We study the $4$-rank of the ideal class group of $K_n := mathbb{Q}(sqrt{-n}, sqrt{n})$. Our main result is that for a positive proportion of the squarefree integers $n$ we have that the $4$-rank of $text{Cl}(K_n)$ equals $omega_3(n) - 1$, where $omega_3(n)$ is the number of prime divisors of $n$ that are $3$ modulo $4$.
{"title":"On Dirichlet biquadratic fields","authors":"'Etienne Fouvry, P. Koymans","doi":"10.5802/jtnb.1220","DOIUrl":"https://doi.org/10.5802/jtnb.1220","url":null,"abstract":"We study the $4$-rank of the ideal class group of $K_n := mathbb{Q}(sqrt{-n}, sqrt{n})$. Our main result is that for a positive proportion of the squarefree integers $n$ we have that the $4$-rank of $text{Cl}(K_n)$ equals $omega_3(n) - 1$, where $omega_3(n)$ is the number of prime divisors of $n$ that are $3$ modulo $4$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46491191","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 place $omega$ of a global function field $K$ over a finite field, with associated affine function ring $R_omega$ and completion $K_omega$, the aim of this paper is to give an effective joint equidistribution result for renormalized primitive lattice points $(a,b)in {R_omega}^2$ in the plane ${K_omega}^2$, and for renormalized solutions to the gcd equation $ax+by=1$. The main tools are techniques of Goronik and Nevo for counting lattice points in well-rounded families of subsets. This gives a sharper analog in positive characteristic of a result of Nevo and the first author for the equidistribution of the primitive lattice points in $ZZ^2$.
{"title":"Effective equidistribution of lattice points in positive characteristic","authors":"Tal Horesh, F. Paulin","doi":"10.5802/jtnb.1222","DOIUrl":"https://doi.org/10.5802/jtnb.1222","url":null,"abstract":"Given a place $omega$ of a global function field $K$ over a finite field, with associated affine function ring $R_omega$ and completion $K_omega$, the aim of this paper is to give an effective joint equidistribution result for renormalized primitive lattice points $(a,b)in {R_omega}^2$ in the plane ${K_omega}^2$, and for renormalized solutions to the gcd equation $ax+by=1$. The main tools are techniques of Goronik and Nevo for counting lattice points in well-rounded families of subsets. This gives a sharper analog in positive characteristic of a result of Nevo and the first author for the equidistribution of the primitive lattice points in $ZZ^2$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44141809","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}
Huy Quoc Dang, Soumya Das, K. Karagiannis, Andrew Obus, Vaidehee Thatte
It is conjectured that if k is an algebraically closed field of characteristic p > 0, then any branched G-cover of smooth projective k-curves where the "KGB" obstruction vanishes and where a p-Sylow subgroup of G is cyclic lifts to characteristic 0. Obus has shown that this conjecture holds given the existence of certain meromorphic differential forms on P_1^k with behavior determined by the ramification data of the cover. We give a more efficient computational procedure to compute these forms than was previously known. As a consequence, we show that all D_25- and D_27-covers lift to characteristic zero.
{"title":"Local Oort groups and the isolated differential data criterion","authors":"Huy Quoc Dang, Soumya Das, K. Karagiannis, Andrew Obus, Vaidehee Thatte","doi":"10.5802/jtnb.1200","DOIUrl":"https://doi.org/10.5802/jtnb.1200","url":null,"abstract":"It is conjectured that if k is an algebraically closed field of characteristic p > 0, then any branched G-cover of smooth projective k-curves where the \"KGB\" obstruction vanishes and where a p-Sylow subgroup of G is cyclic lifts to characteristic 0. Obus has shown that this conjecture holds given the existence of certain meromorphic differential forms on P_1^k with behavior determined by the ramification data of the cover. We give a more efficient computational procedure to compute these forms than was previously known. As a consequence, we show that all D_25- and D_27-covers lift to characteristic zero.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48330810","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}
The main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function $mathcal{L}_p^g(mathbf{f},mathbf{g},mathbf{h})$ associated to a triple of modular forms $(f,g,h)$ of weights $(2,1,1)$, in the case where the classical $L$-function $L(fotimes gotimes h,s)$ - which typically has sign $+1$ - does not vanish at its central critical point $s=1$. When $f$ corresponds to an elliptic curve $E/mathbb{Q}$ and the classical $L$-function vanishes, the Elliptic Stark Conjecture of Darmon-Lauder-Rotger predicts that $mathcal{L}_p^g(mathbf{f},mathbf{g},mathbf{h})(2,1,1)$ is either $0$ (when the order of vanishing of the complex $L$-function is $>2$) or related to logarithms of global points on $E$ and a certain Gross--Stark unit associated to $g$. We complete the picture proposed by the Elliptic Stark Conjecture by providing a formula for the value $mathcal{L}_p^g(mathbf{f},mathbf{g},mathbf{h})(2,1,1)$ in the case where $L(fotimes gotimes h,1)neq 0$.
{"title":"Special values of triple-product p-adic L-functions and non-crystalline diagonal classes","authors":"F. Gatti, Xavier Guitart, Marc Masdeu, V. Rotger","doi":"10.5802/jtnb.1179","DOIUrl":"https://doi.org/10.5802/jtnb.1179","url":null,"abstract":"The main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function $mathcal{L}_p^g(mathbf{f},mathbf{g},mathbf{h})$ associated to a triple of modular forms $(f,g,h)$ of weights $(2,1,1)$, in the case where the classical $L$-function $L(fotimes gotimes h,s)$ - which typically has sign $+1$ - does not vanish at its central critical point $s=1$. When $f$ corresponds to an elliptic curve $E/mathbb{Q}$ and the classical $L$-function vanishes, the Elliptic Stark Conjecture of Darmon-Lauder-Rotger predicts that $mathcal{L}_p^g(mathbf{f},mathbf{g},mathbf{h})(2,1,1)$ is either $0$ (when the order of vanishing of the complex $L$-function is $>2$) or related to logarithms of global points on $E$ and a certain Gross--Stark unit associated to $g$. We complete the picture proposed by the Elliptic Stark Conjecture by providing a formula for the value $mathcal{L}_p^g(mathbf{f},mathbf{g},mathbf{h})(2,1,1)$ in the case where $L(fotimes gotimes h,1)neq 0$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44135053","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 the first part, we revisit the theory of Drinfeld modular curves and π-adic Drinfeld modular forms for GL(2) from the perfectoid point of view. In the second part, we review open problems for families of Drinfeld modular forms for GL(N)
{"title":"Perfectoid Drinfeld Modular Forms","authors":"Marc-Hubert Nicole, Giovanni Rosso","doi":"10.5802/jtnb.1187","DOIUrl":"https://doi.org/10.5802/jtnb.1187","url":null,"abstract":"In the first part, we revisit the theory of Drinfeld modular curves and π-adic Drinfeld modular forms for GL(2) from the perfectoid point of view. In the second part, we review open problems for families of Drinfeld modular forms for GL(N)","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47285316","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}
An asymptotic formula for the variance of squarefree numbers in arithmetic progressions of given modulus was obtained by Nunes (see reference [3]). We improve one of the error terms.
{"title":"A variance for k-free numbers in arithmetic progressions of given modulus","authors":"T. Parry","doi":"10.5802/jtnb.1163","DOIUrl":"https://doi.org/10.5802/jtnb.1163","url":null,"abstract":"An asymptotic formula for the variance of squarefree numbers in arithmetic progressions of given modulus was obtained by Nunes (see reference [3]). We improve one of the error terms.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49381293","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}
Soit $K$ un corps $p$-adique et soit $V$ une representation $p$-adique de $mathcal{G}_K = mathrm{Gal}(bar{K}/K)$. La surconvergence des $(phi,tau)$-modules nous permet d'attacher a $V$ un $phi$-module differentiel a connexion $D_{tau,mathrm{rig}}^dagger(V)$ sur l'anneau de Robba $mathbf{B}_{tau,mathrm{rig},K}^dagger$. On montre dans cet article comment retrouver les invariants $D_{mathrm{cris}}(V)$ et $D_{mathrm{st}}(V)$ a partir de $D_{tau,mathrm{rig}}^dagger(V)$, et comment caracteriser les representations potentiellement semi-stables, ainsi que celles de $E$-hauteur finie, a partir de la connexion. Let $K$ be a $p$-adic field and let $V$ be a $p$-adic representation of $mathcal{G}_K=mathrm{Gal}(bar{K}/K)$. The overconvergence of $(phi,tau)$-modules allows us to attach to $V$ a differential $phi$-module $D_{tau,mathrm{rig}}^dagger(V)$ on the Robba ring $mathbf{B}_{tau,mathrm{rig},K}^dagger$ that comes equipped with a connection. We show in this paper how to recover the invariants $D_{mathrm{cris}}(V)$ and $D_{mathrm{st}}(V)$ from $D_{tau,mathrm{rig}}^dagger(V)$, and give a characterization of both potentially semi-stable representations of $mathcal{G}_K$ and finite $E$-height representations in terms of the connection operator.
{"title":"(ϕ,τ)-modules différentiels et représentations potentiellement semi-stables","authors":"Léo Poyeton","doi":"10.5802/JTNB.1156","DOIUrl":"https://doi.org/10.5802/JTNB.1156","url":null,"abstract":"Soit $K$ un corps $p$-adique et soit $V$ une representation $p$-adique de $mathcal{G}_K = mathrm{Gal}(bar{K}/K)$. La surconvergence des $(phi,tau)$-modules nous permet d'attacher a $V$ un $phi$-module differentiel a connexion $D_{tau,mathrm{rig}}^dagger(V)$ sur l'anneau de Robba $mathbf{B}_{tau,mathrm{rig},K}^dagger$. On montre dans cet article comment retrouver les invariants $D_{mathrm{cris}}(V)$ et $D_{mathrm{st}}(V)$ a partir de $D_{tau,mathrm{rig}}^dagger(V)$, et comment caracteriser les representations potentiellement semi-stables, ainsi que celles de $E$-hauteur finie, a partir de la connexion. \u0000Let $K$ be a $p$-adic field and let $V$ be a $p$-adic representation of $mathcal{G}_K=mathrm{Gal}(bar{K}/K)$. The overconvergence of $(phi,tau)$-modules allows us to attach to $V$ a differential $phi$-module $D_{tau,mathrm{rig}}^dagger(V)$ on the Robba ring $mathbf{B}_{tau,mathrm{rig},K}^dagger$ that comes equipped with a connection. We show in this paper how to recover the invariants $D_{mathrm{cris}}(V)$ and $D_{mathrm{st}}(V)$ from $D_{tau,mathrm{rig}}^dagger(V)$, and give a characterization of both potentially semi-stable representations of $mathcal{G}_K$ and finite $E$-height representations in terms of the connection operator.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46098043","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}
Effective periods are defined by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of Q-rational functions over Q-semi-algebraic domains in R^d. The Kontsevich-Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables and Stokes formula. In this paper, we discuss about possible geometric interpretations of this conjecture, viewed as a generalization of the Hilbert's third problem for compact semi-algebraic sets as well as for rational polyhedron equipped with piece-wise algebraic forms. Based on partial known results for analogous Hilbert's third problems, we study obstructions of possible geometric schemas to prove this conjecture.
{"title":"On the equality of periods of Kontsevich–Zagier","authors":"J. Cresson, Juan Viu-Sos","doi":"10.5802/jtnb.1204","DOIUrl":"https://doi.org/10.5802/jtnb.1204","url":null,"abstract":"Effective periods are defined by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of Q-rational functions over Q-semi-algebraic domains in R^d. The Kontsevich-Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables and Stokes formula. \u0000 \u0000In this paper, we discuss about possible geometric interpretations of this conjecture, viewed as a generalization of the Hilbert's third problem for compact semi-algebraic sets as well as for rational polyhedron equipped with piece-wise algebraic forms. Based on partial known results for analogous Hilbert's third problems, we study obstructions of possible geometric schemas to prove this conjecture.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49431848","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}
Soient $k$ un corps et $f:Xto Y$ un morphisme surjectif de $k$-varietes, avec $dim Y=dgeq1$. On precise des resultats de Poonen en montrant qu'il existe des sous-varietes $X'subset X$ et $Y'=f(X')subset Y$, de dimension $d-1$, telles que le morphisme induit $f':X'to Y'$ soit radiciel. Si $f$ est lisse, on peut exiger que $f'$ soit un isomorphisme, avec $Y'$ lisse si $Y$ l'est. Si $X$ est lisse, il existe un point ferme $x$ de $X$ ayant le meme corps residuel $k'$ que $f(x)$, $k'$ etant de plus separable sur $k$. Enfin, on donne des analogues arithmetiques. Let $k$ be a field and let $f:Xto Y$ be a surjective morphism of $k$-varieties with $dim Y=dgeq1$. Improving on results of Poonen, we prove that there are subvarieties $X'subset X$ and $Y'=f(X')subset Y$, of dimension $d-1$, such that the induced morphism $f':X'to Y'$ is purely inseparable. If $f$ is smooth, $f'$ can be taken to be an isomorphism, with $Y'$ smooth if $Y$ is. If $X$ is smooth, there is a closed point $xin X$ having the same residue field $k'$ as $f(x)$, with $k'$ separable over $k$. We also prove arithmetic analogues.
{"title":"Points rationnels dans leur fibre : compléments à un théorème de Poonen","authors":"Laurent Moret-Bailly","doi":"10.5802/jtnb.1130","DOIUrl":"https://doi.org/10.5802/jtnb.1130","url":null,"abstract":"Soient $k$ un corps et $f:Xto Y$ un morphisme surjectif de $k$-varietes, avec $dim Y=dgeq1$. On precise des resultats de Poonen en montrant qu'il existe des sous-varietes $X'subset X$ et $Y'=f(X')subset Y$, de dimension $d-1$, telles que le morphisme induit $f':X'to Y'$ soit radiciel. Si $f$ est lisse, on peut exiger que $f'$ soit un isomorphisme, avec $Y'$ lisse si $Y$ l'est. Si $X$ est lisse, il existe un point ferme $x$ de $X$ ayant le meme corps residuel $k'$ que $f(x)$, $k'$ etant de plus separable sur $k$. Enfin, on donne des analogues arithmetiques. Let $k$ be a field and let $f:Xto Y$ be a surjective morphism of $k$-varieties with $dim Y=dgeq1$. Improving on results of Poonen, we prove that there are subvarieties $X'subset X$ and $Y'=f(X')subset Y$, of dimension $d-1$, such that the induced morphism $f':X'to Y'$ is purely inseparable. If $f$ is smooth, $f'$ can be taken to be an isomorphism, with $Y'$ smooth if $Y$ is. If $X$ is smooth, there is a closed point $xin X$ having the same residue field $k'$ as $f(x)$, with $k'$ separable over $k$. We also prove arithmetic analogues.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49146064","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 formulate analogues of the Birch and Swinnerton-Dyer conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna attached to elliptic curves $E/mathbf{Q}$ at primes $p$ of good ordinary reduction. Using Iwasawa theory, we then prove under mild hypotheses one of the inequalities predicted by the rank part of our conjectures, as well as the predicted leading coefficient formula up to a $p$-adic unit. Our conjectures are very closely related to conjectures of Birch and Swinnerton-Dyer type formulated by Bertolini-Darmon in 1996 for certain Heegner distributions, and as application of our results we also obtain the proof of an inequality in the rank part of their conjectures.
{"title":"On anticyclotomic variants of the p-adic Birch and Swinnerton-Dyer conjecture","authors":"A. Agboola, Francesc Castella","doi":"10.5802/jtnb.1174","DOIUrl":"https://doi.org/10.5802/jtnb.1174","url":null,"abstract":"We formulate analogues of the Birch and Swinnerton-Dyer conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna attached to elliptic curves $E/mathbf{Q}$ at primes $p$ of good ordinary reduction. Using Iwasawa theory, we then prove under mild hypotheses one of the inequalities predicted by the rank part of our conjectures, as well as the predicted leading coefficient formula up to a $p$-adic unit. Our conjectures are very closely related to conjectures of Birch and Swinnerton-Dyer type formulated by Bertolini-Darmon in 1996 for certain Heegner distributions, and as application of our results we also obtain the proof of an inequality in the rank part of their conjectures.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48918624","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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