armand brumer

Fordham University, Mathematics, Emeritus

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On the group of units of an absolutely cyclic number field of prime degree

Journal of The Mathematical Society of Japan, Jul 1, 1969

Ramification and class towers of number fields

Michigan Mathematical Journal, Jun 1, 1965

Prosaic Abelian Varieties Bad at One Prime

arXiv (Cornell University), May 18, 2023

We say that an abelian variety A /Q of dimension g is prosaic if it is semistable and its points ... more We say that an abelian variety A /Q of dimension g is prosaic if it is semistable and its points of order 2 generate a 2-extension of Q. We focus on prosaic abelian varieties A with good reduction outside one prime p, whose ring of endomorphisms End Q (A) is Z or an order in a totally real field of degree g. We show that p ≡ 1 (8) and that A has totally toroidal reduction at p. We construct indecomposable group schemes over Z[1/p] of exponent 2 whose Galois module is related to the 2-Sylow subgroup H 2 of the class group of Q(√ −p). We find that B[2] is such a group scheme for some B isogenous to A and thereby prove that 2g + 2 ≤ |H 2 |. Moreover if 2g + 4 ≤ |H 2 |, then p has the form a 2 + 64b 2 , with a ≡ ±1 (8). A few examples with real multiplication and one bad prime are given. In contrast, under the Schinzel hypothesis, we exhibit infinite families of prosaic abelian surfaces of conductor pq, for primes p = q. Contents 1. Introduction 1 2. Input from higher arithmetic 3 3. Some 2-primary group schemes and their extensions 6 4. The group schemes Ξ p and Φ d 11 5. Application to abelian varieties 19 6. Real multiplication 23 7. Prosaic families, Richelot isogenies and mild reduction 25 Acknowledgements 29 References 29

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Travaux récents d'Iwasawa et de Leopoldt

Séminaire Bourbaki, 1968

The number of elliptic curves over ℚ with conductorNwith conductorN

Manuscripta Mathematica, Dec 1, 1996

We prove that the number of elliptic curves E/Q with conductor N is O(Nl/2+*). More generally, we... more We prove that the number of elliptic curves E/Q with conductor N is O(Nl/2+*). More generally, we prove that the number of elliptic cur~es E/Q with good reduction outside S i~ 9 1/2+~), where M is the product of the primes in S. Assuming various standard !:onjectures, we show that this hound can be improved to O(M c~ Ioglo~ M). In this note we will give bounds for the number of elliptic curves defined over Q with a given conductor N, and more generally, for the number of elliptic curves over Q with good reduction outside of a given set of prime S. Theorem 1. Let c > O. There is a constant h;l($.

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Lectures on modular forms

These are notes based on a course of lectures given at Princeton University during the Spring Sem... more

The behavior of the Mordell-Weil group of elliptic curves

Bulletin of the American Mathematical Society, 1990

Suppose that E is an elliptic curve defined over Q given by the equation 3 2 (1) y +a { xy + a 3 ... more Suppose that E is an elliptic curve defined over Q given by the equation 3 2 (1) y +a { xy + a 3 y = x + a 2 x + a 4 x + a 6 , where we assume that a t e. Z. The set E(Q) of solutions (x, y) with x , y G Q, together with the point at infinity, forms a finitelygenerated abelian group, the Mordell-Weil group ofE. It is isomorphic to Z r 0 F, where F is finite and where r is the rank of E. The possibilities for the finite group F are completely known . The important question then is to understand the behavior of the rank as E varies over elliptic curves. It is still unknown whether r is unbounded or not. In fact, one opinion is that, in general, an elliptic curve might tend to have the smallest possible rank, namely 0 or 1, compatible with the rank parity predictions of Birch and Swinnerton-Dyer . We present evidence that this may not be the case. Published examples [2, 10] of curves of rank > 2 might suggest that such curves are sparsely distributed. Mestre and Oesterlé found the 436 modular elliptic curves of prime conductor up to 13100, using . There were 80 rank 2 curves among the 233 curves of even rank. This proportion of rank 2 curves seemed too large to conform to the conventional wisdom just stated (see also [18, pages 254-255]). We decided to investigate the ranks of elliptic curves in a systematic way, over a significantly larger range.

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Class Number and Ramification in Number Fields

Nagoya Mathematical Journal, Dec 1, 1963

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Arithmetic of Division Fields

arXiv (Cornell University), Feb 22, 2011

We study the arithmetic of division fields of semistable abelian varieties A /Q . The Galois grou... more We study the arithmetic of division fields of semistable abelian varieties A /Q . The Galois group of Q(A[2])/Q is analyzed when the conductor is odd and squarefree. The irreducible semistable mod 2 representations of small conductor are determined under GRH. These results are used in Paramodular abelian varieties of odd conductor, arXiv:1004.4699.

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The class group of all cyclotomic integers

Journal of Pure and Applied Algebra, Feb 1, 1981

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Galois groups of extensions of algebraic number fields with given ramification

Michigan Mathematical Journal, Apr 1, 1966

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On the paramodularity of typical abelian surfaces (and reduction of G-covariant bilinear forms)

arXiv (Cornell University), May 28, 2018

Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces wit... more

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Addendum to “Structure of hereditary orders”

Bulletin of the American Mathematical Society, 1964

In the hypothesis of Theorem 2, the following condition should be added : Suppose 0, is the uniqu... more In the hypothesis of Theorem 2, the following condition should be added : Suppose 0, is the unique maximal order in D. This is satisfied in two important cases, namely if D~K or if R is complete. One may obtain, by completion, explicit results when this hypothesis is removed, but they are somewhat more complicated and will be described elsewhere.

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Structure of hereditary orders

Bulletin of the American Mathematical Society, Sep 1, 1963

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Corrigendum to &#x201C;Paramodular abelian varieties of odd conductor&quot

Transactions of the American Mathematical Society, May 1, 2019

Frank Calegari was kind enough to point out a phenomenon overlooked in the paramodular conjecture... more

On the size of the Brauer group

Proceedings of the American Mathematical Society, Mar 1, 1968

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The average rank of elliptic curves I

Inventiones Mathematicae, Dec 1, 1992

Recall that r(E) should be the rank of E, according to the Birch-Swinnerton-Dyer conjecture. It w... more Recall that r(E) should be the rank of E, according to the Birch-Swinnerton-Dyer conjecture. It would suffice for our purpose to have the algebraic rank 2 Please consult the "Note added in proof" at the end of the Paper log X This does not affect our bound of 2.3 for the Average Rank since Theorem 4.4 was invoked only with the triangle function as h.

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On the units of algebraic number fields

Mathematika, Dec 1, 1967

Let p be a prime number, Qp the field of p-adic numbers and Ωp the completion of the algebraic cl... more

Large 2-adic Galois image and non-existence of certain abelian surfaces over Q

arXiv (Cornell University), Jan 7, 2017

Motivated by our arithmetic applications, we required some tools that might be of independent int... more Motivated by our arithmetic applications, we required some tools that might be of independent interest. Let E be an absolutely irreducible group scheme of rank p 4 over Zp. We provide a complete description of the Honda systems of p-divisible groups G such that G[p n+1 ]/G[p n ] ≃ E for all n. Then we find a bound for the abelian conductor of the second layer Qp(G[p 2 ])/Qp(G[p]), stronger in our case than can be deduced from Fontaine's bound. Let π : Sp 2g (Zp) → Sp 2g (Fp) be the reduction map and let G be a closed subgroup of Sp 2g (Zp) with G = π(G) irreducible and generated by transvections. We fill a gap in the literature by showing that if p = 2 and G contains a transvection, then G is as large as possible in Sp 2g (Zp) with given reduction G, i.e. G = π −1 (G). One simple application arises when A = J(C) is the Jacobian of a hyperelliptic curve C : y 2 + Q(x)y = P (x), where Q(x) 2 + 4P (x) is irreducible in Z[x] of degree m = 2g + 1 or 2g + 2, with Galois group Sm ⊂ Sp 2g (F 2). If the Igusa discriminant I 10 of C is odd and some prime q exactly divides I 10 , then G = Gal(Q(A[2 ∞ ])/Q) isπ −1 (Sm), whereπ : GSp 2g (Zp) → Sp 2g (Fp). When m = 5, Q(x) = 1 and I 10 = N is a prime, A = J(C) is an example of a favorable abelian surface. We use the machinery above to obtain nonexistence results for certain favorable abelian surfaces, even for large N .

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$Research paper thumbnail of Hyperelliptic $\mathcal{S}_7$-curves of prime conductor$

Hyperelliptic $\mathcal{S}_7$-curves of prime conductor

arXiv (Cornell University), Feb 2, 2020

An abelian threefold A /Q of prime conductor N is favorable if its 2-division field F is an S 7 -... more An abelian threefold A /Q of prime conductor N is favorable if its 2-division field F is an S 7 -extension over Q with ramification index 7 over Q 2 . Let A be favorable and let B be a semistable abelian variety of conductor N d with B[2] filtered by d copies of A [2]. We obtain a class field theoretic criterion on F to guarantee that B is isogenous to A d and a fortiori, A is unique up to isogeny.

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