{"title":"Presentations of Galois groups of maximal extensions with restricted ramification","authors":"Yuan Liu","doi":"10.2140/ant.2025.19.835","DOIUrl":"https://doi.org/10.2140/ant.2025.19.835","url":null,"abstract":"<p>Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>k</mi><mo stretchy=\"false\">)</mo></math>, the Galois group of the maximal extension of a global field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> that is unramified outside a finite set <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math> of places, as <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> varies among a certain family of extensions of a fixed global field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Q</mi></math>. We define a group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>B</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>k</mi><mo>,</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math>, for each finite simple <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>k</mi><mo stretchy=\"false\">)</mo></math>-module <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math>, to generalize the work of Koch and Shafarevich on the pro-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math> completion of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>k</mi><mo stretchy=\"false\">)</mo></math>. We prove that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>k</mi><mo stretchy=\"false\">)</mo></math> always admits a balanced presentation when it is finitely generated. In the setting of the nonabelian Cohen–Lenstra heuristics, we prove that the unramified Galois groups studied by the Liu–Wood–Zureick-Brown conjecture always admit a balanced presentation in the form of the random group in the conjecture. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"8 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Motivic distribution of rational curves and twisted products of toric varieties","authors":"Loïs Faisant","doi":"10.2140/ant.2025.19.883","DOIUrl":"https://doi.org/10.2140/ant.2025.19.883","url":null,"abstract":"<p>This work concerns asymptotical stabilisation phenomena occurring in the moduli space of sections of certain algebraic families over a smooth projective curve, whenever the generic fibre of the family is a smooth projective Fano variety, or not far from being Fano. </p><p> We describe the expected behaviour of the class, in a ring of motivic integration, of the moduli space of sections of given numerical class. Up to an adequate normalisation, it should converge, when the class of the sections goes arbitrarily far from the boundary of the dual of the effective cone, to an effective element given by a motivic Euler product. Such a principle can be seen as an analogue for rational curves of the Batyrev–Manin–Peyre principle for rational points. </p><p> The central tool of this article is the property of equidistribution of curves. We show that this notion does not depend on the choice of a model of the generic fibre, and that equidistribution of curves holds for smooth projective split toric varieties. As an application, we study the Batyrev–Manin–Peyre principle for curves on a certain kind of twisted products.</p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"33 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Smooth cuboids in group theory","authors":"Joshua Maglione, Mima Stanojkovski","doi":"10.2140/ant.2025.19.967","DOIUrl":"https://doi.org/10.2140/ant.2025.19.967","url":null,"abstract":"<p>A smooth cuboid can be identified with a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn><mo>×</mo><mn>3</mn></math> matrix of linear forms in three variables, with coefficients in a field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>. We produce isomorphism invariants of these groups in terms of their <span>adjoint algebras</span>, which also give information on the number of their maximal abelian subgroups. Moreover, when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> is finite, we give a characterization of the isomorphism types of the groups in terms of isomorphisms of elliptic curves and also describe their automorphism groups. We conclude by applying our results to the determination of the automorphism groups and isomorphism testing of finite <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-groups of class <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math> and exponent <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> arising in this way. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"71 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Malle’s conjecture for fair counting functions","authors":"Peter Koymans, Carlo Pagano","doi":"10.2140/ant.2025.19.1007","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1007","url":null,"abstract":"<p>We show that the naive adaptation of Malle’s conjecture to fair counting functions is not true in general. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"97 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Syzygies of tangent-developable surfaces and K3 carpets via secant varieties","authors":"Jinhyung Park","doi":"10.2140/ant.2025.19.1029","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1029","url":null,"abstract":"<p>We give simple geometric proofs of Aprodu, Farkas, Papadima, Raicu and Weyman’s theorem on syzygies of tangent-developable surfaces of rational normal curves and Raicu and Sam’s result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green’s conjecture for general curves of genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi></math> over an algebraically closed field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> k</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> char</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>k</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <mn>0</mn></math> or <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> char</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>k</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">)</mo>\u0000<mo>≥</mo><mo stretchy=\"false\">⌊</mo><mo stretchy=\"false\">(</mo><mi>g</mi>\u0000<mo>−</mo> <mn>1</mn><mo stretchy=\"false\">)</mo><mo>∕</mo><mn>2</mn><mo stretchy=\"false\">⌋</mo></math>. Our approach provides a new way to study tangent-developable surfaces in general. Along the way, we show the arithmetic normality of tangent-developable surfaces of arbitrary smooth projective curves of large degree. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"219 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the D-module of an isolated singularity","authors":"Thomas Bitoun","doi":"10.2140/ant.2025.19.763","DOIUrl":"https://doi.org/10.2140/ant.2025.19.763","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Z</mi></math> be the germ of a complex hypersurface isolated singularity of equation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math>, with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Z</mi></math> at least of dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>. We consider the family of analytic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>D</mi></math>-modules generated by the powers of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn><mo>∕</mo><mi>f</mi></math> and describe it in terms of the pole order filtration on the de Rham cohomology of the complement of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">{</mo><mi>f</mi>\u0000<mo>=</mo> <mn>0</mn><mo stretchy=\"false\">}</mo></math> in the neighbourhood of the singularity. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"18 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Ribbon Schur functors","authors":"Keller VandeBogert","doi":"10.2140/ant.2025.19.771","DOIUrl":"https://doi.org/10.2140/ant.2025.19.771","url":null,"abstract":"<p>We investigate a generalization of the classical notion of a Schur functor associated to a ribbon diagram. These functors are defined with respect to an arbitrary algebra, and in the case that the underlying algebra is the symmetric/exterior algebra, we recover the classical definition of Schur/Weyl functors, respectively. In general, we construct a family of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math>-term complexes categorifying the classical concatenation/near-concatenation identity for symmetric functions, and one of our main results is that the exactness of these <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math>-term complexes is equivalent to the Koszul property of the underlying algebra <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math>. We further generalize these ribbon Schur functors to the notion of a multi-Schur functor and construct a canonical filtration of these objects whose associated graded pieces are described explicitly; one consequence of this filtration is a complete equivariant description of the syzygies of arbitrary Segre products of Koszul modules over the Segre product of Koszul algebras. Further applications to the equivariant structure of derived invariants, symmetric function identities, and Koszulness of certain classes of modules are explored at the end, along with a characteristic-free computation of the regularity of the Schur functor <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝕊</mi></mrow><mrow><mi>λ</mi></mrow></msup></math> applied to the tautological subbundle on projective space. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"28 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Odd moments in the distribution of primes","authors":"Vivian Kuperberg","doi":"10.2140/ant.2025.19.617","DOIUrl":"https://doi.org/10.2140/ant.2025.19.617","url":null,"abstract":"<p>Montgomery and Soundararajan showed that the distribution of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ψ</mi><mo stretchy=\"false\">(</mo><mi>x</mi>\u0000<mo>+</mo>\u0000<mi>H</mi><mo stretchy=\"false\">)</mo>\u0000<mo>−</mo>\u0000<mi>ψ</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math>, for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>0</mn>\u0000<mo>≤</mo>\u0000<mi>x</mi>\u0000<mo>≤</mo>\u0000<mi>N</mi></math>, is approximately normal with mean <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mo>∼</mo>\u0000<mi>H</mi></math> and variance <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mo>∼</mo>\u0000<mi>H</mi><mi>log</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>N</mi><mo>∕</mo><mi>H</mi><mo stretchy=\"false\">)</mo></math>, when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>N</mi></mrow><mrow><mi>δ</mi></mrow></msup>\u0000<mo>≤</mo>\u0000<mi>H</mi>\u0000<mo>≤</mo> <msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>δ</mi></mrow></msup> </math>. Their work depends on showing that sums <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>h</mi><mo stretchy=\"false\">)</mo></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>-term singular series are <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo stretchy=\"false\">(</mo><mo>−</mo><mi>h</mi><mi>log</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>h</mi>\u0000<mo>+</mo>\u0000<mi>A</mi><mi>h</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>k</mi><mo>∕</mo><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msub><mrow><mi>O</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>k</mi><mo>∕</mo><mn>2</mn><mo>−</mo><mn>1</mn><mo>∕</mo><mo stretchy=\"false\">(</mo><mn>7</mn><mi>k</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mi>𝜀</mi></mrow></msup><mo stretchy=\"false\">)</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> is a constant and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> is odd, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>h</mi><mo stretchy=\"false\">)</mo>\u0000<mo>≍</mo> <msup><mrow><mi>h</mi></mrow><mrow><mo stretchy=\"false\">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy=\"false\">)</mo><mo>∕</mo><mn>2</mn></mrow></msup><msup><mrow><mo stretchy=\"false\">(</m","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"124 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Efficient resolution of Thue–Mahler equations","authors":"Adela Gherga, Samir Siksek","doi":"10.2140/ant.2025.19.667","DOIUrl":"https://doi.org/10.2140/ant.2025.19.667","url":null,"abstract":"<p>A Thue–Mahler equation is a Diophantine equation of the form </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mi>F</mi><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000<mo>=</mo>\u0000<mi>a</mi>\u0000<mo>⋅</mo> <msubsup><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub>\u0000</mrow></msubsup><mo>⋯</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mi>v</mi></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>v</mi></mrow></msub>\u0000</mrow></msubsup><mo>,</mo><mspace width=\"1em\"></mspace><mi>gcd</mi><mo> <!--FUNCTION APPLICATION--> </mo><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <mn>1</mn>\u0000</math>\u0000</div>\u0000<p> where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math> is an irreducible binary form of degree at least <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math> with integer coefficients, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>a</mi></math> is a nonzero integer and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo> <!--FUNCTION APPLICATION--></mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>v</mi></mrow></msub></math> are rational primes. Existing algorithms for resolving such equations require computations in the field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi>\u0000<mo>=</mo>\u0000<mi>ℚ</mi><mo stretchy=\"false\">(</mo><mi>𝜃</mi><mo>,</mo><msup><mrow><mi>𝜃</mi></mrow><mrow><mi>′</mi></mrow></msup><mo>,</mo><msup><mrow><mi>𝜃</mi></mrow><mrow><mi>′</mi><mi>′</mi></mrow></msup><mo stretchy=\"false\">)</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜃</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>𝜃</mi></mrow><mrow><mi>′</mi></mrow></msup></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>𝜃</mi></mrow><mrow><mi>′</mi><mi>′</mi></mrow></msup></math> are distinct roots of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi><mo stretchy=\"false\">(</mo><mi>X</mi><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <mn>0</mn></math>. We give a new algorithm that requires computations only in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi>\u0000<mo>=</mo>\u0000<mi>ℚ</mi><mo stretchy=\"false\">(</mo><mi>𝜃</mi><mo stretchy=\"false\">)</mo></math> making it far more suited for higher degree examples. We also introduce a lattice sieving technique reminiscent of the Mordell–Weil sieve that makes it practical to tackle Thue–Mahler equations of higher degree and with larger sets of primes than was previously possible. We give several examples including one of degree <math display=\"inline\" xmlns=\"http://www.w3.org/","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"57 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Automorphisms of del Pezzo surfaces in characteristic 2","authors":"Igor Dolgachev, Gebhard Martin","doi":"10.2140/ant.2025.19.715","DOIUrl":"https://doi.org/10.2140/ant.2025.19.715","url":null,"abstract":"<p>We classify the automorphism groups of del Pezzo surfaces of degrees 1 and 2 over an algebraically closed field of characteristic 2. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"71 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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