Franz Lemmermeyer
This is an updated version of ANT-0166. Generalizing results of Stroeker and Top we show that the 2-ranks of the Tate-Shafarevich groups of the elliptic curves $y^2 = (x+k)(x^2+k^2)$ can become arbitrarily large. We also present a conjecture on the rank of the Selmer groups attached to rational 2-isogenies of elliptic curves.
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In this paper we formulate generalised Isogeny, Mumford-Tate and Shafarevich conjectures and prove that they are equivalent.
We describe the structure of Tate-Shafarevich groups of a constant elliptic curves over function fields by exploiting the volcano structure of isogeny graphs of elliptic curves over finite fields.
We present the results of our search for the orders of Tate-Shafarevich groups for the Neumann-Setzer type elliptic curves.
The Shafarevich-Tate and Selmer groups arise in the context of Kummer theory for elliptic curves. The finiteness of the Shafarevich-Tate group of an elliptic curve $E$ over the field of rational numbers is included in the Birch and Swinnerton-Dyer conjectures, and is still an open question. We present an overview of the Shafarevich-Tate and Selmer groups of an elliptic curve in the framework of Galois cohomology. Known results on the finiteness of the Shafarevich-Tate group are mentioned, including results of Coates and Wiles, Rubin, Gross and Zagier, and Kolyvagin. We then prove the vanishing…
A note on the order of the Tate--Shafarevich group modulo squares
本文研究了阿贝尔簇的Tate--Shafarevich群模有理平方的阶，证明了每个无平方因子的自然数均可作为某个阿贝尔簇Tate--Shafarevich群的非无平方因子部分出现，从而验证了W. Stein的猜想。
We characterize quadratic twists of $y^2=x(x-a^2)(x+b^2)$ with Mordell-Weil groups and $2$-primary part of Shafarevich-Tate groups being isomorphic to $(\mathb Z/2\mathbb Z)^2$ under certain conditions. We also obtain the distribution result of these elliptic curves.
Franz Lemmermeyer's previous work laid the framework for a description of the arithmetic of Pell conics, which is analogous to that of elliptic curves. He describes a group law on conics and conjectures the existence of an analogous Tate--Shafarevich group with order the squared ideals of the narrow class group. In this article, we provide a cohomological definition of the Tate--Shafarevich group and show that its order is as Lemmermeyer conjectured.
The structure of the Tate-Shafarevich groups of a class of elliptic curves over global function fields is determined. These are known to be finite abelian groups from the monograph [1] and hence they are direct sums of finite cyclic groups where the orders of these cyclic components are invariants of the Tate-Shafarevich group. This decomposition of the Tate-Shafarevich groups into direct sums of finite cyclic groups depends on the behaviour of Drinfeld-Heegner points on these elliptic curves. These are points analogous to Heegner points on elliptic curves over the rational numbers.
We study the asymptotic growth of the p-primary component of the Shafarevich-Tate group in the cyclotomic direction at any odd prime of good supersingular reduction, generalizing work of Kobayashi. This explains formulas obtained by Kurihara, Perrin-Riou, and Nasybullin in terms of Iwasawa invariants of modified Selmer groups.
We study a subclass of congruent elliptic curves $E^{(n)}: y^2=x^3-n^2x$, where $n$ is a positive integer congruent to $1\pmod 8$ with all prime factors congruent to $1\pmod 4$. We characterize such $E^{(n)}$ with Mordell-Weil rank zero and $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z/2\mathbb Z \big)^2$. We also discuss such $E^{(n)}$ with 2-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z/2\mathbb Z \big)^{2k}$ with $k\ge2$.
This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann-Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$. Our calculations extend those given by Zagier and Kramarz \cite{ZK} and by Watkins \cite{Wat}. Our main observations concern the asymptotic formula for the frequency of orders of Tate-Shafarevich groups. In the last section we propose a similar asymptotic fo…
The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb Z}/p)$ operates by its standard 2 dimensional representation over ${\mathbb Z}/p$. We establish such a relationship in most cases.
Let $E$ be an elliptic curve over the rational field $\mathbf{Q}$. Let $K$ be a quadratic extension over $\mathbf{Q}$. Let $\mathrm{ST}(E/K)$ dente the Shafarevich-Tate group of $E$ over $K$. We show that (under mild conditions on $E$) for every $r>0$, there are infinitely many quadratic twists $E^d/\mathbf{Q}$ of $E/\mathbf{Q}$ such that $\mathrm{dim}_{\mathbf{F}_2}(\mathrm{ST}(E^d/K)[2]) > r$
A multinorm one torus associated to a commutative étale algebra $L$ over a global field $k$ is of Kummer type if each factor of $L$ is a cyclic Kummer extension. In this paper we compute the Tate-Shafarevich group of such tori based on recent works of Bayer-Fluckiger, T.-Y. Lee and Parimala, and of T.-Y.~Lee. We also implement an effective algorithm using SAGE which computes the Tate-Shafarevich groups when each factor of $L$ is contained in a fixed concrete bicyclic extension of $k$.
We study the distribution of a subclass congruent elliptic curve $E^{(n)}: y^2=x^3-n^2x$, where $n$ is congruent to $1\pmod 8$ with all prime factors congruent to $1\pmod 4$. We prove an independence of residue symbol property. Consequently we get the distribution of rank zero such $E^{(n)}$ with $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z /2\mathbb Z\big)^2$. We also obtain a lower bound of the number of such $E^{(n)}$ with rank zero and $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z /2\mathbb Z\big)^{4}$.
Visible 2-torsion in the Tate-Shafarevich group of an elliptic curve
本文证明椭圆曲线的Tate-Shafarevich群中任意两个2-挠元可在同一阿贝尔曲面中可见，推广了先前仅知单个2-挠元的结果，并解释了Cremona和Mazur论文中的一些原始观察。
The second part of the Birch and Swinnerton-Dyer (BSD) conjecture gives a conjectural formula for the order of the Shafarevich-Tate group of an elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing non-trivial elements of the Shafarevich-Tate group of an elliptic curve by means of the Mordell-Weil group of an ambient curve. In this paper, we generalize Cremona and Mazur's work and give precise conditions under which such a construction can be made for the Shafarevich-Tate gro…
Let A be an abelian surface over a fixed number field. If A is principally polarised, then it is known that the order of the Tate-Shafarevich group of A must, if finite, be a square or twice a square. The situation for A not principally polarised remains unclear. For each k in {1,2,3,5,6,7,10,13} we construct a non-simple non-principally polarised abelian surface over the rationals whose Tate-Shafarevich group has order k times a square. To obtain this result, we explore the invariance under isogeny of the Birch and Swinnerton-Dyer conjecture.
In this paper it is shown that for every prime p>5 the dimension of the p-torsion in the Tate-Shafarevich group of E/K can be arbitrarily large, where E is an elliptic curve defined over a number field K, with [K:Q] bounded by a constant depending only on p. From this we deduce that the dimension of the p-torsion in the Tate-Shafarevich group of A/Q can be arbitrarily large, where A is an abelian variety, with dim A bounded by a constant depending only on p.
Let $\E/\Q$ be a fixed elliptic curve over $\Q$ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W. Duke have obtained an asymptotic formula for the number of primes $p\le x$ such that the reduction of $\E$ modulo p has a trivial Tate-Shafarevich group. Recent results of A. C. Cojocaru and C. David lead to a better error term. We introduce a new argument in the scheme of the proof which gives further improvement.
Let k be a global field and L be a finite dimensional étale algebra over k. In this paper, we assume that L is a product of cyclic extensions of k. Let T_{L/k} be the multinorm-one torus defined by the multinorm equation: N_{L/k} (t) = 1. Let X_c be the variety defined by the equation N_{L/k} (t) = c, for some c in k*. In this paper, we compute the Tate-Shafarevich group and the algebraic Tate-Shafarevich group of the character group of T_{L/k}. These groups measure the obstruction to the local-global principle for existence of rational points of X_c and the obstruction to the weak appraximati…
Congruences of modular forms and modularity of Tate-Shafarevich classes
本文在适当假设下证明了有理数上椭圆曲线的$p$-挠Tate-Shafarevich类在模曲线Jacobi商群中的可见性，验证了Jetchev-Stein猜想，其核心方法是利用Bertolini-Darmon二部Kolyvagin系统的非平凡性来推导Selmer群模$p$的基。
Given an abelian variety J and an abelian subvariety A of J over a number field K, we study the visible elements of the Shafarevich-Tate group of A with respect to J over certain number field extension M of K. The notion of visible elements in Shafarevich-Tate group of an abelian variety was introduced by Mazur. In this article, we study the image of Visible elements of A with respect to J under the natural restriction map of the Galois cohomology of A over K to the Galois cohomology of A over M. In particular, for a fixed odd prime p, we investigate the conditions under which visible elements…
There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.
We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.
Let E be an elliptic curve defined over a number field K. Let h be an element of order 4 in the Shafarevich-Tate group of E. We prove that h is visible in infinitely many abelian surfaces up to isomorphism. This is to say that there are infinitely many abelian surfaces J such that E\hookrightarrow J and h lies in the kernel of the natural map H^1(K,E)\rightarrow H^1(K,J).
Asymptotic formula for Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic extensions
本文在广义Heegner假设下，研究了p-超奇异椭圆曲线在反圆塔Z_p-扩张中数域上的Tate-Shafarevich群的p-准素子群，并给出了其渐近公式。
We present the results of our search for the orders of Tate-Shafarevich groups for the quadratic twists of elliptic curves. We formulate a general conjecture, giving for a fixed elliptic curve $E$ over $\Bbb Q$ and positive integer $k$, an asymptotic formula for the number of quadratic twists $E_d$, $d$ positive square-free integers less than $X$, with finite group $E_d(\Bbb Q)$ and $|\Sha(E_d(\Bbb Q))| = k^2$. This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of the curve $X_0(49)$. In section 8 we exhibit $88$ examples o…
An Asymptotic formula for Tate-Shafarevich groups of CM elliptic curves at supersingular primes
本文在Kobayashi符号Selmer群的附加假设下，对具有复乘的椭圆曲线在反圆$\mathbb{Z}_p$-扩张中间层上的Tate-Shafarevich群证明了渐近公式。
We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their $2$-Selmer group to the $2$-rank of the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ to obtain some examples of elliptic curves with rank one and non-trivial $2$-part of the Tate-Shafarevich group.
For an abelian variety A over a number field k we discuss the divisibility in H^1(k,A) of elements of the subgroup Sha(A/k). The results are most complete for elliptic curves over Q.
Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of $A_s$ should contain an element of order $p$ for a positive proportion of quadratic twists $A_s$ of $A$. We give a general method to prove instances of this conjecture by exploiting independent isogenies of $A$. For each prime $p$, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial $p$-torsion in their Tate-Shafarevich groups. In particular, when the modul…
Rank Jumps and Growth of Shafarevich--Tate Groups for Elliptic Curves in $\mathbb{Z}/p\mathbb{Z}$-Extensions
该研究利用岩泽理论探究素数阶循环扩张中椭圆曲线的秩跳跃，以及$p$-主素Selmer群和Shafarevich–Tate群的增长，改进了相关已有结果。
Nontrivial torsion in the Tate--Shafarevich group of elliptic curves via visibility and twists
本文研究了椭圆曲线在ℓ处加法减少时的可见性定理，推导出合适二次扭变下Sha群中存在非平凡ℓ-torsion，并以ℓ=3为例，展示具有相同BSD不变量等特征的非同构椭圆曲线对，其Tate-Shafarevich群同构且有非平凡3-primary部分。
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and $ι: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field. For a class group character $χ$ over $K$, let $\mathbb{Q}(χ)$ be the field generated by the image of $χ$ and $\mathfrak{p}_χ$ the prime of $\mathbb{Q}(χ)$ above $p$ determined via $ι_p$. Under mild hypotheses, we show that the number of class group characters $χ$ such that the $χ$-isotypic Tate--Shafarevich group of $E$ over $H_{K}$ is finite with trivial $\math…
Abelian fiber spaces and their Tate-Shafarevich twists with sections
本文研究如何从阿贝尔纤维空间出发，构造一个具有有理截面的Tate-Shafarevich扭曲。
Following (and elaborating on) a method of Logan and van Luijk, we exhibit explicit genus-2 curves whose Jacobians have nontrivial 2-torsion in their Tate-Shafarevich groups.
For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup.
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\underline{III}(E)$ be a certain group of equivalence classes of homogeneous spaces of $E$ called its Tate-Shafarevich group. We show in this paper that this group has finite cardinality and discuss its role in the Birch and Swinnerton-Dyer Conjecture. In particular, our result implies the Parity Conjecture, or the Birch and Swinnerton-Dyer Conjecture modulo 2. It also removes the finiteness condition of $\underline{III}(E)$ from previous results in the literature of this subject and makes possible for some computation problems concerning th…
Finitude du groupe de Tate-Shafarevich pour les groupes de type multiplicatif constants sur des corps des fonctions
本文研究了函数域扩张上乘性类型群的Tate-Shafarevich群，证明了在满足两个技术条件时该群有限，并在整数环正则时该群平凡。
Let $K/F$ be a finite Galois extension of number fields with Galois group $G$, let $A$ be an abelian variety defined over $F$, and let ${\cyr W}(A_{^{/ K}})$ and ${\cyr W}(A_{^{/ F}})$ denote, respectively, the Tate-Shafarevich groups of $A$ over $K$ and of $A$ over $F$. Assuming that these groups are finite, we derive, under certain restrictions on $A$ and $K/F$, a formula for the order of the subgroup of ${\cyr W}(A_{^{/ K}})$ of $G$-invariant elements. As a corollary, we obtain a simple formula relating the orders of ${\cyr W}(A_{^{/ K}})$, ${\cyr W}(A_{^{/ F}})$ and ${\cyr W}(A_{^{/ F}}^χ)…
For an elliptic curve $E$ over $\mathbb{Q}$, putting $K=\mathbb{Q}(E[p])$ which is the $p$-th division field of $E$ for an odd prime $p$, we study the ideal class group $\mathrm{Cl}_K$ of $K$ as a $\mathrm{Gal}(K/\mathbb{Q})$-module. More precisely, for any $j$ with $1\leqslant j \leqslant p-2$, we give a condition that $\mathrm{Cl}_K\otimes \mathbb{F}_p$ has the symmetric power $\mathrm{Sym}^j E[p]$ of $E[p]$ as its quotient $\mathrm{Gal}(K/\mathbb{Q})$-module, in terms of Bloch-Kato's Tate-Shafarevich group of $\mathrm{Sym}^j V_p E$. Here $V_p E$ denotes the rational $p$-adic Tate module of …
Let E be an elliptic curve over Q with complex multiplication. The aim of the present paper is to strengthen the theoretical and numerical results of \cite{CZS}. For each prime p, let t_{E/Q, p} denote the Z_p-corank of the p-primary subgroup of the Tate-Shafarevich group of E/Q. For each ε 0, we prove that t_{E/Q, p} is bounded above by (1/2+ε)p for all sufficiently large good ordinary primes p. We also do numerical calculations on one such E of rank 3, and 5 such E of rank 2, showing in all cases that t_{E/Q, p} = 0 for all good ordinary primes p < 30,000. In fact, we show that, with the pos…
On the density of abelian surfaces with Tate- Shafarevich group of order five times a square
本文研究由两条带五阶挠点的椭圆曲线乘积构造的阿贝尔曲面，给出判定其 Tate-Shafarevich 群阶是否为五倍平方数的算法，并在约 2000 万个样本中发现 49.16% 具有非平方阶。
We study analogues for the Tate-Shafarevich group for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields.
For an abelian variety A over a number field k we discuss the maximal divisibile subgroup of H^1(k,A) and its intersection with the subgroup Sha(A/k). The results are most complete for elliptic curves over Q.
Kolyvagin used Heegner points to associate a system of cohomology classes to an elliptic curve over $\Q$ and conjectured that the system contains a non-trivial class. His conjecture has profound implications on the structure of Selmer groups. We provide new computational and theoretical evidence for Kolyvagin's conjecture. More precisely, we explicitly compute Heegner points over ring class fields and use these points to verify the conjecture for specific elliptic curves of rank two. We explain how Kolyvagin's conjecture implies that if the analytic rank of an elliptic curve is at least two th…
Let $A$ be an abelian variety defined over a number field $F$. We prove a control theorem for the fine Selmer group of the abelian variety $A$ which essentially says that the kernel and cokernel of the natural restriction maps in a given $\mathbb{Z}_p$-extension $F_\infty/F$ are finite and bounded. We emphasise that our result does not have any constraints on the reduction of $A$ and the ramification of $F_\infty/F$. As a first consequence of the control theorem, we show that the fine Tate-Shafarevich group over an arbitrary $\mathbb{Z}_p$-extension has trivial $Λ$-corank. We then derive an as…
This paper has been withdrawn and no version of it will ever be published.
We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the Néron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators…
We consider the Kolyvagin cohomology classes associated to an elliptic curve $E$ defined over $\mathbb{Q}$ from a computational point of view. We explain how to go from a model of a class as an element of $(E(L)/pE(L))^{\mathrm{Gal}(L/\mathbb{Q})}$, where $p$ is prime and $L$ is a dihedral extension of $\mathbb{Q}$ of degree $2p$, to a geometric model as a genus one curve embedded in $\mathbb{P}^{p-1}$. We adapt the existing methods to compute Heegner points to our situation, and explicitly compute them as elements of $E(L)$. Finally, we compute explicit equations for several genus one curves …
Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a finite extension $F$ of $K$ where $p$ is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell-Weil ranks of $E$ over a subextension of the cyclotomic $\mathbb{Z}_p$-extension are bounded. Furthermore,…
We prove finiteness results for Tate--Shafarevich groups in degree 2 associated with 1--motives, rely them to Leopoldt's conjecture, and present an example of a semiabelian variety with an infinite Tate--Shafarevich group in degree 2. We also establish an arithmetic duality theorem for 1--motives over number fields which complements earlier results of Harari and Szamuely in this direction.
Let K/F be a finite Galois extension of global fields with Galois group G and let M be a 1-motive over F. We discuss the kernel and cokernel of the restriction map Sha^{i}(F,M) --> Sha^{i}(K,M)^{G} for i=1 and 2, independently of any finiteness hypotheses. We show that these groups are finite and obtain, in particular, formulas for the corresponding quotient of group orders.
We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|\sza(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|\sza(E)| = 1029212^2 = 2^4\cdot 79^2 \cdot 3257^2$. We can use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of $\sza$. For instance, $410536^2$ is the true order of $\sza(E)$ for $E= E_4(21,-233)$ from the table in section 2.3.
On the arithmetic of Tate-Shafarevich groups via Kolyvagin's conjecture
本文通过Heegner点研究半稳定椭圆曲线Tate-Shafarevich群p-primary部分的有限性，给出Kolyvagin猜想的简洁证明，并部分恢复Çiperiani--Wiles定理，方法独立于分圆Iwasawa主猜想。
We show that there exist non-trivial families of algebraic varieties for which all the fibers above rational points (or even above points of odd degree) are torsors of abelian varieties representing nonzero elements of their Shafarevich-Tate groups.
Record values are determined for the order $|\Sha|$ of the Tate--Shafarevich group of an elliptic curve $E$, computed analytically by the Birch--Swinnerton-Dyer conjecture, and for the Goldfeld--Szpiro ratio $G=|\Sha|/\sqrt{N}$, where $N$ is the conductor of $E$. The curves have rank zero and are isogenous to quadratic twists of Frey curves constructed from coprime positive integers $(a,b,c)$ with $a+b=c$ and $c>r^{1.4}$, where the radical $r$ is the product of the primes dividing $abc$. Curves with $|\Sha|>250000^2$ and $G>12$ are found in 20 isogeny classes. Three curves have $G>150$. The la…
Let X_K be a proper, smooth and geometrically connected curve over a global field K. In this paper we generalize a formula of Milne relating the order of the Tate-Shafarevich group of the Jacobian of X_K to the order of the Brauer group of a proper regular model of X_K. We thereby partially answer a question of Grothendieck.
Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and only Tate's conjecture for divisors on $X$ holds and the Tate-Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the formula given for a surfac…
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