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Given two elliptic curves E 1 E_1 and E 2 E_2 defined over the field of rational numbers Q \mathbb {Q} that have good and ordinary reduction at an odd prime p p , and have equivalent, irreducible mod p p Galois representations, we study the variation of the parity of Selmer ranks and analytic ranks of E 1 E_1 and E 2 E_2 over certain number fields.

In this paper we prove a relation between the period of an elliptic curve and the period of its real and imaginary quadratic twists. This relation is often misstated in the literature.

It is conjectured that an elliptic curve E defined over a totally real number field F is modular; i.e., the associated Z-adic representation pE := Pe,i of I> := Gal(F/F), for some rational prime /, is isomorphic to the Z-adic representation pn := pnj of TF associated to some automorphic representation tt of GL(2)/F (see §2 below for details). This conjecture was proved when F = Q (see [BCDT], [W]). The Birch and Swinnerton-Dyer conjecture says in particular that (for more precise details see [M] and also §2 below)

This note gives explicit equations for the elliptic curves (in characteristic not 2 or 3) with mod 2 representation isomorphic to that of a given one.

This paper gives a new proof of Kamienny’s Criterion using the method of Coleman and Chabauty.

Let E E be an elliptic curve over a number field k k such that End k ⁡ E \operatorname {End}_{k}E = Z = {\mathbf Z} and let w ( k ) w(k) denote the number of roots of unity in k k . Ross proposed a question: Is E E isogenous over k k to an elliptic curve E ′ / k E’/k such that E ′ ( k ) t o r s E’(k)_{tors} is cyclic of order dividing w ( k ) w(k) ? A counter-example of this question is given. We show that E E is isogenous to E ′ / k E’/k such that E ′ ( k ) t o r s ⊂ Z / w ( k ) 2 Z E’(k)_{tors}\,\subset {\mathbf Z}/w(k)^2{\mathbf Z} . In case E E has complex multiplication and End k ⁡ E = Z \operatorname {End}_kE={\mathbf Z} , we obtain certain criteria whether or not E E is isogenous to E ′ / k E’/k such that E ′ ( k ) t o r s ⊂ Z / 2 Z E’(k)_{tors}\,\subset {\mathbf Z}/2{\mathbf Z} .

We consider the degree of irrationality d r ( S ) d_r(S) of some algebraic surface S S . Firstly we give an estimate of d r ( S ) d_r(S) for a surface S S with a structure of a fiber space. Secondly we prove the existence of a nonsingular curve of genus 3 on E × E E\times E for a certain elliptic curve E E with complex multiplications. As a corollary, we obtain that d r ( E × E ) = 3 d_r(E\times E)=3 .

We give a strong maximum principle for some nonlinear parabolic and elliptic systems with convex invariant regions. We also obtain a version of the Hopf boundary lemma for the systems.

Let p be an odd prime and let ( ⋅ p ) (\tfrac { \cdot }{p}) be the Legendre symbol. It is shown how to evaluate the character sum Σ x = 0 p − 1 ( f ( x ) p ) \Sigma _{x = 0}^{p - 1}(\tfrac {{f(x)}}{p}) , for certain quartic polynomials f ( x ) f(x) . For example, it is shown that \[ ∑ x = 0 p − 1 ( x 4 − 8 x 3 + 12 x 2 − 16 x + 4 p ) = { 2 ( 2 p ) x 1 − 1 , a m p ; if p ≡ 1 ( mod 4 ) , − 1 , a m p ; if p ≡ 3 ( mod 4 ) , \sum \limits _{x = 0}^{p - 1} {\left ( {\frac {{{x^4} - 8{x^3} + 12{x^2} - 16x + 4}}{p}} \right )} = \left \{ {\begin {array}{*{20}{c}} {2\left ( {\frac {2}{p}} \right ){x_1} - 1,} \hfill & {{\text {if}}\;p \equiv 1\;\pmod 4,} \hfill \\ { - 1,} \hfill & {{\text {if}}\;p \equiv 3\;\pmod 4,} \hfill \\ \end {array} } \right . \] where x 1 {x_1} is defined for primes p ≡ 1 ( mod 4 ) p \equiv 1\;\pmod 4 by \[ p = x 1 2 + y 1 2 , x 1 ≡ − 1 ( mod 4 ) . p = x_1^2 + y_1^2,\quad {x_1} \equiv - 1\quad \pmod 4. \]

The critical points of a function restricted to the solution set of a linear elliptic equation are characterized. An extension of the Lagrange multiplier method is given. Existence and the relation to elliptic eigenvalue problems are discussed.

We show that there are only finitely many elliptic curves of bounded degree and height, provided that one takes a naive height defined in terms of minimal Weierstrass equations. We show that the corresponding statement is false if instead one uses the Faltings-Parshin modular height. Several theorems in algebraic number theory assert that there are only finitely many "objects" of bounded "degree" and "size". Two important examples of such theorems are the following: Theorem 1 (Hermite-Minkowski [2], p. 619). There are only finitely many number fields of bounded degree and absolute discriminant. Theorem 2 (Northcott [5]). There are only finitely many algebraic numbers of bounded degree and height. In [9] the author gave a "proof' of an analogous result for elliptic curves of bounded degree and height, where the degree was the degree of the field of definition and the height was the (Faltings-Parshin) modular height. Daniel Bertrand (personal communication) has noted that the proof in [9] is not correct. Further, even the statement of Corollary 2.5 in [9] is somewhat ambiguous. In this note we will formulate and prove a precise theorem for elliptic curves analogous to Theorems 1 and 2. We will use a naive definition for the height of an elliptic curve, and will indicate why our result is false if instead one uses the modular height. It is important to fix precisely what is meant by the minimal field of definition of an elliptic curve. We begin by letting ( [E, K). K is a number field E is a K-isomorphism class of elliptic curves Thus & consists of all elliptic curves with all possible fields of definition. There is a natural partial ordering on 8, defined by (E',K') -< (E,K) if K' C K and E'xK, K E. Received by the editors May 27, 1988 and, in revised form, July 28, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary I1 G05; Secondary 14K07, 14K 15. Research supported by NSF grant #DMS-86 12393 and a Sloan Foundation Fellowship. ( 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

We present a family of elliptic curves defined over the rationals Q {\mathbf {Q}} such that each curve admits only good or multiplicative reduction and for every integer n n there is a curve whose Tate-Shafarevitch group over Q {\mathbf {Q}} has more than n n elements of order 2. Previously known examples of large Tate-Shafarevitch groups were constructed by forcing many places of additive reduction.

A general vanishing theorem is proved for elliptic operators. This result is then used to give a simple proof of the fact that the arithmetic genus vanishes for complex manifolds of odd dimension n n with nowhere zero ( n , 0 ) \left ( {n,0} \right ) form.

The existence of infinitely many elliptic curves with no rational points except the origin ∞ \infty is proved by refining a theorem of Davenport-Heilbronn. The existence of infinitely many quadratic fields with the Iwasawa invariant λ 3 = 0 {\lambda _3} = 0 is proved at the same time.

Seemingly disparate comparison theorems for nonselfadjoint elliptic equations have been established by C. A. Swanson and the author. The present paper establishes a class of comparison theorems involving an arbitrary vector valued function p(x). Special choices of p(x) yield both of the comparison theorems cited above. Several papers have recently concerned themselves with generalizations of Sturm's comparison theorem to nonselfadjoint elliptic equations. The first such theorem was due to Protter [1], and subsequent generalizations have been established by Swanson [2 ] and the author [3 ]. An interesting aspect of [2 ] and [3 ] is the seemingly disparate hypotheses which the principal theorems require. The purpose of this paper is to establish a class of comparison theorems for nonselfadjoint elliptic equations which contain the results of [2] and [3] as special cases. Our method consists of combining appropriate generalizations of techniques used by Picard [4] and Picone [5]. Let D denote a smooth bounded domain in Rn and let u(x) and v(x) be nontrivial real valued solutions of the elliptic equations ) a Eij(x)-+ 2 Z bi(x) + c(x)u = 0, -i=1 aXi axi i(l (x) (2) -E aij(x)+ 2 E Zfl(x) -+ y(x)v = 0. i, j= i 0xj d9xi i=1 d9xi It is assumed that the coefficients a,j(x), aij(x), bi(x) and fi(x) are of class C' in D while c(x) and y(x) are to be continuous in D7 (some of these requirements can be weakened somewhat as will be clear from the manipulations below). Furthermore (1) and (2) are to be uniformly elliptic in D. Received by the editors September 4, 1970 and, in revised form, November 16, 1970. A MS 1969 subject classifications. Primary 3511.

1. A. A. Albert, Finite division algebras and finite planes, Combinatorial analysis, Proc. Sympos. Appl. Math. Vol. 10, pp. 53-70, Amer. Math. Soc., Providence, R. I., 1960. 2. R. Baer, Projectivities with fixed points on every line of the plane, Bull. Amer. Math. Soc. 52 (1946), 273-286. 3. Marshall Hall, Jr., The theory of groups, Macmillan, New York, 1959. 4. D. Knuth, Finite semifields and projective planes, Ph.D. Dissertation, California Institute of Technology, Pasadena, Calif., 1963.

In this paper, we study the singular elliptic equation L u + K ( x ) u p = 0 Lu + K(x){u^p} = 0 , where L L is a uniformly elliptic operator of divergence form, p > 1 p > 1 and K ( x ) K(x) has a singularity at the origin. We prove that this equation does not possess any positive (local) solution in any punctured neighborhood of the origin if there exist two constants C 1 {C_1} , C 2 {C_2} such that C 1 | x | σ ⩾ K ( x ) ⩾ C 2 | x | σ {C_1}|x{|^\sigma } \geqslant K(x) \geqslant {C_2}|x{|^\sigma } near the origin for some σ ⩽ − 2 \sigma \leqslant - 2 (with no other condition on the gradient of K K ). In fact, an integral condition is derived.

where a and b are positive and continuously differentiable on a closed interval I and f and g are continuous on I. The theorem states that if a-b ? 0 and f < g on I and xi, X2 are zeros of u in I, then v has a zero in [Xl, X2]. This theorem was generalized to self-adjoint second order elliptic partial differential equations by Hartman and Wintner [1]. The purpose of this note is to present a new and simple proof of this generalization. Let u and v be solutions of the elliptic equations

The purpose of this note is to give a concise proof of a comparison theorem for selfadjoint, second order elliptic equations which yields stronger results than those previously derived in [1], [2] and [3]. All coefficients and domains are to be sufficiently smooth so that the variational techniques of Courant [4] can be applied. Specifically, it is assumed that the first eigenfunction of the selfadjoint boundary value problem

In this note we give a short proof of a theorem of Kotake and Narasimhan to the effect that if A A is a strongly elliptic operator of order 2 m 2m with analytic coefficients and | | A j u | | ≦ C j + 1 ( 2 m j ) ! ||{A^j}u|| \leqq {C^{j + 1}}(2mj)! , where | | | | ||\;|| is some suitable norm, then u u is analytic. (Actually Kotake and Narasimhan prove the theorem when A A is elliptic, but the trick we use here requires some specialization.) This is applied to derive a short proof of a theorem proved by Gårding and Malgrange, in the constant coefficients case, concerning conditionally elliptic operators.

We give a proof, which makes use of the RieszThorin theorem, for a smoothness theorem for solutions of elliptic systems in divergence form with bounded measurable coefficients. The results imply an important theorem in two dimensions due to Morrey [3]. Meyers has used a similar technique to get these results for elliptic equations [4]. Let Q be a compact domain in Rn with smooth boundary and consider the m x m real elliptic system given by: (Lu)k = E aj ul i, j,l axi aX j where a Ak are bounded measurable functions on Q satisfying the inequality Z Z alravlk _: Z (>Z)2 i, j, k, 1 i,k E=Ho' (Q) is the usual complex Sobolev space of in complex functiQ,ns with p-integrable derivatives which are 0 on the boundary of Q, and 11I, denotes norms in EP or norms of operators on EP. THEOREM. There exist q0>2 depending on Q, n and the number K= maxf aI ,c616i1 such that L: Ho', q (Q) is invertible for qol(qo-1) < q