Hershel M. Farkas

Professor of Mathematics

telephone: 972 2 6584386

email: farkas@math.huji.ac.il

RESEARCH INTERESTS

The research in the area of function theory on compact Riemann surfaces has focused on the Schottky problem which asks for a description of the space of Jacobians of compact Riemann surfaces of genus g in the Siegel upper half plane of degree g.

Contributions have been made to Schottky-Jung theory.

More recently, I have been studying modular forms and their applications to geometry and number theory. Here the main contribution has been obtaining concrete representations of holomorphic mappings of modular curves into projective spaces of low dimensions and the realization of their automorphism groups as the restriction of global isometries of the projective space to the image of the curve. In addition, these mappings have given generalizations of the Platonic solids, interesting theta constant identities, and connections with classical combinatorial number theory.

PUBLICATIONS

• Special Divisors and analytic subloci of Teichmuller Space, Amer. J. Math. 88 (1966), pp. 881-901.
  
• Automorphisms of Compact Riemann surfaces and the vanishing of theta constants, Bull of AMS 73 (1967), pp. 231-232.
  
• Relations between two kinds of theta constants on a Riemann surface, Proc. of the Nat. Acad. Sci. U.S.A. (1968), pp. 52-55 (with H. E. Rauch).
  
• Special divisors on compact Riemann surfaces, Proc. of AMS (1968), pp. 315-318.
  
• Weierstrass points and analytic submanifolds of Teichmueller space, Proc. of AMS (1969), pp. 35-38.
  
• Theta constants, moduli, and compact Riemann surfaces. Proc. of the Nat. Acad. Sci. U.S.A. (1969), pp. 320-325.
  
• Two kinds of theta constants and period relations on a Riemann Surface, Proc. of the Nat. Acad. Sci. U.S.A. (1969), pp. 679-686.
  
• Theta constants of two kinds on a compact Riemann surface of genus 2, Jul. D'Analyse (1970), pp. 381-407.
  
• On the Schottky relation and its generalization to arbitrary genus, Ann. of Math. (1970), pp. 57-86.
  
• Period relations of Schottky type on Riemann surfaces, Ann. of Math. (1970), pp. 434-461 (with H. E. Rauch).
  
• Relations between quadratic differentials, \un{Advances in the theory of Riemann Surfaces}, Ann. of Math. Studies (66) Princeton Univ. Press. (1971), pp. 141-156.
  
• Abelian differentials with double zeros, Proc. of AMS (1971), pp. 155-162.
  
• Period Relations for hyperelliptic Riemann surfaces, Israel J. Math. (1971), pp. 289-301.
  
• On the Poincare relation, \un{Contributions to Analysis}, Academic Press N.Y. (1974), pp. 125-132 (with H. E. Rauch, H. Martens, L. Ehrenpreis).
  
• Elliptic functions and modular forms, \un{Contributions to Analysis}, Academic Press, N.Y. (1974), pp. 133-145.
  
• Some Refinements of the Poincare period relations, Discontinuous Groups and Riemann Surfaces, Ann. of Math. Studies (79), Princeton Univ. Press (1974), pp. 105-112 (with L. Ehrenpreis).
  
• Remarks on automorphisms of compact Riemann surfaces, Discontinuous Groups and Riemann Surfaces , Ann. of Math. Studies (79), \un{Princeton} Univ. Press. (1974), pp. 121 - 145.
  
• Unramified double coverings of hyperelliptic surfaces, Jnl. D'Analyse (1976), pp. 150-155.
  
• Singular points of theta functions, quadric relations, and holomorphic differentials with prescribed zeros. Complex analysis, Joensuu} 1978, Proceedings, Lecture Notes in Mathematics, Springer Verlag (1979), pp. 108-122.
  
• Vanishing of theta constants, Riemann Surfaces and Related Topics: Proc. of the 1978 Stony Brook Conference, Annals of Mathematics Studies, \#97, Princeton Univ. Press (1981) pp. 101-106.
  
• Relations between quadratic differentials II, Compositio Mathematica, Vol. 44 pp. 67-77, (1981)
  
• H. E. Rauch, Theta Function Practioner. Differential Geometry and Complex Analysis, 1985, Springer Verlag.
  
• Fay's Tri-secant formula. Jnl D'Analyse \#44, (1984) pp. 205-217.
  
• The (g-1) support cover of the Canonical Locus (with M. Fried), Jnl. D'Analyse \#46 (1986), pp. 148-157.
  
• Unramified Double coverings of hyperelliptic surfaces II. Proc. of A.M.S. Vol. 101, 3 (1987) pp. 470-474.
  
• Unramified coverings of hyperelliptic surfaces. Proc. of Conf. Md. 1986. Lecture Notes in Math. 1275 pp. 113-131.
  
• Remarks on Clifford Indices Jnl f. die Reine und Angewandte Math. (391) 1988 pp. 213-220.
  
• Schottky-Jung theory . Proceedings of Symposia in Pure Mathematics, Volume 49 (1989) pp. 459-483
  
• Identities on Compact Riemann Surfaces. Geometric and Algebraical Aspects in Several Complex Variables, Cetraro (Italy) ,(1989) pp. 97-106
  
• The Trisecant Formula and Hyperelliptic Surfaces , Contemporary Mathematics, Volume 136 (1992) pp.161-169
  
• Branched Two-Sheeted Covers (with Irwin Kra) Israel Journal of Mathematics, vol 74 (1991) pp. 169-197
  
• Automorphic Forms for Subgroups of the Modular Group (with Irwin Kra), Israel Journal of Mathematics 82 (1993) 87-131
  
• New Theta Constant Identities (with Y. Kopeliovich) Israel Journal of Mathematics 82 (1993) 133-140
  
• New Theta Constant Identities II (with Y. Kopeliovich) Proc. of AMS 123 , 4 , (1995) 1009-1020
  
• Automorphic Forms for Subgroups of the Modular Group II (with Irwin Kra) ,Jnl D'Analyse Mathematique, vol 70 (1996) pp. 91-156
  
• Uniformizations of Modular Curves (with I. Kra and Y. Kopeliovich) Comm. Anal. and Geom. vol 4 no 2 1996 pp. 207-259
  
• Theta Constant Identities with Applications to Combinatorial Number Theory (with Irwin Kra )Contemp. Math vol 211, 1997, pp. 227-252
  
• Generalizations of the $\lambda $ Function , Israel Mathematical Conference Proceedings Vol. 9 1996, pp.231-239.
  
• Factorization of $N^2+1$ and the Modular Group (with R. Brooks) (preprint)
  
• Number Theory, Theta Identities and Modular Curves ( with R. Brooks and Irwin Kra ) Contemp. Math vol 201, 1997, pp. 125-154
  
• A Variation on a Theorem of Euler, ICMP, Proc. of the Ashkelon Conf on Complex Function Theory vol 11 (1997)
  
• A Variation on a Theorem of Euler II (preprint)
  
• Ramanujan Partition Identities (with I. Kra) Complex Geometry of Groups, Contemporary Mathematics (240), Proc of Conf in Olmue, Chile Jan. 5-8 1998, pp. 111-130 (1999).
  
• On the Quintuple Product Identity, (with I. Kra ) Proc. AMS vol 127 1999 pp.771-778
  
• A Function Theoretic approach to the Ramanujan Partition Identities with Applications to Combinatorial Number Theory, (with I. Kra ) Complex Geometry of Groups, Contemporary Mathematics (240), Proc of Conf in Olmue, Chile Jan. 5-8 1998, pp. 131-158 (1999) .
  
• Cosines and Triangular Numbers (with I. Kra) Revue Roumaine de Math. et Appl. , vol 46, nr. 1, pp. 37-43 (2001)
  
• Weierstrass Points and the Theta Divisor, Lectures in Memory of Lars V. Ahlfors, Israel Mathematics Conference Proceedings (14) pp. 65-79 (2000).
  
• Partitions and Theta Constant Identities (with I. Kra) Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis, Contemporary Mathematics (251) pp. 197-203 (2000)
  
• Special Sets of Points on Compact Riemann Surfaces (with I. Kra). In the Tradition of Ahlfors and Bers, Contemporary Mathematics (256) Ahlfors-Bers Colloquium, Nov. 6-8 1998 pp.75-94 (2000).
  
• Three Term Theta Identities (with I. Kra) In the Tradition of Ahlfors and Bers, Contemporary Mathematics (256) Ahlfors-Bers Colloquium, Nov. 6-8 1998 pp.95-101 (2000).
  
• On Theta Constant Identities and the Evaluation of Trigonometric Sums (with I. Kra ) Complex Manifolds and Hyperbolic Geometry, II Iberoamerican Congress on Geometry, Jan. 4-9, 2001 Contemporary Mathematics (311) pp. 115-131 (2002).
  
• Identities in the Theory of Theta Constants (with I. Kra) In the Tradition of Ahlfors and Bers III, Contemporary Mathematics 355 Ahlfor-Bers Colloquium October 18-21 2001, pp.231-238 (2004).
  
• On an Arithmetical Function, Ramanujan Journal , vol 8 no 3 pp. 309-315 (2004).
  
• Logarithmic Derivatives of Theta Functions (with Y. Godin), Israel Journal of Mathematics vol 148 pp 253-265 (2005)
  
• On an Arithmetical Function II , Contemporary Mathematics 382, Complex Analysis and Dynamical Systems II, conference in honor of L. Zalcman's 60th birthday (2005) pp.121-130.
  
• Partition Theory, Rogers Ramanujan through Gollnitz Gordon (preprint).
  
• Sums of Squares and Triangular Numbers, Online Journal of Analytic Combinatorics, April 2006.
  
• Variants of the 3N+1 Conjecture and Multiplicative Semigroups, Contemporary Mathematics 387, Geometry, Spectral Theory, Groups and Dynamics, Proc. of Conf. in memory of Robert Brooks pp.121-127. (2005).
  
• Vanishing Theta Nulls and Jacobians, Contemporary Mathematics 397, The Geometry of Riemann Surfaces and Abelian Varieties, III Iberoamerican Congress on Geometry , Proc. of Conference in honor of Sevin Recillas (2006) pp.37-53.
  
• Theta Functions in Complex Analysis and Number Theory, Surveys in Number Theory, Developments in Mathematics (17) Springer (2008) pp. 57-87.
  
• Fagnano's Theorem and Tetrahedrons, preprint.
  
• An Elementary Proof of Thomae's Formula (with A. Eisenmann), Online Journal of Analytic Combinatorics (2008).
  
• Some Identities on Z_n Curves, in preparation.
  
• On Thomae Type Formulae for Singular Z_n Curves, in preparation.
  
• Generalizations of Hutchinson's Curve and the Thomae Formula, Partitions q-Series and Modular Forms , Developments in Mathematics (23) Springer (2010) pp. 69-81.
  
• Thomae's Formula for $Z_n$ Curves (with David Ebin), Journal D'Analyse Mathematique, vol. 111 pp. 289-320 (2010)
  
• Theta Functions, Wronskians and Weierstrass Points for Linear Spaces of Meromorphic Functions, The Mathematical Legacy of Leon Ehrenpreis, Springer Proc. in Mathematics vol 16 (2012) pp. 95-110.
  
• A Matrix related to the theorem of Fermat and the Goldbach conjecture, From Fourier Analysis and Number Theory to Radon Transforms and Geometry In Memory of leon Ehrenpreis, Springer DEVM series vol. 28 pp. 217-232 (2012).
  

BOOKS

• Advances in the theory of Riemann Surfaces (Edit ed with L. V. Ahlfors, L. Bers, R. C. Gunning, I. Kra and H. E. Rauch). Ann. of Math. Studies (66), Princeton Univ. Press. 1971
  
• Theta Functions with Applications to Riemann Sur faces (with H. E. Rauch) Williams and Wilkins, Baltimore, Md. (1974)
  
• Riemann Surfaces (with I. Kra) Springer-Verlag ( 1980).
  
• Differential Geometry and Complex Analysis, (Edited with I. Chavel) Springer Verlag (1985).
  
• Theta Constants, Riemann Surfaces and the Modular Group (with I. Kra), Graduate Studies in Mathematics, vol 37 American Mathematical Society, 2001. Information on this book is obtainable at
  http://www.math.sunysb.edu/~irwin/bookinfo.html
  
• Generalizations of Thomae's Formula for Z_n Curves (with S. Zemel) Springer, DEVM series, (2010)
  
• From Fourier Analysis and Number Theory to Radon Transforms and Geometry In Memory of Leon Ehrenpreis, Springer DEVM series vol. 28 ( edited with R. Gunning, M. Knopp B.A. Taylor ) 2012.