We provide a new, elementary proof of the multiplicative independence of pairwise distinct $mathrm{GL}_2^+(mathbb{Q})$-translates of the modular $j$-function, a result due originally to Pila and Tsimerman. We are thereby able to generalise this result to a wider class of modular functions. We show that this class includes a set comprising modular functions which arise naturally as Borcherds lifts of certain weakly holomorphic modular forms. For modular functions $f in overline{mathbb{Q}}(j)$ belonging to this class, we deduce, for each $n geq 1$, the finiteness of $n$-tuples of distinct $f$-special points that are multiplicatively dependent and minimal for this property. This generalises a theorem of Pila and Tsimerman on singular moduli. We then show how these results relate to the Zilber--Pink conjecture for subvarieties of the mixed Shimura variety $Y(1)^n times mathbb{G}_{mathrm{m}}^n$ and prove some special cases of this conjecture.
{"title":"Multiplicative independence of modular functions","authors":"G. Fowler","doi":"10.5802/jtnb.1167","DOIUrl":"https://doi.org/10.5802/jtnb.1167","url":null,"abstract":"We provide a new, elementary proof of the multiplicative independence of pairwise distinct $mathrm{GL}_2^+(mathbb{Q})$-translates of the modular $j$-function, a result due originally to Pila and Tsimerman. We are thereby able to generalise this result to a wider class of modular functions. We show that this class includes a set comprising modular functions which arise naturally as Borcherds lifts of certain weakly holomorphic modular forms. For modular functions $f in overline{mathbb{Q}}(j)$ belonging to this class, we deduce, for each $n geq 1$, the finiteness of $n$-tuples of distinct $f$-special points that are multiplicatively dependent and minimal for this property. This generalises a theorem of Pila and Tsimerman on singular moduli. We then show how these results relate to the Zilber--Pink conjecture for subvarieties of the mixed Shimura variety $Y(1)^n times mathbb{G}_{mathrm{m}}^n$ and prove some special cases of this conjecture.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49378113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is the third part of a series of papers discussing the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=39$, we show that $mathbb{Z}/Nmathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.
{"title":"On the cyclic torsion of elliptic curves over cubic number fields (II)","authors":"Jian Wang","doi":"10.5802/jtnb.1100","DOIUrl":"https://doi.org/10.5802/jtnb.1100","url":null,"abstract":"This is the third part of a series of papers discussing the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=39$, we show that $mathbb{Z}/Nmathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46889672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $m$ be a positive integer and let $E$ be an elliptic curve over $mathbb{Q}$ with the property that $mmid#E(mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability that $m$ divides the the order of the torsion subgroup of $E(mathbb{Q})$: we find it is nonzero for all $m in { 1, 2, dots, 10, 12, 16}$ and we compute it exactly when $m in { 1,2,3,4,5,7 }$. As a supplement, we give an asymptotic count of elliptic curves with extra level structure when the parametrizing modular curve is torsion free of genus zero.
{"title":"On a probabilistic local-global principle for torsion on elliptic curves","authors":"J. Cullinan, Meagan Kenney, J. Voight","doi":"10.5802/jtnb.1193","DOIUrl":"https://doi.org/10.5802/jtnb.1193","url":null,"abstract":"Let $m$ be a positive integer and let $E$ be an elliptic curve over $mathbb{Q}$ with the property that $mmid#E(mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability that $m$ divides the the order of the torsion subgroup of $E(mathbb{Q})$: we find it is nonzero for all $m in { 1, 2, dots, 10, 12, 16}$ and we compute it exactly when $m in { 1,2,3,4,5,7 }$. As a supplement, we give an asymptotic count of elliptic curves with extra level structure when the parametrizing modular curve is torsion free of genus zero.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43663203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the existence of local constancy phenomena for reductions in a general prime power setting of two-dimensional irreducible crystalline representations. Up to twist, these representations depend on two parameters: a trace $a_p$ and a weight $k$. We find an (explicit) local constancy result with respect to $a_p$ using Fontaine's theory of $(varphi, Gamma)$-modules and its crystalline refinement due to Berger via Wach modules and their continuity properties. The local constancy result with respect to $k$ (for $a_pnot=0$) will follow from a local study of Colmez's rigid analytic space parametrizing trianguline representations. This work extends some results of Berger obtained in the semi-simple residual case.
我们证明了在二维不可约晶体表示的一般素数幂集中，约化的局部恒定现象的存在。直到扭曲，这些表示取决于两个参数：轨迹$a_p$和权重$k$。使用Fontaine的$（varphi，Gamma）$-模理论及其由Berger via Wach模及其连续性性质引起的结晶精化，我们发现了关于$a_p$的（显式）局部恒定性结果。关于$k$（对于$a_pnot=0$）的局部恒定性结果将来自对Colmez刚性分析空间参数化三角线表示的局部研究。这项工作推广了Berger在半简单残差情况下得到的一些结果。
{"title":"Local constancy for reductions of two-dimensional crystalline representations","authors":"Emiliano Torti","doi":"10.5802/jtnb.1205","DOIUrl":"https://doi.org/10.5802/jtnb.1205","url":null,"abstract":"We prove the existence of local constancy phenomena for reductions in a general prime power setting of two-dimensional irreducible crystalline representations. Up to twist, these representations depend on two parameters: a trace $a_p$ and a weight $k$. We find an (explicit) local constancy result with respect to $a_p$ using Fontaine's theory of $(varphi, Gamma)$-modules and its crystalline refinement due to Berger via Wach modules and their continuity properties. The local constancy result with respect to $k$ (for $a_pnot=0$) will follow from a local study of Colmez's rigid analytic space parametrizing trianguline representations. This work extends some results of Berger obtained in the semi-simple residual case.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42703629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove or conjecture several relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or conjecturally equivalently algebraic). These are the first known relations between multizetas, which are not with prime field coefficients. We seem to have one universal family. We also find that interestingly the mechanism with which the relations work is quite different from the rational function field case, raising interesting questions about the expected motivic interpretation in higher genus. We provide some data in support of the guesses.
{"title":"Zeta-like Multizeta Values for higher genus curves","authors":"J. Rodr'iguez, D. Thakur","doi":"10.5802/jtnb.1169","DOIUrl":"https://doi.org/10.5802/jtnb.1169","url":null,"abstract":"We prove or conjecture several relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or conjecturally equivalently algebraic). These are the first known relations between multizetas, which are not with prime field coefficients. We seem to have one universal family. We also find that interestingly the mechanism with which the relations work is quite different from the rational function field case, raising interesting questions about the expected motivic interpretation in higher genus. We provide some data in support of the guesses.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45371359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a pair of elliptic curves $E_1,E_2$ over a field $k$, we have a natural map $text{CH}^1(E_1)_0otimestext{CH}^1(E_2)_0totext{CH}^2(E_1times E_2)$, and a conjecture due to Beilinson predicts that the image of this map is finite when $k$ is a number field. We construct a $2$-parameter family of elliptic curves that can be used to produce examples of pairs $E_1,E_2$ where this image is finite. The family is constructed to guarantee the existence of a rational curve passing through a specified point in the Kummer surface of $E_1times E_2$.
{"title":"Rational Equivalences on Products of Elliptic Curves in a Family","authors":"Jonathan R. Love","doi":"10.5802/JTNB.1148","DOIUrl":"https://doi.org/10.5802/JTNB.1148","url":null,"abstract":"Given a pair of elliptic curves $E_1,E_2$ over a field $k$, we have a natural map $text{CH}^1(E_1)_0otimestext{CH}^1(E_2)_0totext{CH}^2(E_1times E_2)$, and a conjecture due to Beilinson predicts that the image of this map is finite when $k$ is a number field. We construct a $2$-parameter family of elliptic curves that can be used to produce examples of pairs $E_1,E_2$ where this image is finite. The family is constructed to guarantee the existence of a rational curve passing through a specified point in the Kummer surface of $E_1times E_2$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48881437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given any integer $N>1$ prime to $3$, we denote by $C_N$ the elliptic curve $x^3+y^3=N$. We first study the $3$-adic valuation of the algebraic part of the value of the Hasse-Weil $L$-function $L(C_N,s)$ of $C_N$ over $mathbb{Q}$ at $s=1$, and we exhibit a relation between the $3$-part of its Tate-Shafarevich group and the number of distinct prime divisors of $N$ which are inert in the imaginary quadratic field $K=mathbb{Q}(sqrt{-3})$. In the case where $L(C_N,1)neq 0$ and $N$ is a product of split primes in $K$, we show that the order of the Tate-Shafarevich group as predicted by the conjecture of Birch and Swinnerton-Dyer is a perfect square.
{"title":"Tamagawa number divisibility of central L-values of twists of the Fermat elliptic curve","authors":"Yukako Kezuka","doi":"10.5802/jtnb.1183","DOIUrl":"https://doi.org/10.5802/jtnb.1183","url":null,"abstract":"Given any integer $N>1$ prime to $3$, we denote by $C_N$ the elliptic curve $x^3+y^3=N$. We first study the $3$-adic valuation of the algebraic part of the value of the Hasse-Weil $L$-function $L(C_N,s)$ of $C_N$ over $mathbb{Q}$ at $s=1$, and we exhibit a relation between the $3$-part of its Tate-Shafarevich group and the number of distinct prime divisors of $N$ which are inert in the imaginary quadratic field $K=mathbb{Q}(sqrt{-3})$. In the case where $L(C_N,1)neq 0$ and $N$ is a product of split primes in $K$, we show that the order of the Tate-Shafarevich group as predicted by the conjecture of Birch and Swinnerton-Dyer is a perfect square.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43286412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a recent work of Darmon, Pozzi and Vonk, the authors consider a particular $p$-adic family of Hilbert Eisenstein series $E_k(1,brch)$ associated with an odd character $brch$ of the narrow ideal class group of a real quadratic field $F$ and compute the first derivative of a certain one-variable twisted triple product $p$-adic $L$-series attached to $E_k(1,brch)$ and an elliptic newform $f$ of weight $2$ on $Gamma_0(p)$. In this paper, we generalize their construction to include the cyclotomic variable and thus obtain a two-variable twisted triple product $p$-adic $L$-series. Moreover, when $f$ is associated with an elliptic curve $E$ over $Q$, we prove that the first derivative of this $p$-adic $L$-series along the weight direction is a product of the $p$-adic logarithm of a Stark-Heegner point of $E$ over $F$ introduced by Darmon and the cyclotomic $p$-adic $L$-function for $E$.
{"title":"Restriction of Eisenstein series and Stark–Heegner points","authors":"Ming-Lun Hsieh, Shunsuke Yamana","doi":"10.5802/jtnb.1182","DOIUrl":"https://doi.org/10.5802/jtnb.1182","url":null,"abstract":"In a recent work of Darmon, Pozzi and Vonk, the authors consider a particular $p$-adic family of Hilbert Eisenstein series $E_k(1,brch)$ associated with an odd character $brch$ of the narrow ideal class group of a real quadratic field $F$ and compute the first derivative of a certain one-variable twisted triple product $p$-adic $L$-series attached to $E_k(1,brch)$ and an elliptic newform $f$ of weight $2$ on $Gamma_0(p)$. In this paper, we generalize their construction to include the cyclotomic variable and thus obtain a two-variable twisted triple product $p$-adic $L$-series. Moreover, when $f$ is associated with an elliptic curve $E$ over $Q$, we prove that the first derivative of this $p$-adic $L$-series along the weight direction is a product of the $p$-adic logarithm of a Stark-Heegner point of $E$ over $F$ introduced by Darmon and the cyclotomic $p$-adic $L$-function for $E$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46274875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study $p$-adic Hodge theory for families of Galois representations over pseudorigid spaces. Such spaces are non-archimedean analytic spaces which may be of mixed characteristic, and which arise naturally in the study of eigenvarieties at the boundary of weight space. We construct overconvergent $(varphi,Gamma)$-modules for Galois representations over pseudorigid spaces, and we show that such $(varphi,Gamma)$-modules have finite cohomology. As a consequence, we deduce that the cohomology groups yield coherent sheaves, and we give partial results extending triangulations defined away from closed subspaces.
{"title":"Galois representations over pseudorigid spaces","authors":"Rebecca Bellovin","doi":"10.5802/jtnb.1246","DOIUrl":"https://doi.org/10.5802/jtnb.1246","url":null,"abstract":"We study $p$-adic Hodge theory for families of Galois representations over pseudorigid spaces. Such spaces are non-archimedean analytic spaces which may be of mixed characteristic, and which arise naturally in the study of eigenvarieties at the boundary of weight space. We construct overconvergent $(varphi,Gamma)$-modules for Galois representations over pseudorigid spaces, and we show that such $(varphi,Gamma)$-modules have finite cohomology. As a consequence, we deduce that the cohomology groups yield coherent sheaves, and we give partial results extending triangulations defined away from closed subspaces.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45356733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field $F'$ such that $E$ has semistable reduction at every prime of $F'$ above $p$ and is supersingular at at least one prime above $p$. Under appropriate hypotheses, we compute the Akashi series of the signed Selmer groups of $E$ over a $mathbb{Z}_p^d$-extension over a finite extension $F$ of $F'$. As a by-product, we also compute the Euler characteristics of these Selmer groups.
{"title":"Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p","authors":"Antonio Lei, M. Lim","doi":"10.5802/jtnb.1185","DOIUrl":"https://doi.org/10.5802/jtnb.1185","url":null,"abstract":"Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field $F'$ such that $E$ has semistable reduction at every prime of $F'$ above $p$ and is supersingular at at least one prime above $p$. Under appropriate hypotheses, we compute the Akashi series of the signed Selmer groups of $E$ over a $mathbb{Z}_p^d$-extension over a finite extension $F$ of $F'$. As a by-product, we also compute the Euler characteristics of these Selmer groups.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45244768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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