Pub Date : 2024-03-13DOI: 10.1112/s0010437x23007972
Harold Blum, Yuchen Liu, Lu Qi
We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.
{"title":"Convexity of multiplicities of filtrations on local rings","authors":"Harold Blum, Yuchen Liu, Lu Qi","doi":"10.1112/s0010437x23007972","DOIUrl":"https://doi.org/10.1112/s0010437x23007972","url":null,"abstract":"<p>We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140117466","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}
Pub Date : 2024-03-07DOI: 10.1112/s0010437x23007728
Raju Krishnamoorthy, Jinbang Yang, Kang Zuo
Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let ${mathbb {L}}$ be a rank $2$, geometrically irreducible lisse $overline {{mathbb {Q}}}_ell$-sheaf on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $Esubset overline {mathbb {Q}}_{ell }$, and has bad, infinite reduction at some closed point $x$ of
{"title":"Constructing abelian varieties from rank 2 Galois representations","authors":"Raju Krishnamoorthy, Jinbang Yang, Kang Zuo","doi":"10.1112/s0010437x23007728","DOIUrl":"https://doi.org/10.1112/s0010437x23007728","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$U$</span></span></img></span></span> be a smooth affine curve over a number field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$K$</span></span></img></span></span> with a compactification <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$X$</span></span></img></span></span> and let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb {L}}$</span></span></img></span></span> be a rank <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$2$</span></span></img></span></span>, geometrically irreducible lisse <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$overline {{mathbb {Q}}}_ell$</span></span></img></span></span>-sheaf on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$U$</span></span></img></span></span> with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$Esubset overline {mathbb {Q}}_{ell }$</span></span></img></span></span>, and has bad, infinite reduction at some closed point <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$x$</span></span></img></span></span> of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140053721","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}
Pub Date : 2024-03-05DOI: 10.1112/s0010437x23007704
Sebastián Herrero, Ricardo Menares, Juan Rivera-Letelier
We show that for every finite set of prime numbers $S$, there are at most finitely many singular moduli that are $S$-units. The key new ingredient is that for every prime number $p$, singular moduli are $p$-adically disperse. We prove analogous results for the Weber modular functions, the $lambda$-invariants and the McKay–Thompson series associated with the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit.
{"title":"There are at most finitely many singular moduli that are S-units","authors":"Sebastián Herrero, Ricardo Menares, Juan Rivera-Letelier","doi":"10.1112/s0010437x23007704","DOIUrl":"https://doi.org/10.1112/s0010437x23007704","url":null,"abstract":"<p>We show that for every finite set of prime numbers <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304200415819-0744:S0010437X23007704:S0010437X23007704_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$S$</span></span></img></span></span>, there are at most finitely many singular moduli that are <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304200415819-0744:S0010437X23007704:S0010437X23007704_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$S$</span></span></img></span></span>-units. The key new ingredient is that for every prime number <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304200415819-0744:S0010437X23007704:S0010437X23007704_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>, singular moduli are <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304200415819-0744:S0010437X23007704:S0010437X23007704_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-adically disperse. We prove analogous results for the Weber modular functions, the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304200415819-0744:S0010437X23007704:S0010437X23007704_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$lambda$</span></span></img></span></span>-invariants and the McKay–Thompson series associated with the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034860","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}
Pub Date : 2024-03-05DOI: 10.1112/s0010437x23007698
Matthew Northey, Pankaj Vishe
We prove the Hasse principle for a smooth projective variety $Xsubset mathbb {P}^{n-1}_mathbb {Q}$ defined by a system of two cubic forms $F,G$ as long as $ngeq 39$. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defined over $mathbb {Q}$.
{"title":"On the Hasse principle for complete intersections","authors":"Matthew Northey, Pankaj Vishe","doi":"10.1112/s0010437x23007698","DOIUrl":"https://doi.org/10.1112/s0010437x23007698","url":null,"abstract":"<p>We prove the Hasse principle for a smooth projective variety <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304180407639-0000:S0010437X23007698:S0010437X23007698_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$Xsubset mathbb {P}^{n-1}_mathbb {Q}$</span></span></img></span></span> defined by a system of two cubic forms <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304180407639-0000:S0010437X23007698:S0010437X23007698_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$F,G$</span></span></img></span></span> as long as <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304180407639-0000:S0010437X23007698:S0010437X23007698_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$ngeq 39$</span></span></img></span></span>. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defined over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304180407639-0000:S0010437X23007698:S0010437X23007698_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {Q}$</span></span></img></span></span>.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034864","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}
Pub Date : 2024-02-23DOI: 10.1112/s0010437x23007674
Joost Nuiten
The straightening–unstraightening correspondence of Grothendieck–Lurie provides an equivalence between cocartesian fibrations between $(infty, 1)$-categories and diagrams of $(infty, 1)$-categories. We provide an alternative proof of this correspondence, as well as an extension of straightening–unstraightening to all higher categorical dimensions. This is based on an explicit combinatorial result relating two types of fibrations between double categories, which can be applied inductively to construct the straightening of a cocartesian fibration between higher categories.
{"title":"On straightening for Segal spaces","authors":"Joost Nuiten","doi":"10.1112/s0010437x23007674","DOIUrl":"https://doi.org/10.1112/s0010437x23007674","url":null,"abstract":"<p>The straightening–unstraightening correspondence of Grothendieck–Lurie provides an equivalence between cocartesian fibrations between <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240222174850284-0238:S0010437X23007674:S0010437X23007674_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(infty, 1)$</span></span></img></span></span>-categories and diagrams of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240222174850284-0238:S0010437X23007674:S0010437X23007674_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$(infty, 1)$</span></span></img></span></span>-categories. We provide an alternative proof of this correspondence, as well as an extension of straightening–unstraightening to all higher categorical dimensions. This is based on an explicit combinatorial result relating two types of fibrations between double categories, which can be applied inductively to construct the straightening of a cocartesian fibration between higher categories.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139948376","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}
Pub Date : 2024-02-22DOI: 10.1112/s0010437x23007686
Spencer Leslie, Aaron Pollack
We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by ${pm }1$. We analyze the minimal modular form $Theta _{F_4}$ on the double cover of $F_4$, following Loke–Savin and Ginzburg. Using $Theta _{F_4}$, we define a modular form of weight $tfrac {1}{2}$ on (the double cover of) $G_2$. We prove that the Fourier coefficients of this modular form on $G_2$ see the $2$-torsion in the narrow class groups of totally real cubic fields.
{"title":"Modular forms of half-integral weight on exceptional groups","authors":"Spencer Leslie, Aaron Pollack","doi":"10.1112/s0010437x23007686","DOIUrl":"https://doi.org/10.1112/s0010437x23007686","url":null,"abstract":"<p>We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${pm }1$</span></span></img></span></span>. We analyze the minimal modular form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$Theta _{F_4}$</span></span></img></span></span> on the double cover of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$F_4$</span></span></img></span></span>, following Loke–Savin and Ginzburg. Using <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$Theta _{F_4}$</span></span></img></span></span>, we define a modular form of weight <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$tfrac {1}{2}$</span></span></img></span></span> on (the double cover of) <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$G_2$</span></span></img></span></span>. We prove that the Fourier coefficients of this modular form on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$G_2$</span></span></img></span></span> see the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$2$</span></span></img></span></span>-torsion in the narrow class groups of totally real cubic fields.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139928034","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}
Pub Date : 2024-02-12DOI: 10.1112/s0010437x23007637
Maarten Mol
Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.
{"title":"Stratification of the transverse momentum map","authors":"Maarten Mol","doi":"10.1112/s0010437x23007637","DOIUrl":"https://doi.org/10.1112/s0010437x23007637","url":null,"abstract":"<p>Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772847","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}
Pub Date : 2024-02-08DOI: 10.1112/s0010437x23007649
Ashvin A. Swaminathan
Let $F$ be a separable integral binary form of odd degree $N geq 5$. A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree-$N$superelliptic equation$y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $mathscr {F}_N(f_0)$ of degree-$N$ superelliptic equations with fixed leading coefficient $f_0 in mathbb {Z} smallsetminus pm mathbb {Z}^2$, ordered by height. For every sufficiently large $N$, we prove that among equations in the family
{"title":"Most odd-degree binary forms fail to primitively represent a square","authors":"Ashvin A. Swaminathan","doi":"10.1112/s0010437x23007649","DOIUrl":"https://doi.org/10.1112/s0010437x23007649","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$F$</span></span></img></span></span> be a separable integral binary form of odd degree <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$N geq 5$</span></span></img></span></span>. A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$N$</span></span></img></span></span> <span>superelliptic equation</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$y^2 = F(x,z)$</span></span></img></span></span> has finitely many primitive integer solutions. In this paper, we consider the family <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathscr {F}_N(f_0)$</span></span></img></span></span> of degree-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$N$</span></span></img></span></span> superelliptic equations with fixed leading coefficient <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$f_0 in mathbb {Z} smallsetminus pm mathbb {Z}^2$</span></span></img></span></span>, ordered by height. For every sufficiently large <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$N$</span></span></img></span></span>, we prove that among equations in the family <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline9.png\"><span data-mathjax-type","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772611","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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