolvent field. Specifically we show that this study is equivalent to the study of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>F</mml:mi> <mml:mi>p</mml:mi> </mml:msub></mml:math>-number fields, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>F</mml:mi> <mml:mi>p</mml:mi> </mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub><mml:mo>⋊</mml:mo><mml:msub><mml:mi>C</mml:mi> <mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow> </mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math> with fixed resolvent field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ℚ</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>ζ</mml:mi> <mml:mi>p</mml:mi> </mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math>.

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2025-06-23

Giovanni Bosco | Journal de Théorie des Nombres de Bordeaux

We give a complete classification of all the potentially crystalline <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math>-adic representations of the absolute Galois group of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℚ</mml:mi> <mml:mn>3</mml:mn> </mml:msub></mml:math> that are isomorphic to the Tate module … We give a complete classification of all the potentially crystalline <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math>-adic representations of the absolute Galois group of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℚ</mml:mi> <mml:mn>3</mml:mn> </mml:msub></mml:math> that are isomorphic to the Tate module of an elliptic curve defined over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℚ</mml:mi> <mml:mn>3</mml:mn> </mml:msub></mml:math>. These representations are described in terms of their associated filtered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-modules. The most interesting cases occur when the potential good reduction is wild.

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2025-06-23

Paolo Dolce | Journal de Théorie des Nombres de Bordeaux

Without assuming the Northcott property we provide an upper bound on the number of “big solutions” of a special system of Diophantine inequalities over proper adelic curves. This system is … Without assuming the Northcott property we provide an upper bound on the number of “big solutions” of a special system of Diophantine inequalities over proper adelic curves. This system is interesting since it is a stronger version of Roth’s inequality for adelic curves.

Faber's socle intersection numbers via Gromov–Witten theory of elliptic curve

2025-06-23

Xavier Blot , Sergey Shadrin , I. J. Singh | Bulletin of the London Mathematical Society

Abstract The goal of this very short note is to give a new proof of Faber's formula for the socle intersection numbers in the tautological ring of . This new … Abstract The goal of this very short note is to give a new proof of Faber's formula for the socle intersection numbers in the tautological ring of . This new proof exhibits a new beautiful tautological relation that stems from the recent work of Oberdieck–Pixton on the Gromov–Witten theory of the elliptic curve via a refinement of their argument, and some straightforward computation with the double ramification cycles that enters the recursion relations for the Hamiltonians of the KdV hierarchy.

Equivariant Parabolic connections and stack of roots

2025-06-23

Sujoy Chakraborty , Arjun Paul | Geometriae Dedicata

None

2025-06-23

Martin Djukanović | Journal de Théorie des Nombres de Bordeaux

We exhibit sufficient and necessary conditions under which, over a field of characteristic zero, an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-isogeny between two elliptic curves without complex multiplication induces curves <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> of genus … We exhibit sufficient and necessary conditions under which, over a field of characteristic zero, an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-isogeny between two elliptic curves without complex multiplication induces curves <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> of genus two whose Jacobian is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-isogenous to the product of the two elliptic curves. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mo>≤</mml:mo><mml:mn>25</mml:mn></mml:mrow></mml:math>, we also present a partial list of one-dimensional families of curves <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> whose Jacobian is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-split and has <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-isogenous components.

None

2025-06-23

Kim Klinger-Logan , Kalani Thalagoda , Tian An Wong | Journal de Théorie des Nombres de Bordeaux

We construct a generalization of the Dedekind–Rademacher cocycle to congruence subgroups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="normal">SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>ℂ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, and derive some of its basic properties. In particular, we show that it … We construct a generalization of the Dedekind–Rademacher cocycle to congruence subgroups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="normal">SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>ℂ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, and derive some of its basic properties. In particular, we show that it parametrizes a family of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math>-values and prove the integrality of these values.

None

2025-06-23

Guillem Tarrach | Journal de Théorie des Nombres de Bordeaux

We compute the non-Eisenstein systems of Hecke eigenvalues contributing to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-arithmetic homology of irreducible smooth mod <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> representations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>π</mml:mi></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="normal">GL</mml:mi> <mml:mn>2</mml:mn> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>ℚ</mml:mi> <mml:mi>p</mml:mi> … We compute the non-Eisenstein systems of Hecke eigenvalues contributing to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-arithmetic homology of irreducible smooth mod <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> representations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>π</mml:mi></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="normal">GL</mml:mi> <mml:mn>2</mml:mn> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>ℚ</mml:mi> <mml:mi>p</mml:mi> </mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> and to the cohomology of their duals. We show that in most cases they are associated to odd irreducible 2-dimensional Galois representations whose local component at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> corresponds under the mod <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> local Langlands correspondence to a smooth representation that contains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>π</mml:mi></mml:math> as a subrepresentation.

Erratum: Unramified extensions of quadratic number fields with certain perfect Galois groups

2025-06-21

Joachim König | International Journal of Number Theory

On certain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math>-maximal curves of the form y3 = f(x)

2025-06-20

Guilherme Dias , Saeed Tafazolian | Finite Fields and Their Applications

Murmurations and Sato–Tate Conjecture for High Rank Zetas of elliptic curves II: Beyond Riemann Hypothesis

2025-06-20

Zhan Shi , Lin Weng | International Journal of Data Science in the Mathematical Sciences

On L-values of elliptic curves twisted by cubic Dirichlet characters

2025-06-20

David Kurniadi Angdinata | Canadian Journal of Mathematics

The stable and augmented base locus under finite morphisms

2025-06-19

Tanuj Gomez | manuscripta mathematica

Abstract We study the pullback of the stable and augmented base locus under a finite surjective morphism between normal varieties over a perfect field. Abstract We study the pullback of the stable and augmented base locus under a finite surjective morphism between normal varieties over a perfect field.

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Correlated Gromov-Witten Invariants

2025-06-18

Thomas Blomme , Francesca Carocci | Symmetry Integrability and Geometry Methods and Applications

We introduce a geometric refinement of Gromov-Witten invariants for $\mathbb P^1$-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore, we prove a … We introduce a geometric refinement of Gromov-Witten invariants for $\mathbb P^1$-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore, we prove a refinement of the degeneration formula keeping track of the correlation. Finally, combining certain invariance properties of the correlated invariant, a local computation and the refined degeneration formula we follow floor diagram techniques to prove regularity results for the generating series of the invariants in the case of $\mathbb P^1$-bundles over elliptic curves. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces.

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Derived categories of families of Fano threefolds

2025-06-18

Alexander Kuznetsov | Algebraic geometry

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Degrees of isogenies over prime degree number fields of non-CM elliptic curves with rational j-invariant

2025-06-18

I. H. Novak | Journal of Number Theory

Fine Selmer groups of CM elliptic curves

2025-06-18

Meng Fai Lim , Jun Wang , Yan Dong | Journal of Number Theory

A note on spherical bundles on K3 surfaces (with an appendix by Genki Ouchi)

2025-06-18

C.H. Li , S. H. Liu | Algebraic geometry

Nontautological cycles on moduli spaces of smooth pointed curves

2025-06-17

Dario Faro , Carolina Tamborini | Bulletin of the London Mathematical Society

Abstract In recent work by Arena, Canning, Clader, Haburcak, Li, Mok, and Tamborini, it was proven that for infinitely many values of and , there exist nontautological algebraic cohomology classes … Abstract In recent work by Arena, Canning, Clader, Haburcak, Li, Mok, and Tamborini, it was proven that for infinitely many values of and , there exist nontautological algebraic cohomology classes on the moduli space of smooth genus , ‐pointed curves. Here we show how a generalization of their technique allows to cover most of the remaining cases, proving the existence of nontautological algebraic cohomology classes on the moduli space for all but finitely many values of and .

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On the $p$-adic weight-monodromy conjecture for complete intersections in toric varieties

2025-06-17

Federico Binda , Hiroki Kato , Alberto Vezzani | Inventiones mathematicae

The tame Nori fundamental group

2025-06-17

Indranil Biswas , Manish Kumar , A. J. Parameswaran | Journal of Algebra

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None

2025-06-17

Jeffrey D. Achter , Sebastian Casalaina‐Martin , Charles Vial | Annales de l’institut Fourier

Albanese varieties provide a standard tool in algebraic geometry for converting questions about general varieties into questions about Abelian varieties. A result of Serre provides the existence of an Albanese … Albanese varieties provide a standard tool in algebraic geometry for converting questions about general varieties into questions about Abelian varieties. A result of Serre provides the existence of an Albanese variety for any geometrically connected and geometrically reduced scheme of finite type over a field, and a result of Grothendieck–Conrad establishes that Albanese varieties are stable under base change of field provided the scheme is, in addition, proper. A result of Raynaud shows that base change can fail for Albanese varieties without this properness hypothesis. In this paper we show that Albanese varieties of geometrically connected and geometrically reduced schemes of finite type over a field are stable under separable field extensions. We also show that the failure of base change in general is explained by the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi><mml:mo>/</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:math>-image for purely inseparable extensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi><mml:mo>/</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:math>.

On Miyanishi conjecture for quasi-projective varieties

2025-06-17

Takao Asano | Expositiones Mathematicae

Nakano positivity of direct image sheaves of adjoint line bundles with mild singularities

2025-06-17

Yongpan Zou | Bulletin de la Société mathématique de France

TROPICAL WEIL’S RECIPROCITY LAW AND WEIL’S PAIRING

2025-06-14

Nikita Kalinin , Matthew Magin | Journal of Mathematical Sciences

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On the finiteness of maps into simple abelian varieties satisfying certain tangency conditions

2025-06-13

Finn Bartsch | Bulletin of the London Mathematical Society

Abstract We show that given a simple abelian variety and a normal variety defined over a finitely generated field of characteristic zero, the set of non‐constant morphisms satisfying certain tangency … Abstract We show that given a simple abelian variety and a normal variety defined over a finitely generated field of characteristic zero, the set of non‐constant morphisms satisfying certain tangency conditions imposed by a Campana orbifold divisor on is finite. To do so, we study the geometry of the scheme parameterizing such morphisms from a smooth curve and show that it admits a quasi‐finite non‐dominant morphism to .

Families of elliptic curves over the four-pointed configuration space and exceptional sequences for the braid group on four strands

2025-06-12

William Chen , Nick Salter | Proceedings of the American Mathematical Society

We show that the configuration space of four unordered points in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> … We show that the configuration space of four unordered points in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with barycenter 0 is isomorphic to the space of triples <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper E comma upper Q comma omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(E,Q,\omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an elliptic curve, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q element-of upper E Superscript ring"> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>E</mml:mi> <mml:mo>∘</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">Q\in E^\circ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a nonzero point, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a nonzero holomorphic differential on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. At the level of fundamental groups, our construction unifies two classical exceptional exact sequences involving the braid group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B 4"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">B_4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>: namely, the sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 right-arrow upper F 2 right-arrow upper B 4 right-arrow upper B 3 right-arrow 1"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false">→</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1\rightarrow F_2\rightarrow B_4\rightarrow B_3\rightarrow 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 2"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a free group of rank 2, related to Ferrari’s solution of the quartic, and the sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 right-arrow double-struck upper Z right-arrow upper B 4 right-arrow upper A u t Superscript plus Baseline left-parenthesis upper F 2 right-parenthesis right-arrow 1"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:msup> <mml:mi>Aut</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1\rightarrow \mathbb {Z} \rightarrow B_4\rightarrow \operatorname {Aut}^+(F_2)\rightarrow 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Dyer-Formanek-Grossman [Arch. Math. (Basel) 38 (1982), pp. 404–409].

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On the maximum gonality of a curve over a finite field

2025-06-12

Xander Faber , Jon Grantham , Everett W. Howe | Algebra & Number Theory

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Arithmetic geometry of character varieties with regular monodromy

2025-06-12

Masoud Kamgarpour , GyeongHyeon Nam , Anna Puskás | Representation Theory of the American Mathematical Society

We count points on a family of smooth character varieties with regular semisimple and regular unipotent monodromies. We show that these varieties are polynomial count and obtain an explicit expression … We count points on a family of smooth character varieties with regular semisimple and regular unipotent monodromies. We show that these varieties are polynomial count and obtain an explicit expression for their <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polynomials using complex representation theory of finite reductive groups. As an application, we give an example of a cohomologically rigid representation which is not physically rigid.

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On a conjecture of Levesque and Waldschmidt

2025-06-10

Tobias Hilgart , Volker Ziegler | Glasnik Matematicki

One of the first parametrised Thue equations, \[ \hspace{-18ex}\left| X^3 - (n-1)X^2 Y - (n+2) XY^2 - Y^3 \right| = 1, \] over the integers was solved by E. Thomas … One of the first parametrised Thue equations, \[ \hspace{-18ex}\left| X^3 - (n-1)X^2 Y - (n+2) XY^2 - Y^3 \right| = 1, \] over the integers was solved by E. Thomas in 1990. If we interpret this as a norm-form equation, we can write this as \[ \hspace{-18ex} \left| N_{K/\mathbb{Q}}\left( X - \lambda_0 Y \right) \right| = \left| \left( X-\lambda_0 Y \right) \left( X-\lambda_1 Y \right) \left( X-\lambda_2 Y \right) \right| =1 \] if $\lambda_0, \lambda_1, \lambda_2$ are the roots of the defining irreducible polynomial, and $K$ is the corresponding number field. Levesque and Waldschmidt twisted this norm-form equation by an exponential parameter $s$ and looked, among other things, at the equation \[ \hspace{-18ex} \left| N_{K/\mathbb{Q}}\left( X - \lambda_0^s Y \right) \right| = 1. \] They solved this effectively and conjectured that introducing a second exponential parameter $t$ and looking at $\left| N_{K/\mathbb{Q}}\left( X - \lambda_0^s\lambda_1^t Y \right) \right| = 1$ does not change the effective solvability. We want to partially confirm this if \[ \hspace{-18ex} \min\left\{ \left| 2s-t \right|, \left| 2t-s \right|, \left| s+t \right| \right\} \gt \varepsilon \cdot \max\left\{ \left|s\right|, \left|t\right| \right\} \gt 2, \] i.e., the two exponents do not almost cancel in specific cases.

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Partitions into triples with equal products and families of elliptic curves

2025-06-10

Ahmed El Amine Youmbai , Arman Shamsi Zargar , Maksym Voznyy | Glasnik Matematicki

Let ${\mathcal{S}_{\ell}}(M,N)$ denote a set of $\ell$ (distinct) triples of positive integers having the same sum $M$ and the same product $N$. For each $2\leq\ell\leq 4$ we establish a connection … Let ${\mathcal{S}_{\ell}}(M,N)$ denote a set of $\ell$ (distinct) triples of positive integers having the same sum $M$ and the same product $N$. For each $2\leq\ell\leq 4$ we establish a connection between a subset of ${\mathcal{S}_{\ell}}(M,N)$ with (integral) parametric elements and a family of elliptic curves. When $\ell=2$ and $3$, we use certain known subsets of ${\mathcal{S}_{\ell}}(M,N)$ with parametric elements and respectively find families of elliptic curves of generic rank $\geq 5$ and $\geq 6$, while for $\ell=4$ we first obtain a subset of ${\mathcal{S}_{\ell}}(M,N)$ with parametric elements, then construct a family of elliptic curves of generic rank $\geq 8$. Finally, we perform a computer search within these families to find specific curves with rank $\geq 11$ and in particular we found two curves of rank $14$.

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On the Kähler cone of irreducible symplectic orbifolds

2025-06-06

Grégoire Menet , Ulrike Rieß | Mathematische Zeitschrift

Classification of Minimal Abelian Coulomb Branches

2025-01-09

Antoine Bourget , Quentin Lamouret , Sinan Moura Soysüren +1 more

Obtaining the classification of 3d $\mathcal{N}=4$ quivers whose Coulomb branches have an isolated singularity is an essential step in understanding moduli spaces of vacua of supersymmetric field theories with 8 … Obtaining the classification of 3d $\mathcal{N}=4$ quivers whose Coulomb branches have an isolated singularity is an essential step in understanding moduli spaces of vacua of supersymmetric field theories with 8 supercharges in any dimension. In this work, we derive a full classification for such Abelian quivers with arbitrary charges, and identify all possible Coulomb branch geometries as quotients of $\mathbb{H}^n$ by $\mathrm{U}(1)$ or a finite cyclic group. We give two proofs, one which uses the decay and fission algorithm, and another one relying only on explicit computations involving 3d mirror symmetry. In the process, we put forward a method for computing the 3d mirror of any $\mathrm{U}(1)^r$ gauge theory, which is sensitive to discrete gauge factors in the mirror theory. This constitutes a confirmation for the decay and fission algorithm.

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Arithmetic geometry of the moduli stack of Weierstrass fibrations over $\mathbb{P}^1$

2021-07-26

Jun–Yong Park , Johannes Schmitt

Coarse moduli spaces of Weierstrass fibrations over the (unparameterized) projective line were constructed by the classical work of [Miranda] using Geometric Invariant Theory. In our paper, we extend this treatment … Coarse moduli spaces of Weierstrass fibrations over the (unparameterized) projective line were constructed by the classical work of [Miranda] using Geometric Invariant Theory. In our paper, we extend this treatment by using results of [Romagny] regarding group actions on stacks to give an explicit construction of the moduli stack $\mathcal{W}_n$ of Weierstrass fibrations over an unparameterized $\mathbb{P}^{1}$ with discriminant degree $12n$ and a section. We show that it is a smooth algebraic stack and prove that for $n \geq 2$, the open substack $\mathcal{W}_{\mathrm{min},n}$ of minimal Weierstrass fibrations is a separated Deligne-Mumford stack over any base field $K$ with $\mathrm{char}(K) \neq 2,3$ and not dividing $n$. Arithmetically, for the moduli stack $\mathcal{W}_{\mathrm{sf},n}$ of stable Weierstrass fibrations, we determine its motive in the Grothendieck ring of stacks to be $\{\mathcal{W}_{\mathrm{sf},n}\} = \mathbb{L}^{10n - 2}$ in the case that $n$ is odd, which results in its weighted point count to be $\#_q(\mathcal{W}_{\mathrm{sf},n}) = q^{10n - 2}$ over $\mathbb{F}_q$. In the appendix, we show how our methods can be applied similarly to the classical work of [Silverman] on coarse moduli spaces of self-maps of the projective line, allowing us to construct the natural moduli stack and to compute its motive.

Integral differential forms for superelliptic curves

2020-03-27

Sabrina Kunzweiler , Stefan Wewers

Given a superelliptic curve $Y_K : y^n = f(x)$ over a local field $K$, we describe the theoretical background and an implementation of a new algorithm for computing the $\mathcal{O}_K$-lattice … Given a superelliptic curve $Y_K : y^n = f(x)$ over a local field $K$, we describe the theoretical background and an implementation of a new algorithm for computing the $\mathcal{O}_K$-lattice of integral differential forms on $Y_K$. We build on the results of Obus and the second author, which describe arbitrary regular models of the projective line using only valuations. One novelty of our approach is that we construct an $\mathcal{O}_K$-model of $Y_K$ with only rational singularities, but which may not be regular.

Arithmetic special cycles and Jacobi forms

2020-02-10

Siddarth Sankaran

We consider families of special cycles, as introduced by Kudla, on Shimura varieties attached to anisotropic quadratic spaces over totally real fields. By augmenting these cycles with Green currents, we … We consider families of special cycles, as introduced by Kudla, on Shimura varieties attached to anisotropic quadratic spaces over totally real fields. By augmenting these cycles with Green currents, we obtain classes in the arithmetic Chow groups of the canonical models of these Shimura varieties (viewed as arithmetic varieties over their reflex fields). The main result of this paper asserts that generating series built from these cycles can be identified with the Fourier expansions of non-holomorphic Hilbert-Jacobi modular forms. This result provides evidence for an arithmetic analogue of Kudla's conjecture relating these cycles to Siegel modular forms.