BROOKS JETCHEV WESOLOWSKI ISOGENY GRAPHS OF ORDINARY ABELIAN - PowerPoint PPT Presentation

ERNEST HUNTER DIMITAR BENJAMIN BROOKS JETCHEV WESOLOWSKI ISOGENY GRAPHS OF ORDINARY ABELIAN VARIETIES PRESENTED AT ECC 2017, NIJMEGEN, THE NETHERLANDS BY BENJAMIN WESOLOWSKI FROM EPFL, SWITZERLAND

AN INTRODUCTION TO ISOGENY GRAPHS

ISOGENY GRAPHS OF ORDINARY ELLIPTIC CURVES An isogeny is a morphism This vertex represents an elliptic between two elliptic curves, curve E 0 over a finite field F with finite kernel. The degree of an isogeny is the size of the kernel (our isogenies are separable…) This edge is an isogeny of E 0 degree 𝓶 , a prime number E 1 Another elliptic curve over F

ISOGENY GRAPHS OF ORDINARY ELLIPTIC CURVES Any isogeny has a dual of E 0 the same degree (here, 𝓶 ) going in the opposite direction E 1

ISOGENY GRAPHS OF ORDINARY ELLIPTIC CURVES Neighbours of E 0 have more neighbours From E 0 , there are other E 3 isogenies of degree 𝓶 , going to other curves Any isogeny has a dual of E 2 E 0 the same degree (here, 𝓶 ) going in the opposite direction So we represent it by an E 1 undirected edge

ISOGENY GRAPHS OF ORDINARY ELLIPTIC CURVES E 0 Once all the possible neighbours have been reached, we obtain the connected graph of 𝓶 -isogenies of E 0

ISOGENY GRAPHS OF ORDINARY ELLIPTIC CURVES E 0 This one is a typical example!

ISOGENY GRAPHS OF ORDINARY ELLIPTIC CURVES A cycle E 0 This one is a typical example!

ISOGENY GRAPHS OF ORDINARY ELLIPTIC CURVES A cycle E 0 D i s j o i n t i s o r m o o o t r e p d h o i c n c t o h p e i e c s y o c f l e a t r e e This one is a typical example!

ISOGENY GRAPHS OF ORDINARY ELLIPTIC CURVES Level 0, surface Level 1 E 0 o r fl o 2 , e l e v L

Level 0, surface Level 1 E 0 An isogeny volcano o r fl o 2 , e l e v L

Level 0, surface Level 1 E 0 (sometimes “isogeny tutu") o r fl o 2 , e l e v L

ISOGENY GRAPHS OF ORDINARY ELLIPTIC CURVES Level 0, surface Level 1 E 0 Why is this useful? o r fl o 2 , e l e v L By inspecting solely the structure of the graph, one can infer that E 0 is at “level 1 ” at 𝓶 … which tells a lot about the endomorphism ring of E 0 !

APPLICATIONS ▸ Computing the endomorphism ring of an elliptic curve [Kohel, 1996], ▸ Counting points [Fouquet and Morain, 2002], ▸ Random self-reducibility of the discrete logarithm problem [Jao et al., 2005] (worst case to average case reduction) ▸ Accelerating the CM method [Sutherland, 2012], ▸ Computing modular polynomials [Bröker et al., 2012]

GENERALISING TO ORDINARY ABELIAN VARIETIES… ▸ These applications motivate the search for a generalisation to other abelian varieties… An abelian variety is a geometric object (curve, surface…) which is also an abelian group (there is an addition law on the points). Elliptic curves = abelian varieties of dimension 1.

GENERALISING TO ORDINARY ABELIAN VARIETIES… ▸ These applications motivate the search for a generalisation to other abelian varieties… A principally polarised abelian surface over a finite field F Isogeny of type ( 𝓶 , 𝓶 )

GENERALISING TO ORDINARY ABELIAN VARIETIES… ▸ These applications motivate the search for a generalisation to other abelian varieties…

GENERALISING TO ORDINARY ABELIAN VARIETIES… ▸ These applications motivate the search for a generalisation to other abelian varieties…

GENERALISING TO ORDINARY ABELIAN VARIETIES… How to study ( 𝓶 , 𝓶 )-isogeny graphs? ➡ Focus on interesting subgraphs ➡ Decompose ( 𝓶 , 𝓶 )-isogenies into simpler ones

ENDOMORPHISM RINGS

ENDOMORPHISM RING AND ALGEBRA ▸ Let 𝓑 be an ordinary abelian variety of dimension g over a finite field F = 𝔾 q . K ⊃ 𝓟 ≅ End( 𝓑 ) ▸ The endomorphisms of 𝓑 form a ring 2 End( 𝓑 ). K 0 ▸ The algebra K = End( 𝓑 ) ⊗ ℚ is a number g field of degree 2 g (a CM-field). ℚ ▸ End( 𝓑 ) is isomorphic to an order 𝓟 of K (i.e., a lattice of dimension 2 g in K , that is also a subring).

THE CASE OF ELLIPTIC CURVES ▸ If 𝓑 = E is an elliptic curve, the dimension is g = 1. K ⊃ 𝓟 ≅ End( E ) ▸ K has a maximal order 𝓟 K , the ring of integers of K . 2 K 0 = ℚ ▸ Any order of K is of the form 𝓟 = ℤ + f 𝓟 K , for a positive integer f , the conductor .

THE CASE OF ELLIPTIC CURVES The “levels” of the volcano of ℓ -isogenies tell how many times ℓ divides the conductor. Here, ( f , ℓ ) = 1. End ≅ ℤ + f 𝓟 K End ≅ ℤ + ℓ f 𝓟 K End ≅ ℤ + ℓ 2 f 𝓟 K

THE CASE OF ELLIPTIC CURVES Only an ℓ -isogeny can change the valuation at ℓ of the conductor. Horizontal 𝓶 -isogeny Descending End ≅ ℤ + f 𝓟 K 𝓶 -isogeny End ≅ ℤ + ℓ f 𝓟 K Ascending 𝓶 -isogeny End ≅ ℤ + ℓ 2 f 𝓟 K

CLASSIFICATION OF ORDERS ▸ This classification of orders in quadratic fields is the key to the volcanic structures for elliptic curves. ▸ Analog in dimension g > 1? For any number field K 0 and quadratic extension K / K 0 , we prove the following classification Any order 𝓟 of K containing 𝓟 K0 is of the form 𝓟 = 𝓟 K0 + 𝔤𝓟 K for an ideal 𝔤 of 𝓟 K0 , the conductor of 𝓟 .

CLASSIFICATION OF ORDERS Any order 𝓟 of K containing 𝓟 K0 is of the form 𝓟 = 𝓟 K0 + 𝔤𝓟 K for an ideal 𝔤 of 𝓟 K0 , the conductor of 𝓟 . ▸ This is exactly 𝓟 = ℤ + f 𝓟 K when K 0 = ℚ ! ▸ When 𝓟 contains 𝓟 K0 , we say that 𝓟 has maximal real multiplication (RM). ▸ For K 0 = ℚ , any order has maximal RM since 𝓟 K0 = ℤ .

VOLCANOES AGAIN?

𝖒 -ISOGENIES ▸ For an elliptic curve, the conductor is an integer f , which decomposes as a product of prime numbers: we then look at ℓ -isogenies where ℓ is a prime number ▸ For g > 1 and maximal RM, the conductor is an ideal 𝔤 of 𝓟 K0 , and decomposes into prime ideals… ▸ Notion of 𝖒 -isogenies, where 𝖒 is a prime ideal of 𝓟 K0 ? An 𝖒 -isogeny from 𝓑 is an isogeny whose kernel is a proper, 𝓟 K0 -stable subgroup of 𝓑 [ 𝖒 ]. ▸ Coincides with the “ 𝖒 -isogenies” defined in [Ionica and Thomé, 2014] when g = 2

VOLCANOES AGAIN? If 𝓑 has maximal RM (locally at ℓ ), and 𝖒 is a prime ideal of 𝓟 K0 above ℓ , is the graph of 𝖒 -isogenies a volcano? Theorem: yes!… at least when 𝖒 is principal, and all the units of 𝓟 K are totally real! ‣ First observed in some particular case in [Ionica and Thomé, 2014] ‣ When 𝖒 is generated by a totally positive unit, independently proven in [Martindale, 2017]

VOLCANOES AGAIN? If 𝓑 has maximal RM (locally at ℓ ), and 𝖒 is a prime ideal of 𝓟 K0 above ℓ , is the graph of 𝖒 -isogenies a volcano? Theorem: yes!… at least when 𝖒 is principal, and all the units of 𝓟 K are totally real! End ≅ 𝓟 K0 + 𝔤𝓟 K End ≅ 𝓟 K0 + 𝖒𝔤𝓟 K End ≅ 𝓟 K0 + 𝖒 2 𝔤𝓟 K

VOLCANOES AGAIN? If 𝖒 is not principal? The graph is oriented! End ≅ 𝓟 K0 + 𝔤𝓟 K End ≅ 𝓟 K0 + 𝖒𝔤𝓟 K End ≅ 𝓟 K0 + 𝖒 2 𝔤𝓟 K End ≅ 𝓟 K0 + 𝖒 3 𝔤𝓟 K

VOLCANOES AGAIN? If 𝓟 K has complex units ? Multiplicities appear For instance, K = ℚ ( ζ 5 ), K 0 = ℚ ( ζ 5 + ζ 5 ) , and 𝖒 = 2 𝓟 K0 . -1 End ≅ 𝓟 K0 + 𝔤𝓟 K 5 End ≅ 𝓟 K0 + 𝖒𝔤𝓟 K End ≅ 𝓟 K0 + 𝖒 2 𝔤𝓟 K

MAIN STEPS OF THE PROOF

COUNTING VERTICES AT EACH LEVEL ▸ First ingredient: we can count the number of vertices on each level using the class number formula. End ≅ 𝓟 K0 + 𝔤𝓟 K Level 0 End ≅ 𝓟 K0 + 𝖒𝔤𝓟 K Level 1 End ≅ 𝓟 K0 + 𝖒 2 𝔤𝓟 K Level 2

COUNTING VERTICES AT EACH LEVEL ▸ First ingredient: we can count the number of vertices on each level using the class number formula. End { #(level 0) = # Cl( 𝓟 K0 + 𝔤𝓟 K ) Level 0 #Cl ( 𝓟 K0 + 𝖒𝔤𝓟 K ) #(level 1) = ? Level 1 Level 2

COUNTING VERTICES AT EACH LEVEL ▸ First ingredient: we can count the number of vertices on each level using the class number formula. End { #(level 0) = # Cl( 𝓟 K0 + 𝔤𝓟 K ) Level 0 #(level 1) = ? Level 1 ‣ #(level 1) = (N( 𝖒 ) − 1) ∙ #(level 0) if 𝖒 splits in K Level 2 ‣ #(level 1) = N( 𝖒 ) ∙ #(level 0) if 𝖒 ramifies in K ‣ #(level 1) = (N( 𝖒 ) + 1) ∙ #(level 0) if 𝖒 is inert in K

COUNTING VERTICES AT EACH LEVEL ▸ First ingredient: we can count the number of vertices on each level using the class number formula. End { #(level 0) = # Cl( 𝓟 K0 + 𝔤𝓟 K ) Level 0 #(level 1) = ? Warning: these are simplified formulas (need Level 1 extra assumptions on the units of 𝓟 K ) ‣ #(level 1) = (N( 𝖒 ) − 1) ∙ #(level 0) if 𝖒 splits in K Level 2 ‣ #(level 1) = N( 𝖒 ) ∙ #(level 0) if 𝖒 ramifies in K ‣ #(level 1) = (N( 𝖒 ) + 1) ∙ #(level 0) if 𝖒 is inert in K

COUNTING VERTICES AT EACH LEVEL ▸ First ingredient: we can count the number of vertices on each level using the class number formula. #(level 0) = # Cl( 𝓟 K0 + 𝔤𝓟 K ) Level 0 { ‣ (N( 𝖒 ) − 1) ∙ #(level 0) #(level 1) = ‣ N( 𝖒 ) ∙ #(level 0) Level 1 ‣ (N( 𝖒 ) + 1) ∙ #(level 0) #(level 2) = N( 𝖒 ) ∙ #(level 1) Level 2 #(level i + 1) = N( 𝖒 ) ∙ #(level i) for i ≥ 1