A central result in extremal combinatorics is textit{the ErdH os--Ko--Rado Theorem} which investigates the maximum size of $cA subset binom{[n]}{k}$ such that for every choice of sets $A_1, A_2in cA$ we have $|A_1cap A_2| geq t$. In this talk we consider a version with two families. Two families $cA$ and $cB subset binom{[n]}{k}$ are {em cross $t$-intersecting} if for every choice of sets $A in cA$ and $B in cB$ we have $|A cap B| geq t$. The following `cross $t$-intersecting version' of the ErdH os-Ko-Rado Theorem was conjectured: For all $n geq (t+1)(k-t+1)$ the maximum value of $|cA||cB|$ for two cross $t$-intersecting families $cA, cB subsetbinom{[n]}{k}$ is $binom{n-t}{k-t}^2$.We verified a strongly related textit{$p$-weighted version} of the above conjecture for $t geq 14$. For $0
Let $E$ be a relative quadratic extension of a number field $F$, the 2-Sylow subgroup of the class group of $F$ elementary abelian, and the $S$-ideal class number of $F$ odd, where $S$ is a set consisting of all infinite primes and some finite primes of $F$ ramifying in $E$. In this paper, we compute the 2-rank and 4-rank of the class group of $E$.
The Jacobi group, the semi-direct product of the symplectic group and the Heisenberg group is one of the simplest and most important examples of a non-reductive Lie group. Its associated non-reductive homogeneous space is the so-called Siegel-Jacobi space that is very important arithmetically and geometrically. In this lecture, I develop the theory of harmonic analysis on the Siegel-Jacobi space. I plan to introduce some results about harmonic analysis on the Siegel-Jacobi space obtained by J.-H. Yang, E. Balslev, F. Gay-Balmaz, C. Tronci, S. Berceanu, A. Gheorge, H. Ochiai and M. Itoh.
Let X be a compact Riemann surface (or equivalently, a function field over the complex numbers, or equivalently a smooth projective curve over the complex numbers) of genus at least 2. A cyclic subgroup of prime order p of Aut(X) is called properly (p,h)-gonal if it has a fixed point and the quotient surface has genus h. We show that if p>6h+6>11, then a properly (p,h)-gonal subgroup of Aut(X) is unique.
Let be the Möbius function on the polynomial ring Fq[T] over the nite field Fq of q elements. We compute the (asymptotic) mean value and variance of Möbius function in short intervals as q→∞.
For any positive integer n and real number λ≧1, let Kn (λ) be the set of all real numbers Ɵ such that for any λ’<λ, there exist infinitely many algebraic numbers α with degree at most n satisfying | Ɵ -α| < H(α)-(n+1)λ’. Moreover we let Kn ‘(λ) be the subset of Kn (λ) consisting of all Ɵ not in Kn (λ’) for any λ’<λ. Baker and Schmidt proved that the Hausdorff dimensions of Kn (λ) and Kn ‘(λ) are both 1/λ. We will give similar results of Baker and Schmidt in function field version.
Let $X$ be a Mumford curve. We say that an elliptic curve is an optimal quotient of $X$ is there is a finite morphism $Xto E$ such that the homomorphism $pi: Jac(X)to E$ induced by the Albanese functoriality has connected and reduced kernel. We consider the functorially induced map $pi_ast: Phi_Xto Phi_E$ on component groups of the Neron models of $Jac(X)$ and $E$. We show that in general this map need not be surjective, which answers negatively a question of Ribet and Takahashi. Using rigid-analytic techniques, we give some conditions under which $pi_ast$ is surjective, and discuss arithmetic applications to modular curves. This is a joint work with Joe Rabinoff.
Let Gamma(T) be the full congruence subgroup of level T in the Drinfeld modular group Gamma:=GL(2, F_q[T]). We (joint work with my PhD student Enrico Varela) consider the ring of modular forms Gamma(T) as a module under the quotient group G=Gamma/Gamma(T), which is isomorphic with GL(2,F_q). We have some results about Jordan-Hölder series of the space of forms of fixed weight k, considered as a G-module.
Let $p$ and $q$ be distinct primes with $gcd(p-1,q-1)=4$, Whiteman gave the sixteen cyclotomic numbers depending solely upon one of the two decompositions: $pq=a^2+4b^2=a'^2+4b'^2, aequiv a'equiv 1pmod 4$. In this paper, we will determine the sixteen cyclotomic numbers of order four depending on unique $a, b$ if we choose a common primitive root of $p$ and $q$.
Modular curves parametrize elliptic curves with prescribed torsion structures. In this talk I will investigate torsion of elliptic curves over number fields by applying canonical models of modular curves.
In the past modular curves of various type (classical, Drinfeld, Shimura) have been used successfully to construct high genus curves with many rational points over finite fields of square cardinality. In this talk I will explain how Drinfeld modular varieties can be used similarly to obtain high genus curves over any non-prime finite field with many rational points. This way we obtain lower bounds for the Ihara constant A(q) for all non-prime q, which are better then all previously known bounds.
This talk comprises two applications of shift operators to characterization of continuous functions and ergodic functions defined on the integer ring of a non-Archimedean local field of positive characteristic. In the first part of the talk, we establish that digit expansion of shift operators becomes an orthonormal basis for the space of continuous functions on ${F}_q[[T]],$ including a closed-form expression for expansion coefficients, and we establish that this is also true for $p$-adic integers, excluding the coefficient formula. In the second part, we obtain the necessary and sufficient conditions for ergodicity of 1-Lipschitz functions represented on ${F}_2[[T]]$ by digit shift operators, recalling the cases with the Carlitz polynomials and digit derivatives.
For any function field k, together with a fixed place called the infinity, we give explicit construction of ''heta series'' from Eichler orders of definite quaternion algebra over k. These theta series are recognized as automorphic forms of Drinfeld type for GL2 over k. Description of the space generated by these theta series inside all Drinfeld type automorphic forms is given. This gives an answer to the basis problem for Drinfeld type automorphic forms.
Given two non-isogenous rank $r$ Drinfeld $A$-modules $phi$ and $phi'$ over $K$, where $F = F_q(T)$, $A = F_q[T]$, and $K$ is a finite extension of $F$, we obtain a partially explicit upper bound (dependent only on $phi$ and $phi'$) on the degree of primes $p$ of $K$ such that $P_p(phi) not= P_p(phi')$, where $P_p(ast)$ denotes the characteristic polynomial of Frobenius at $p$ on a Tate module of $ast$. The bounds are completely explicit in terms of the defining coefficients of $phi$ and $phi'$, except for one term, which can be made explicit in the rank 2 case by determining the Newton polygons of exponential functions attached to $phi$ and the successive minima of the lattice associated to $phi$ by uniformization. An ingredient in the proof of the explicit isogeny theorem for general rank is an explicit bound for the different divisor of torsion fields of Drinfeld modules which detects primes of potentially good reduction.
We give asymptotic formulas for the number of biquadratic extensions of $F_q[T]$ that admit a quadratic extension which is a Galois extension of $F_q[T]$ with a prescribed Galois group.
In this talk we will present recent results that show the existence of infinitely many cuspidal Drinfeld eigenforms that possess A-expansions. In addition, we will give examples (some of which conjectural) of vectorial Drinfeld modular forms with A-expansions.
In this talk, we construct infinite families of elliptic curves over quartic number fields with torsion group $mathbb Z/Nmathbb Z$ with $N = 20,24$. Moreover, we prove that for each elliptic curve $E_t$ in the constructed families, the Galois group $Gal(L/mathbb Q)$ is isomorphic to the Dihedral group $D_4$ of order 8 for the Galois closure $L$ of $K$ over $mathbb Q$, where $K$ is the defining field of $(E_t ,Q_t)
VOD the MathNet Korea
A central result in extremal combinatorics is textit{the ErdH os--Ko--Rado Theorem} which investigates the maximum size of $cA subset binom{[n]}{k}$ such that for every choice of sets $A_1, A_2in cA$ we have $|A_1cap A_2| geq t$. In this talk we consider a version with two families. Two families $cA$ and $cB subset binom{[n]}{k}$ are {em cross $t$-intersecting} if for every choice of sets $A in cA$ and $B in cB$ we have $|A cap B| geq t$. The following `cross $t$-intersecting version' of the ErdH os-Ko-Rado Theorem was conjectured: For all $n geq (t+1)(k-t+1)$ the maximum value of $|cA||cB|$ for two cross $t$-intersecting families $cA, cB subsetbinom{[n]}{k}$ is $binom{n-t}{k-t}^2$.We verified a strongly related textit{$p$-weighted version} of the above conjecture for $t geq 14$. For $0
Let $E$ be a relative quadratic extension of a number field $F$, the 2-Sylow subgroup of the class group of $F$ elementary abelian, and the $S$-ideal class number of $F$ odd, where $S$ is a set consisting of all infinite primes and some finite primes of $F$ ramifying in $E$. In this paper, we compute the 2-rank and 4-rank of the class group of $E$.
The Jacobi group, the semi-direct product of the symplectic group and the Heisenberg group is one of the simplest and most important examples of a non-reductive Lie group. Its associated non-reductive homogeneous space is the so-called Siegel-Jacobi space that is very important arithmetically and geometrically. In this lecture, I develop the theory of harmonic analysis on the Siegel-Jacobi space. I plan to introduce some results about harmonic analysis on the Siegel-Jacobi space obtained by J.-H. Yang, E. Balslev, F. Gay-Balmaz, C. Tronci, S. Berceanu, A. Gheorge, H. Ochiai and M. Itoh.
Let X be a compact Riemann surface (or equivalently, a function field over the complex numbers, or equivalently a smooth projective curve over the complex numbers) of genus at least 2. A cyclic subgroup of prime order p of Aut(X) is called properly (p,h)-gonal if it has a fixed point and the quotient surface has genus h. We show that if p>6h+6>11, then a properly (p,h)-gonal subgroup of Aut(X) is unique.
Let be the Möbius function on the polynomial ring Fq[T] over the nite field Fq of q elements. We compute the (asymptotic) mean value and variance of Möbius function in short intervals as q→∞.
For any positive integer n and real number λ≧1, let Kn (λ) be the set of all real numbers Ɵ such that for any λ’<λ, there exist infinitely many algebraic numbers α with degree at most n satisfying | Ɵ -α| < H(α)-(n+1)λ’. Moreover we let Kn ‘(λ) be the subset of Kn (λ) consisting of all Ɵ not in Kn (λ’) for any λ’<λ. Baker and Schmidt proved that the Hausdorff dimensions of Kn (λ) and Kn ‘(λ) are both 1/λ. We will give similar results of Baker and Schmidt in function field version.
Let $X$ be a Mumford curve. We say that an elliptic curve is an optimal quotient of $X$ is there is a finite morphism $Xto E$ such that the homomorphism $pi: Jac(X)to E$ induced by the Albanese functoriality has connected and reduced kernel. We consider the functorially induced map $pi_ast: Phi_Xto Phi_E$ on component groups of the Neron models of $Jac(X)$ and $E$. We show that in general this map need not be surjective, which answers negatively a question of Ribet and Takahashi. Using rigid-analytic techniques, we give some conditions under which $pi_ast$ is surjective, and discuss arithmetic applications to modular curves. This is a joint work with Joe Rabinoff.
Let Gamma(T) be the full congruence subgroup of level T in the Drinfeld modular group Gamma:=GL(2, F_q[T]). We (joint work with my PhD student Enrico Varela) consider the ring of modular forms Gamma(T) as a module under the quotient group G=Gamma/Gamma(T), which is isomorphic with GL(2,F_q). We have some results about Jordan-Hölder series of the space of forms of fixed weight k, considered as a G-module.
Let $p$ and $q$ be distinct primes with $gcd(p-1,q-1)=4$, Whiteman gave the sixteen cyclotomic numbers depending solely upon one of the two decompositions: $pq=a^2+4b^2=a'^2+4b'^2, aequiv a'equiv 1pmod 4$. In this paper, we will determine the sixteen cyclotomic numbers of order four depending on unique $a, b$ if we choose a common primitive root of $p$ and $q$.
Modular curves parametrize elliptic curves with prescribed torsion structures. In this talk I will investigate torsion of elliptic curves over number fields by applying canonical models of modular curves.
In the past modular curves of various type (classical, Drinfeld, Shimura) have been used successfully to construct high genus curves with many rational points over finite fields of square cardinality. In this talk I will explain how Drinfeld modular varieties can be used similarly to obtain high genus curves over any non-prime finite field with many rational points. This way we obtain lower bounds for the Ihara constant A(q) for all non-prime q, which are better then all previously known bounds.
This talk comprises two applications of shift operators to characterization of continuous functions and ergodic functions defined on the integer ring of a non-Archimedean local field of positive characteristic. In the first part of the talk, we establish that digit expansion of shift operators becomes an orthonormal basis for the space of continuous functions on ${F}_q[[T]],$ including a closed-form expression for expansion coefficients, and we establish that this is also true for $p$-adic integers, excluding the coefficient formula. In the second part, we obtain the necessary and sufficient conditions for ergodicity of 1-Lipschitz functions represented on ${F}_2[[T]]$ by digit shift operators, recalling the cases with the Carlitz polynomials and digit derivatives.
For any function field k, together with a fixed place called the infinity, we give explicit construction of ''heta series'' from Eichler orders of definite quaternion algebra over k. These theta series are recognized as automorphic forms of Drinfeld type for GL2 over k. Description of the space generated by these theta series inside all Drinfeld type automorphic forms is given. This gives an answer to the basis problem for Drinfeld type automorphic forms.
Given two non-isogenous rank $r$ Drinfeld $A$-modules $phi$ and $phi'$ over $K$, where $F = F_q(T)$, $A = F_q[T]$, and $K$ is a finite extension of $F$, we obtain a partially explicit upper bound (dependent only on $phi$ and $phi'$) on the degree of primes $p$ of $K$ such that $P_p(phi) not= P_p(phi')$, where $P_p(ast)$ denotes the characteristic polynomial of Frobenius at $p$ on a Tate module of $ast$. The bounds are completely explicit in terms of the defining coefficients of $phi$ and $phi'$, except for one term, which can be made explicit in the rank 2 case by determining the Newton polygons of exponential functions attached to $phi$ and the successive minima of the lattice associated to $phi$ by uniformization. An ingredient in the proof of the explicit isogeny theorem for general rank is an explicit bound for the different divisor of torsion fields of Drinfeld modules which detects primes of potentially good reduction.
We give asymptotic formulas for the number of biquadratic extensions of $F_q[T]$ that admit a quadratic extension which is a Galois extension of $F_q[T]$ with a prescribed Galois group.
In this talk we will present recent results that show the existence of infinitely many cuspidal Drinfeld eigenforms that possess A-expansions. In addition, we will give examples (some of which conjectural) of vectorial Drinfeld modular forms with A-expansions.
In this talk, we construct infinite families of elliptic curves over quartic number fields with torsion group $mathbb Z/Nmathbb Z$ with $N = 20,24$. Moreover, we prove that for each elliptic curve $E_t$ in the constructed families, the Galois group $Gal(L/mathbb Q)$ is isomorphic to the Dihedral group $D_4$ of order 8 for the Galois closure $L$ of $K$ over $mathbb Q$, where $K$ is the defining field of $(E_t ,Q_t)
VOD the MathNet Korea
A central result in extremal combinatorics is textit{the ErdH os--Ko--Rado Theorem} which investigates the maximum size of $cA subset binom{[n]}{k}$ such that for every choice of sets $A_1, A_2in cA$ we have $|A_1cap A_2| geq t$. In this talk we consider a version with two families. Two families $cA$ and $cB subset binom{[n]}{k}$ are {em cross $t$-intersecting} if for every choice of sets $A in cA$ and $B in cB$ we have $|A cap B| geq t$. The following `cross $t$-intersecting version' of the ErdH os-Ko-Rado Theorem was conjectured: For all $n geq (t+1)(k-t+1)$ the maximum value of $|cA||cB|$ for two cross $t$-intersecting families $cA, cB subsetbinom{[n]}{k}$ is $binom{n-t}{k-t}^2$.We verified a strongly related textit{$p$-weighted version} of the above conjecture for $t geq 14$. For $0
Let $E$ be a relative quadratic extension of a number field $F$, the 2-Sylow subgroup of the class group of $F$ elementary abelian, and the $S$-ideal class number of $F$ odd, where $S$ is a set consisting of all infinite primes and some finite primes of $F$ ramifying in $E$. In this paper, we compute the 2-rank and 4-rank of the class group of $E$.
The Jacobi group, the semi-direct product of the symplectic group and the Heisenberg group is one of the simplest and most important examples of a non-reductive Lie group. Its associated non-reductive homogeneous space is the so-called Siegel-Jacobi space that is very important arithmetically and geometrically. In this lecture, I develop the theory of harmonic analysis on the Siegel-Jacobi space. I plan to introduce some results about harmonic analysis on the Siegel-Jacobi space obtained by J.-H. Yang, E. Balslev, F. Gay-Balmaz, C. Tronci, S. Berceanu, A. Gheorge, H. Ochiai and M. Itoh.
Let X be a compact Riemann surface (or equivalently, a function field over the complex numbers, or equivalently a smooth projective curve over the complex numbers) of genus at least 2. A cyclic subgroup of prime order p of Aut(X) is called properly (p,h)-gonal if it has a fixed point and the quotient surface has genus h. We show that if p>6h+6>11, then a properly (p,h)-gonal subgroup of Aut(X) is unique.
Let be the Möbius function on the polynomial ring Fq[T] over the nite field Fq of q elements. We compute the (asymptotic) mean value and variance of Möbius function in short intervals as q→∞.
For any positive integer n and real number λ≧1, let Kn (λ) be the set of all real numbers Ɵ such that for any λ’<λ, there exist infinitely many algebraic numbers α with degree at most n satisfying | Ɵ -α| < H(α)-(n+1)λ’. Moreover we let Kn ‘(λ) be the subset of Kn (λ) consisting of all Ɵ not in Kn (λ’) for any λ’<λ. Baker and Schmidt proved that the Hausdorff dimensions of Kn (λ) and Kn ‘(λ) are both 1/λ. We will give similar results of Baker and Schmidt in function field version.
Let $X$ be a Mumford curve. We say that an elliptic curve is an optimal quotient of $X$ is there is a finite morphism $Xto E$ such that the homomorphism $pi: Jac(X)to E$ induced by the Albanese functoriality has connected and reduced kernel. We consider the functorially induced map $pi_ast: Phi_Xto Phi_E$ on component groups of the Neron models of $Jac(X)$ and $E$. We show that in general this map need not be surjective, which answers negatively a question of Ribet and Takahashi. Using rigid-analytic techniques, we give some conditions under which $pi_ast$ is surjective, and discuss arithmetic applications to modular curves. This is a joint work with Joe Rabinoff.
Let Gamma(T) be the full congruence subgroup of level T in the Drinfeld modular group Gamma:=GL(2, F_q[T]). We (joint work with my PhD student Enrico Varela) consider the ring of modular forms Gamma(T) as a module under the quotient group G=Gamma/Gamma(T), which is isomorphic with GL(2,F_q). We have some results about Jordan-Hölder series of the space of forms of fixed weight k, considered as a G-module.
Let $p$ and $q$ be distinct primes with $gcd(p-1,q-1)=4$, Whiteman gave the sixteen cyclotomic numbers depending solely upon one of the two decompositions: $pq=a^2+4b^2=a'^2+4b'^2, aequiv a'equiv 1pmod 4$. In this paper, we will determine the sixteen cyclotomic numbers of order four depending on unique $a, b$ if we choose a common primitive root of $p$ and $q$.
Modular curves parametrize elliptic curves with prescribed torsion structures. In this talk I will investigate torsion of elliptic curves over number fields by applying canonical models of modular curves.
In the past modular curves of various type (classical, Drinfeld, Shimura) have been used successfully to construct high genus curves with many rational points over finite fields of square cardinality. In this talk I will explain how Drinfeld modular varieties can be used similarly to obtain high genus curves over any non-prime finite field with many rational points. This way we obtain lower bounds for the Ihara constant A(q) for all non-prime q, which are better then all previously known bounds.
This talk comprises two applications of shift operators to characterization of continuous functions and ergodic functions defined on the integer ring of a non-Archimedean local field of positive characteristic. In the first part of the talk, we establish that digit expansion of shift operators becomes an orthonormal basis for the space of continuous functions on ${F}_q[[T]],$ including a closed-form expression for expansion coefficients, and we establish that this is also true for $p$-adic integers, excluding the coefficient formula. In the second part, we obtain the necessary and sufficient conditions for ergodicity of 1-Lipschitz functions represented on ${F}_2[[T]]$ by digit shift operators, recalling the cases with the Carlitz polynomials and digit derivatives.
For any function field k, together with a fixed place called the infinity, we give explicit construction of ''heta series'' from Eichler orders of definite quaternion algebra over k. These theta series are recognized as automorphic forms of Drinfeld type for GL2 over k. Description of the space generated by these theta series inside all Drinfeld type automorphic forms is given. This gives an answer to the basis problem for Drinfeld type automorphic forms.
Given two non-isogenous rank $r$ Drinfeld $A$-modules $phi$ and $phi'$ over $K$, where $F = F_q(T)$, $A = F_q[T]$, and $K$ is a finite extension of $F$, we obtain a partially explicit upper bound (dependent only on $phi$ and $phi'$) on the degree of primes $p$ of $K$ such that $P_p(phi) not= P_p(phi')$, where $P_p(ast)$ denotes the characteristic polynomial of Frobenius at $p$ on a Tate module of $ast$. The bounds are completely explicit in terms of the defining coefficients of $phi$ and $phi'$, except for one term, which can be made explicit in the rank 2 case by determining the Newton polygons of exponential functions attached to $phi$ and the successive minima of the lattice associated to $phi$ by uniformization. An ingredient in the proof of the explicit isogeny theorem for general rank is an explicit bound for the different divisor of torsion fields of Drinfeld modules which detects primes of potentially good reduction.
We give asymptotic formulas for the number of biquadratic extensions of $F_q[T]$ that admit a quadratic extension which is a Galois extension of $F_q[T]$ with a prescribed Galois group.
In this talk we will present recent results that show the existence of infinitely many cuspidal Drinfeld eigenforms that possess A-expansions. In addition, we will give examples (some of which conjectural) of vectorial Drinfeld modular forms with A-expansions.
In this talk, we construct infinite families of elliptic curves over quartic number fields with torsion group $mathbb Z/Nmathbb Z$ with $N = 20,24$. Moreover, we prove that for each elliptic curve $E_t$ in the constructed families, the Galois group $Gal(L/mathbb Q)$ is isomorphic to the Dihedral group $D_4$ of order 8 for the Galois closure $L$ of $K$ over $mathbb Q$, where $K$ is the defining field of $(E_t ,Q_t)$# and $Q_t$ is a point of $E_t$ of order $N$.
We give a simple criterion for two v-adic Galois representations of a global field K to be locally isomorphic at another place u in terms of their reductions mod v. Such a study is motivated by the Rasmussen-Tamagawa conjecture for Abelian varieties.
Let $k$ be an integer such that $kequiv 0 pmod {q-1}$. Then there exist unique integers $l_k$ and $r_k$ such that $0leq r_kleq q$ and $k=(q^2-1)l_k +(q-1)r_k$. For each integer $mgeq-l_k$,there exists a unique weakly holomorphic Drinfeld modular form $f_{k,m}$ of weight $k$ for $GL_2(A)$ with a $t$-expansion of the form begin{equation*}f_{k,m}(z)=t^{-(q-1)m} +O(t^{(q-1)(l_k+1)}) end{equation*}. In this talk, we review some properties of these forms $f_{k,m}$ and prove that the zeros of $f_{k,m}(z)$ in the fundamental domain $frak{F}$ are on the unit circle $|z|=1$ under a certain condition and in addition, if $q$ is odd then the zeros of $f_{k,m}$ are transcendental over $K$ or elliptic points. Where $frak{F}:={zin Omega~:~ |z|=text{inf}{|z-a|~ : ~ a in Bbb{F}_q[T]}geq 1}$. This is a work with Bohae Im We verified a strongly related textit{$p$-weighted version} of the above conjecture for $t geq 14$. For $0
Information Center for Mathematical Sciences KAIST
335 Gwahangno (373-1 Guseong-dong), Yuseong-gu, Daejeon 305-701, Republic of Korea
TEL +82-42-869-8195~6, FAX +82-42-869-5722, e-mail : mathnet@mathnet.or.kr
Copyright (c) 2009 by ICMS. All Rights Reserved
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
.jpg){kind=small}
