A higher-order generalization of Jacobi’s derivative formula and its algebraic geometric analogue

Grant, David

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 2, pp. 361-386.

Nous généralisons la formule de la dérivée de Jacobi en écrivant, pour un $m$ impair, un déterminant de taille $m$ composé de dérivées d’ordre supérieur évaluées en $0$ des fonctions thêta d’une variable avec vecteurs caractéristiques à coordonnées dans $\frac{1}{2m}\mathbb{Z}$ comme une constante explicite multipliée par une puissance de la fonction $\eta$ de Dedekind. Nous déduisons ce résultat de sa version algébro-géométrique, qui est valable si la caractéristique ne divise pas $6m$.

We generalize Jacobi’s derivative formula for odd $m$ by writing an $m\times m$ determinant of higher order derivatives at 0 of theta functions in 1 variable with characteristic vectors with entries in $\frac{1}{2m}\mathbb{Z}$ as an explicit constant times a power of Dedekind’s $\eta$-function. We do so by deriving it from an algebraic geometric version that holds in characteristic not dividing $6m$.

Classification : 14K25, 14H42
Mots clés : Theta functions, elliptic curves

[1] Anderson, Greg W. Lacunary Wronskians on genus one curves, J. Number Theory, Volume 115 (2005) no. 2, pp. 197-214 | DOI | MR | Zbl

[2] Ayad, Mohamed Points $\Delta$-entiers sur les courbes elliptiques, J. Number Theory, Volume 38 (1991) no. 3, pp. 323-337 | DOI | MR | Zbl

[3] Ayad, Mohamed Points $S$-entiers des courbes elliptiques, Manuscr. Math., Volume 76 (1992) no. 3-4, pp. 305-324 | DOI | MR | Zbl

[4] Bannai, Kenichi; Kobayashi, Shinichi Algebraic theta functions and the $p$-adic interpolation of Eisenstein-Kronecker numbers, Duke Math. J., Volume 153 (2010) no. 2, pp. 229-295 | MR | Zbl

[5] Boxall, John; Grant, David Theta functions and singular torsion on elliptic curves, Number theory for the millennium I, A K Peters, 2002, pp. 111-126 | Zbl

[6] Boxall, John; Grant, David Singular torsion on elliptic curves, Math. Res. Lett., Volume 10 (2003) no. 5-6, pp. 847-866 | DOI | MR | Zbl

[7] Connell, Ian Elliptic Curve Handbook (unpublished)

[8] Farkas, Hershel M.; Kra, Irwin Theta constants, Riemann surfaces and the modular group. An introduction with applications to uniformization theorems, partition identities and combinatorial number theory, Graduate Studies in Mathematics, 37, American Mathematical Society, 2001 | Zbl

[9] Grant, David Resultants of Division Polynomials. I: Reciprocity and Elliptic Units (in preparation)

[10] Grant, David Resultants of Division Polynomials. II: Singular torsion on Elliptic Curves (in preparation)

[11] Grant, David Some product formulas for theta functions in one and two variables, Acta Arith., Volume 102 (2002) no. 3, pp. 223-238 | DOI | MR | Zbl

[12] Grant, David A generalization of Jacobi’s derivative formula to dimension two. II, Acta Arith., Volume 190 (2019) no. 4, pp. 403-420 | DOI | MR | Zbl

[13] Grushevsky, Samuel; Salvati Manni, Riccardo Two generalizations of Jacobi’s derivative formula, Math. Res. Lett., Volume 12 (2005) no. 5-6, pp. 921-932 | DOI | MR | Zbl

[14] Hartshorne, Robin Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | Zbl

[15] Hindry, Marc Polynômes de Cassels $P_N$ et $Q_N$ et résultant (https://webusers.imj-prg.fr/~marc.hindry/cassels.pdf)

[16] Igusa, Jun-ichi On Jacobi’s derivative formula and its generalizations, Am. J. Math., Volume 102 (1980) no. 2, pp. 409-446 | DOI | MR | Zbl

[17] Landau, Edmund Elementary Number Theory, American Mathematical Society, 1999 | Zbl

[18] Lang, Serge Elliptic Functions, Graduate Texts in Mathematics, 112, Springer, 1987 | MR | Zbl

[19] Mazur, Barry; Tate, John The $p$-adic sigma function, Duke Math. J., Volume 62 (1991) no. 3, pp. 663-688 | MR | Zbl

[20] Milas, Antun; Mortenson, Eric; Ono, Ken Number theoretic properties of wronskians of Andrews-Gordon series, Int. J. Number Theory, Volume 4 (2008) no. 2, pp. 323-337 | DOI | MR | Zbl

[21] Mumford, David On the equations defining abelian varieties I, Invent. Math., Volume 1 (1966), pp. 287-354 | DOI | MR | Zbl

[22] Mumford, David The Tata Lectures on Theta. I: Introduction and motivation: Theta functions in one variable. Basic results on theta functions in several variables, Progress in Mathematics, 28, Birkhäuser, 1983 | Zbl

[23] Mumford, David The Tata Lectures on Theta. III, Progress in Mathematics, 97, Birkhäuser, 1991 | MR | Zbl

[24] Schur, Issai Über die Gaußschen Summen, Gött. Nachr., Volume 1921 (1921), pp. 147-153 | Zbl

[25] Serre, Jean-Pierre Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1971), pp. 259-331 | DOI | Zbl

[26] Serre, Jean-Pierre Algebraic Groups and Class Fields, Graduate Texts in Mathematics, 117, Springer, 1988 | MR | Zbl

[27] Silverman, Joseph H. The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, 2000 | Zbl

[28] Weber, Heinrich Elliptische Functionen und Algebraische Zahlen, Vieweg u. Sohn., 1891 | Zbl

[29] Zemel, Shaul A direct evaluation of the periods of the Weierstrass zeta function, Ann. Univ. Ferrara, Sez. VII, Sci. Mat., Volume 60 (2014), pp. 495-505 | DOI | MR | Zbl