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(with E. Rains, T. Scholl, S. Sharif, A. Silverberg)
Algebraic maps constant on isomorphism
classes of unpolarized abelian varieties are constant. To appear in *Algebra & Number Theory.*

(with B. Mazur)
Arithmetic conjectures suggested by the statistical
behavior of modular symbols. To appear in *Experimental Mathematics.*

(with B. Mazur)
Big fields that are not large, *Proc. Amer. Math. Soc.* (Series B) **7** (2020) 159-169.

(with B. Mazur, and an appendix by M. Larsen)
Diophantine stability.
* American J. Math.* **140** (2018) 571-616.

(with B. Mazur)
Refined class number
formulas for **G**_{m}.
* Journal de Théorie des Nombres de Bordeaux* **28** (2016) 185–211.

(with B. Mazur)
Controlling
Selmer groups in the higher core rank case.
* Journal de Théorie des Nombres de Bordeaux* **28** (2016) 145–183.

(with B. Mazur)
Selmer companion curves.
*Transactions Amer. Math. Soc.* **367** (2015) 401-421.

(with Z. Klagsbrun and B. Mazur)
A Markov model for Selmer
ranks in families of twists.
*Compositio Math.* **150** (2014) 1077-1106.

(with R. Greenberg, A. Silverberg, and M. Stoll)
On elliptic curves with an isogeny of degree 7. * American J. Math.* **136** (2014) 77-109.

(with J. B. Friedlander, H. Iwaniec, and B. Mazur)
The spin of prime ideals.
*Inventiones math.* **193** (2013) 697-749.

(with Z. Klagsbrun and B. Mazur)
Disparity in Selmer ranks of quadratic twists of elliptic curves. *Annals of Math.* **178** (2013) 287-320.

(with D. Boneh and A. Silverberg)
Finding
composite order ordinary elliptic curves using the Cocks-Pinch method.
*Journal of Number Theory* **131** (2011) 832-841.

(with B. Mazur)
Refined class number
formulas and Kolyvagin systems.
*Compositio Mathematica* **147** (2011) 54-76.

(with B. Mazur)
Ranks of twists of
elliptic curves and Hilbert's Tenth Problem.
*Inventiones mathematicae* **181** (2010) 541-575.

(with A. Silverberg)
Choosing
the correct elliptic curve in the CM method.
*Mathematics of Computation* **79** (2010) 545-561.

(with A. Silverberg)
Point counting on reductions of CM elliptic curves.
*Journal of Number Theory* **129** (2009) 2903-2923.

(with A. Silverberg)
Using
abelian varieties to improve pairing-based cryptography.
*Journal of Cryptology* **22** (2009) 330-364.

(with B. Mazur)
Growth of Selmer rank in nonabelian extensions of number fields.
*Duke Math. Journal* **143** (2008) 437-461.

(with A. Silverberg)
Compression in
finite fields and torus-based cryptography.
*SIAM Journal on Computing* **37** (2008) 1401-1428.

(with B. Mazur)
Finding large Selmer rank via an arithmetic theory of local constants.
*Annals of Mathematics* **166** (2007) 581-614.

(with B. Mazur and A. Silverberg)
Twisting commutative algebraic groups.
*Journal of Algebra* **314** (2007) 419-438.

(with A. Silverberg)
Twists of elliptic curves of rank at least four. In:
Ranks of elliptic curves and random matrix theory, Conrey et al., eds.,
*London Math. Soc. Lecture Notes* **341** (2007) 177-188.

Fudge factors in the Birch and Swinnerton-Dyer conjecture. In:
Ranks of elliptic curves and random matrix theory, Conrey et al., eds.,
*London Math. Soc. Lecture Notes* **341** (2007) 233-236.

Appendix to:
Anticyclotomic Iwasawa theory of CM elliptic curves, by A. Agboola and B. Howard.
*Annales de l'Institut Fourier* **56** (2006) 1001-1048.

(with B. Mazur)
Finding large Selmer groups.
*J. Differential Geometry* **70** (2005) 1–22

(with B. Mazur)
Organizing the arithmetic of elliptic curves.
*Advances in Math.* **198** (2005) 504-546.

(with M. van Dijk, R. Granger, D. Page, A. Silverberg, M. Stam, D. Woodruff)
Practical cryptography in high dimensional tori.
In: Advances in Cryptology - EUROCRYPT 2005.
*Lect. Notes in Comp. Sci.* **3494** (2005) 234-250.

(with R. Pollack)
The main conjecture for CM elliptic curves at supersingular primes.
* Ann. of Math.* **159** (2004) 447-464.

(with A. Silverberg)
Using primitive subgroups to do more with fewer bits.
In: Algorithmic Number Theory - ANTS VI.
*Lect. Notes in Comp. Sci.* **3076** (2004) 18-41.

(with B. Mazur)
Studying the Growth of Mordell-Weil.
* Doc. Math.* Extra volume: Kato's Fiftieth Birthday (2003) 585-607.

(with A. Silverberg)
Torus-based cryptography.
In: Advances in Cryptology - CRYPTO 2003.
*Lect. Notes in Comp. Sci.* **2729** (2003) 349-365.

(with B. Mazur)
Elliptic curves and class field theory.
*Proceedings of the ICM, Beijing 2002* vol. 2, 185-196.

(with A. Silverberg)
Ranks of elliptic curves.
*Bull. Amer. Math. Soc.* **39** (2002) 455-474.

(with A. Silverberg)
Supsersingular abelian varieties in cryptology.
In: Advances in Cryptology - CRYPTO 2002.
*Lect. Notes in Comp. Sci.* **2442** (2002) 336-353.

(with A. Silverberg)
Rank frequencies for quadratic twists of elliptic curves.
*Exp. Math.* **10** (2001) 559-569.

(with A. Silverberg)
Mod 2 representations of elliptic curves.
*Proc. Amer. Math. Soc.* **129** (2001) 53-57.

(with A. Silverberg)
Ranks of elliptic curves in families of quadratic twists.
*Exp. Math.* **9** (2000) 583-590.

A Stark conjecture "over **Z**" for abelian *L*-functions with multiple zeros.
*Annales de l'institut Fourier* **46** (1996) 33-62.

(with A. Silverberg)
A report on Wiles' Cambridge lectures.
*Bull. Amer. Math. Soc.* **31** (1994) 15-38.

*p*-adic *L*-functions and rational points on elliptic curves with complex multiplication.
*Invent. Math.* **107** (1992) 323-350.

The "main conjectures" of Iwasawa theory for imaginary quadratic fields.
*Invent. Math.* **103** (1991) 25-68.

On the main conjecture of Iwasawa theory for imaginary quadratic fields.
*Invent. Math.* **93** (1988) 701-713.

Tate-Shafarevich groups and *L*-functions of elliptic curves with complex multiplication.
*Invent. Math.* **89** (1987) 527-559.

Global units and ideal class groups.
*Invent. Math.* **89** (1987) 511-526.

Local units, elliptic units, Heegner points and elliptic curves.
*Invent. Math.* **88** (1987) 405-422.

Congruences for special values of *L*-functions of elliptic curves with complex multiplication.
*Invent. Math.* **71** (1983) 339-364.

Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer.
*Invent. Math.* **64** (1981) 455-470.